CHAPTER VII

The cohomology of coherent sheaves

1. Basic Cechˇ cohomology We begin with the general set-up. (i) X any topological space

= U an open covering of X U { α}α∈S a presheaf of abelian groups on X. F Define: (ii)

Ci( , ) = group of i-cochains with values in U F F = (U U ). F α0 ∩···∩ αi α0,...,αYi∈S

We will write an i-cochain s = s(α0,...,αi), i.e.,

s(α ,...,α ) = the component of s in (U U ). 0 i F α0 ∩··· αi (iii) δ : Ci( , ) Ci+1( , ) by U F → U F i+1 δs(α ,...,α )= ( 1)j res s(α ,..., α ,...,α ), 0 i+1 − 0 j i+1 Xj=0 b where res is the restriction map

(U U ) (U U ) F α ∩···∩ Uαj ∩···∩ αi+1 −→ F α0 ∩··· αi+1 and means “omit”. Forb i = 0, 1, 2, this comes out as

δs(cα , α )= s(α ) s(α ) if s C0 0 1 1 − 0 ∈ δs(α , α , α )= s(α , α ) s(α , α )+ s(α , α ) if s C1 0 1 2 1 2 − 0 2 0 1 ∈ δs(α , α , α , α )= s(α , α , α ) s(α , α , α )+ s(α , α , α ) s(α , α , α ) if s C2. 0 1 2 3 1 2 3 − 0 2 3 0 1 3 − 0 1 2 ∈ One checks very easily that the composition δ2:

Ci( , ) δ Ci+1( , ) δ Ci+2( , ) U F −→ U F −→ U F is 0. Hence we define:

211 212 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

s(σβ0, σβ1) defined here

U σβ0 Uσβ1

Vβ1

Vβ0

ref s(β0, β1) defined here Figure VII.1

(iv) Zi( , ) = Ker δ : Ci( , ) Ci+1( , ) U F U F −→ U F = group of i-cocycles,  Bi( , ) = Image δ : Ci−1( , ) Ci( , ) U F U F −→ U F = group of i-coboundaries  Hi( , )= Zi( , )/Bi( , ) U F U F U F = i-th Cech-cohomologyˇ group with respect to . U For i = 0, 1, this comes out: H0( , ) =group of maps α s(α) (U ) such that U F 7→ ∈ F α s(α )= s(α ) in (U U ) 1 0 F α0 ∩ α1 =Γ(X, ) if is a sheaf . ∼ F F H1( , ) =group of cochains s(α , α ) such that U F 0 1 s(α0, α2)= s(α0, α1)+ s(α1, α2) modulo the cochains of the form s(α , α )= t(α ) t(α ). 0 1 0 − 1 Next suppose = U and = V are two open coverings and that is a refinement U { α}α V { β}β∈T V of , i.e., for all V , V U for some α S. Fixing a map σ : T S such that V U , U β ∈V β ⊂ α ∈ → β ⊂ σ(β) define (v) the refinement homomorphism ref : Hi( , ) Hi( , ) U,V U F −→ V F by the homomorphism on i-cochains:

σ refU,V (s)(β0,...,βi) = res s(σβ0,...,σβi) (using res: (U U ) (V V ) and checking that δ refσ = F σβ0 ∩···∩ σβi → F β0 ∩···∩ βi ◦ U,V refσ δ, so that ref on cochains induces a map ref on cohomology groups.) (cf. Figure U,V ◦ VII.1) 1. BASIC CECHˇ COHOMOLOGY 213

Now one might fear that the refinement map depends on the choice of σ : T S, but here we → encounter the first of a series of nice identities that make cohomology so elegant — although “ref” on cochains depends on σ, “ref” on cohomology does not. (vi) Suppose σ, τ : T S satisfy V U U . Then → β ⊂ σβ ∩ τβ a) for all 1-cocycles s for the covering , U σ refU,V s(α0, α1)= s(σα0, σα1) = s(σα , τα ) s(σα , τα ) 0 1 − 1 1 = s(σα , τα )+ s(τα , τα ) s(σα , τα ) { 0 0 0 1 }− 1 1 = s(τα , τα )+ s(σα , τα ) s(σα , τα ) 0 1 0 0 − 1 1 = refτ s(α , α )+ s(σα , τα ) s(σα , τα ) U,V 0 1 0 0 − 1 1 i.e., the two ref’s differ by the coboundary δt, where t(α) = res s(σα, τα) (V ). ∈ F α More generally, one checks easily that b) if s Zi( , ), then ∈ U F refσ s refτ s = δt U,V − U,V where i−1 t(α ,...,α )= ( 1)js(σα ,...,σα , τα ,...,τα ). 0 i−1 − 0 j j i−1 Xj=0 For general presheaves and topological spaces X, one finally passes to the limit via ref over F finer and finer coverings and defines: (vii) 1 Hi(X, ) = lim Hi( , ). F U F −→U Here are three important variants of the standard Cechˇ complex. The first is called the alternating cochains: Ci ( , ) = group of i-cochains s as above such that: alt U F a) s(α ,...,α )=0if α = α for some i = j 0 n i j 6 b) s(α ,...,α ) = sgn(π) s(α ,...,α ) for all permutations π. π0 πn · 0 n For i = 1, one sees that every 1-cocycle is automatically alternating; but for i > 1, this is no i i+1 longer so. One checks immediately that δ(Calt) Calt , hence we can form the cohomology of  ⊂ the complex (Calt, δ). By another beautiful identity, it turns out that the cohomology of the   subcomplex Calt and the full complex C are exactly the same! This can be proved as follows: a) Order the set S of open sets Uα. b) Define Dn Cn to be the group of cochains s (k) ⊂ such that s(α ,...,α )=0 if α < < α . 0 n 0 · · · n−k 1This group, the Cechˇ cohomology, is often written Hˇ i(X, F) to distinguish it from the “derived functor” cohomology. In most cases they are however canonically isomorphic and as we will not define the latter, we will not use the ˇ . 214 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

c) Note that δ(Dn ) Dn+1, that (k) ⊂ (k) (0) = Dn Dn Dn (n) ⊂ (n−1) ⊂···⊂ (0) and that Cn = Cn Dn . ∼ alt ⊕ (0) d) Therefore    Hn(complex C , δ) = Hn(complex C , δ) Hn(complex D , δ) ∼ alt ⊕ (0) so it suffices to prove that the last is 0. We use the elementary fact: if (G, d) is a complex of abelian groups and Hn Gn is a subcomplex, then (Hn, d) exact and n n n ⊂ n n (G /H , d) exact imply (G , d) exact. Thus we need only check that (D(k)/D(k+1), δ) is exact for each k. e) We may interpret:

s(α0,...,αn) defined only if Dn /Dn = cochains s α < < α and . (k) (k+1)  0 · · · n−k−1  =0 if α < < α  0 n−k  · · · Define   n n−1 n−1 h: D(k)/Dn D /D (k+1) −→ (k) (k+1) as follows: Assume α < < α ; set 0 · · · n−k−2 hs(α ,...,α ) =0 if α = α , some 0 i n k 2 0 n−1 n−k−1 i ≤ ≤ − − =( 1)i+1s(α ,...,α , α , α ,...,α ) − 0 i n−k−1 i+1 n if α < α < α , some 0 i n k 3 i n−k−1 i+1 ≤ ≤ − − =0 if αn−k−1 < αi. One checks with some patience that hδ + δh = identity   hence (D(k)/D(k+1), δ) is exact. The second variant is local cohomology. Suppose Y X is a closed subset and that the ⊂ covering has the property: U X Y = U . \ α α[∈S0 Consider the subgroups: Ci ( , )= s Ci( , ) s(α ,...,α )=0if α ,...,α S . S0 U F { ∈ U F | 0 i 0 i ∈ 0} i i+1 i One checks that δ(CS ) CS , hence one can define HS ( , ) = cohomology of complex  0 ⊂ 0 0 U F (C , δ). Passing to a limit with refinements ( , T0 refines , S0 if ρ: T S such that S0 V U ∃ → V U and ρ(T ) S ), one gets Hi (X, ) much as above. β ⊂ ρβ 0 ⊂ 0 Y F The third variation on the same theme is the hypercohomology of a complex of presheaves:

 d d dm−1 : 0 0 0 1 1 m 0 F −→ F −→ F −→ · · · −→ F −→ (i.e., d d = 0, for all i). If = U is an open covering, we get a double complex i+1 ◦ i U { α}α∈S Cij = Ci( , j ) U F where δ : Cij Ci+1,j is the Cechˇ coboundary 1 −→ δ : Cij Ci,j+1 is given by applying d to the cochain. 2 −→ j 1. BASIC CECHˇ COHOMOLOGY 215

Then δ1δ2 = δ2δ1 and if we set C(n) = Ci,j i+Xj=n and use d = δ +( 1)iδ : C(n) C(n+1) as differential, then d2 = 0. This is called the associated 1 − 2 → “total complex”. Define   Hn( , )= n-th cohomology group of complex (C( ), d). U F  Passing to a limit with refinements, one gets Hn(X, ). This variant is very important in the F De Rham theory (cf. VIII.3 below). § The most important property of Cechˇ cohomology is the long exact cohomology sequence. Suppose 0 0 −→ F1 −→ F2 −→ F3 −→ is a short exact sequence of presheaves (which means that 0 (U) (U) (U) 0 −→ F1 −→ F2 −→ F3 −→ is exact for every open U). Then for every covering , we get a big diagram relating the cochain U complexes: ......

   0 / Ci−1( , ) / Ci−1( , ) / Ci−1( , ) / 0 U F1 U F2 U F3 δ δ δ    0 / Ci( , ) / Ci( , ) / Ci( , ) / 0 U F1 U F2 U F3 δ δ δ    0 / Ci+1( , ) / Ci+1( , ) / Ci+1( , ) / 0 U F1 U F2 U F3    ...... with exact rows, i.e., a short exact sequence of complexes of abelian groups. By a standard fact in homological algebra, this always leads to a long exact sequence relating the cohomology groups of the three complexes. In this case, this gives: δ 0 H0( , ) H0( , ) H0( , ) H1( , ) −→ U F1 −→ U F2 −→ U F3 −→ U F1 H1( , ) H1( , ) δ H2( , ) . −→ U F2 −→ U F3 −→ U F1 −→ · · · Moreover, we may pass to the limit over refinements, getting:

0 H0(X, ) H0(X, ) H0(X, ) δ H1(X, ) −→ F1 −→ F2 −→ F3 −→ F1 δ H1(X, ) H1(X, ) H2(X, ) . −→ F2 −→ F3 −→ F1 −→ · · · In almost all applications, we are only interested in the cohomology of sheaves and unfor- tunately short exact sequences of sheaves are seldom exact as sequences of presheaves. Still, in reasonable cases the long exact cohomology sequence continues to hold. The problem can be analyzed as follows: let 0 0 −→ F1 −→ F2 −→ F3 −→ be a short exact sequence of sheaves. If we define a subpresheaf ∗ by F3 ⊂ F3 ∗(U) = Image [ (U) (U)] , F3 F2 −→ F3 216 VII.THECOHOMOLOGYOFCOHERENTSHEAVES then 0 ∗ 0 −→ F1 −→ F2 −→ F3 −→ is an exact sequence of presheaves, hence we get a long exact sequence: δ Hi(X, ) Hi(X, ) Hi(X, ∗) Hi+1(X, ) ··· −→ F1 −→ F2 −→ F3 −→ F1 −→ · · · Now is the sheafification of ∗ so a long exact sequence for the cohomology of the sheaves F3 F3 follows if we can prove the more general assertion: Fi for all presheaves , the canonical maps F ( ) Hi(X, ) Hi(X, sh( )) ∗ F −→ F are isomorphisms. Breaking up sh( ) into a diagram of presheaves: F → F 0 / / / sh / / 0 K F HHH s9 F C HH# sss ′ s : L vv F LLL vv LL% 0 v 0 ( = kernel, = cokernel, ′ = image) and applying twice the long exact sequence for K C F presheaves, ( ) follows from: ∗ ( ) If is a presheaf such that sh( ) = (0), then Hi(X, ) = (0). ∗∗ F F F The standard case where ( ) and hence ( ) is satisfied is for paracompact Hausdorff spaces2 X: ∗∗ ∗ we will use this fact once in (3.11) below and VIII.3 in comparing classical and algebraic De § Rham cohomology for complex varieties. Schemes however are far from Hausdorff so we need to take a different tack. In fact, suppose X is a scheme (separated as usual) and 0 0 −→ F1 −→ F2 −→ F3 −→ is a short exact sequence of quasi-coherent sheaves. Then in the above notations: ∗(U) ≈ (U), all affine U F3 −→ F3 so (U)= (U) = (0), all affine U. K C Now if is any affine open covering of X, then X separated implies U U affine for U α0 ∩···∩ αi all α ,...,α , hence Ci( , ) = Ci( , ) = (0), hence Hi( , ) = Hi( , ) = (0). Since affine 0 i U K U C U K U C coverings are cofinal among all coverings, Hi(X, ) = Hi(X, ) = (0), hence Hi(X, ∗) ≈ K C F3 −→ Hi(X, ) and we get a long exact sequence for the cohomology of the ’s for much more F3 Fi elementary reasons! What are the functorial properties of cohomology groups? Here are three important kinds:

2The proof is as follows: We may compute Hi(X, F) by locally finite coverings U so let U be one and let s ∈ Ci(U, F). A paracompact space is normal so one easily constructs a covering V with the same index set I such that V α ⊂ Uα, ∀α ∈ I. Now for all x ∈ X, the local finiteness of U shows that ∃ neighborhood Nx of x such that

a) x ∈ Uα0 ∩···∩ Uαi =⇒ Nx ⊂ Uα0 ∩···∩ Uαi and resNx s(α0,...,αi) = 0. Shrinking Nx, we can also assume that Nx meets only a finite set of Uα’s hence there is a smaller neighborhood Mx ⊂ Nx of x such that:

b) Mx ⊂ some Vα and if Mx ∩ Vβ 6= ∅, then Mx ⊂ Vβ . Let W = {Mx}x∈X . Then W refines V and it

follows immediately that refV,W (s) ≡ 0 as a cochain. 1. BASIC CECHˇ COHOMOLOGY 217

a) If f : X Y is a continuous map of topological spaces, (resp. ) a presheaf on X → F G (resp. Y ), and α: a homomorphism covering f, (i.e., a set of homomorphisms: G → F α(U): (U) (f −1(U)), all open U Y G −→ F ⊂ commuting with restriction), then we get canonical maps: (f, α)∗ : Hi(Y, ) Hi(X, ), all i. G −→ F b) If we have two short exact sequences of presheaves and a commutative diagram:

0 / / / / 0 F1 F2 F3 α α α 1  2  3  0 / / / / 0, G1 G2 G3 then the δ’s in the long exact cohomology sequences give a commutative diagram:

δ Hi(X, ) / Hi+1(X, ) F3 F1 α α 3   1 δ Hi(X, ) / Hi+1(X, ), G3 G1 (i.e., the Hi(X, )’s together are a “cohomological δ-functor”). F c) If and are two presheaves of abelian groups, define a presheaf by ( )(U)= F G F⊗G F⊗G (U) (U). Then there is a bilinear map: F ⊗ G Hi(X, ) Hj(X, ) Hi+j(X, ) F × G −→ F ⊗ G called cup product, and written . ∪ To construct the map in (a), take the obvious map of cocycles and check that it commutes with δ; (b) is a straightforward computation; as for (c), define on couples by: ∪ (s t)(α ,...,α ) = res s(α ,...,α ) res t(α ,...,α ) ∪ 0 i+j 0 i ⊗ i i+j and check that δ(s t)= δs t + ( 1)is δt. It is not hard to check that is associative and ∪ ∪ − ∪ ∪ has a certain skew-commutativity property: c′) If s Hki (X, ), i = 1, 2, 3, then in the group Hk1+k2+k3 (X, ) we have i ∈ Fi F1 ⊗ F2 ⊗ F3 (s s ) s = s (s s ). 1 ∪ 2 ∪ 3 1 ∪ 2 ∪ 3 c′′) If Symm2 is the quotient presheaf of by the subsheaf of elements a b b a, F F ⊗ F ⊗ − ⊗ and s Hki (X, ), i = 1, 2, then in the group Hk1+k2 (X, Symm2 ) we have i ∈ F F s s = ( 1)k1k2 s s . 1 ∪ 2 − 2 ∪ 1 The proofs are left to the reader. The cohomology exact sequence leads to the method of computing cohomology by acyclic resolutions: suppose a sheaf is given and we construct a long exact sequence of sheaves F 0 , −→ F −→ G0 −→ G1 −→ G2 −→ · · · such that: a) Hi(X, ) = (0), i 1, k 0. Gk ≥ ≥ b) If = Ker( ) and = cokernel as presheaf ( ) so that = Kk Gk+1 → Gk+2 Ck Gk−1 → Gk Kk sh( ), then assume Ck Hi(X, ) ≈ Hi(X, ), i 0, k 0. Ck −→ Kk ≥ ≥ 218 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Then Hi(X, ) is isomorphic to the i-th cohomology group of the complex: F 0 (X) (X) (X) . −→ G0 −→ G1 −→ G2 −→ · · · To see this, use induction on i. We may split off the first part of our resolution like this: i) 0 0, exact as presheaves. →F→G0 →C0 → ii) 0 , exact as sheaves. → K0 → G1 → G2 → G3 →··· So by the cohomology sequence of (i) and induction applied to the resolution (ii): a)

H0(X, ) = Ker H0( ) H0( ) F ∼ G0 −→ C0 0 0 = Ker H ( 0) H ( 0) ∼ G −→ K 0 0 = Ker H ( 0) H ( 1)  ∼ G −→ G b)  

H1(X, ) = Coker H0( ) H0( ) F ∼ G0 −→ C0 0 0 = Coker H ( 0) H ( 0) ∼ G −→ K 0 0 0 = Coker H ( 0) Ker H ( 1) H ( 2) ∼ G −→ G −→ G 1 0 = H (the complex H ( )).  ∼ G c)

Hi(X, ) = Hi−1(X, ) F ∼ C0 = Hi−1(X, ) ∼ K0 i 0 = H (the complex H ( )), i 2. ∼ G ≥ If is a sheaf, we have seen that H0(X, ) is just Γ(X, ) or (X). H1(X, ) also has a F F F F F simple interpretation in terms of “twisted structures” over X. Define

A principal -sheaf F = a sheaf of sets , plus an action of on G F G (i.e., (U) acts on (U) commuting with restriction) F G such that a covering U of X where: ∃ { α} res ( , as sheaf with -action) Uα G F = res ( , with -action on itself by translation) . ∼ Uα F F Then if is a sheaf: F ( ) H1(X, ) = set of principal -sheaves, modulo isomorphism . ∗ F ∼ { F } subset of those principal -sheaves which are trivial H1( , ) = F . U F ∼ on the open sets U of the covering  α U  In fact, a) Given , let φ : ≈ be an -isomorphism. Then on U U , φ G α G|Uα −→ F|Uα F α ∩ β α ◦ φ−1 : is an -automorphism. If it carries the 0-section to β F|Uα∩Uβ −→ F|Uα∩Uβ F s(α, β) (U U ), it will be the map x x + s(α, β). One checks that s is ∈ F α ∩ β 7→ a 1-cocycle, hence it defines a cohomology class in H1( , ), and by refinement in U F H1(X, ). F 2.THECASEOFSCHEMES:SERRE’STHEOREM 219

b) Conversely, given σ H1(X, ), represent σ by a 1-cocycle s(α, β) for a covering U . ∈ F { α} Define a sheaf by Gσ collections of elements t (V U ) such that (V )= α ∈ F ∩ α . Gσ res t + s(α, β) = res t in (V U U )  α β F ∩ α ∩ β  Intuitively, is obtained by “glueing” the sheaves together by translation by Gσ F|Uα s(α, β) on U U . α ∩ β We leave it to the reader to check that is independent of the choice of s and that the Gσ constructions (a) and (b) are inverse to each other. The same ideas exactly allow you to prove: If is a sheaf of rings on X and ∗ = subsheaf of units in , then OX OX OX set of sheaves of -modules, locally isomorphic H1(X, ∗ ) = OX OX ∼ to itself, modulo isomorphism  OX  (cf. III.6) § and

If X is locally connected and (Z/nZ)X = sheafification of the constant presheaf Z/nZ, then set of covering spaces π : Y X with Z/nZ 1 → H (X, (Z/nZ)X ) = acting on Y , permuting freely and transitively . ∼   the points of each set π−1(x), x X  ∈    2. The case of schemes: Serre’s theorem From now on, we assume that X is a scheme3 and that is a quasi-coherent sheaf. The F main result is this:

Theorem 2.1 (Serre). Let and be two affine open coverings of X, with refining . U V V U Then res : Hi( , ) Hi( , ) U,V U F −→ V F is an isomorphism.

The proof consists in two steps. The first is a general criterion for ref to be an isomorphism. The second is an explicit computation for modules and distinguished affine coverings. The general criterion is this:

Proposition 2.2. Let X be any topological space, a sheaf of abelian groups on X, and F U and two open coverings of X. Suppose refines . For every finite subset S = α ,...,α V V U 0 { 0 p}⊂ S, let U = U U S0 α0 ∩···∩ αp and let U denote the covering of US induced by . Assume: V| S0 0 V i H ( U , U ) = (0), all S0, i> 0. V| S0 F| S0 Then ref : Hi( , ) Hi( , ) is an isomorphism for all i. U,V U F → V F

3Our approach works only because all our schemes are separated. In the general case, Cechˇ cohomology is not good and either derived functors (via Grothendieck) or a modified Cechˇ complex (via Lubkin or Verdier ) must be used. 220 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Proof. The technique is to compare the two Cechˇ cohomologies via a big double complexes:

Cp,q = (U U V V ). F α0 ∩···∩ αp ∩ β0 ∩···∩ βq α0,...,αYp∈S β0,...,βYq∈T

By ignoring either the α’s or the β’s and taking δ in the β’s or α’s, we get two coboundary maps:

δ : Cp,q Cp+1,q 1 −→ p+1 δ s(α ,...,α , β ,...,β )= ( 1)js(α ,..., α ,...,α , β ,...,β ) 1 0 p+1 0 q − 0 j p+1 0 q Xj=0 b and

δ : Cp,q Cp,q+1 2 −→ q+1 δ s(α ,...,α , β ,...,β )= ( 1)js(α ,...,α , β ,..., β ,...,β ). 2 0 p 0 q+1 − 0 p 0 j q+1 Xj=0 b

One checks immediately that these satisfy δ2 = δ2 = 0 and δ δ = δ δ . As in 1, we get a “total 1 2 1 2 2 1 § complex” by setting:

C(n) = Cp,q p+q=n p,qX≥0

and with d = δ + ( 1)pδ : C(n) C(n+1) as differential. Here is the picture: 1 − 2 →

O

0,2 / C O O

δ2 δ 0,1 1 / 1,1 / C O C O O

δ2 δ2

δ1 δ1 C0,0 / C1,0 / C2,0 / 2.THECASEOFSCHEMES:SERRE’STHEOREM 221 where C0,2 = (U V V V ) F α ∩ β0 ∩ β1 ∩ β2 α∈S β0,βY1,β2∈T C0,1 = (U V V ) F α ∩ β0 ∩ β1 α∈S β0,βY1∈T C1,1 = (U U V V ) F α0 ∩ α1 ∩ β0 ∩ β1 α0,α1∈S β0,βY1∈T C0,0 = (U V ) F α ∩ β α∈S βY∈T C1,0 = (U U V ) F α0 ∩ α1 ∩ β α0,α1∈S βY∈T C2,0 = (U U V ). F α0 ∩ α1 ∩ β α0,α1,α2∈S βY∈T We need to observe four things about this situation: (A) The columns of this double complex are just products of the Cechˇ complexes for the p,0 p,1 coverings U for various S0 S: in fact the p-th column C C is the V| S0 ⊂ → → ··· product of these complexes for all S0 with #S0 = p+1. By assumption these complexes have no cohomology beyond the first place, hence the δ2-cohomology of the columns Ker δ : Cp,q Cp,q+1 / Image δ : Cp,q−1 Cp,q 2 −→ 2 −→ is (0) for all p 0, all q > 0.   ≥ (B) The rows of this double complex are similarly products of the Cechˇ complexes for the coverings V for various T0 T . Now VT some Vβ some Uα, hence the covering U| T0 ⊂ 0 ⊂ ⊂ V of VT includes among its open sets the whole space VT . For such silly coverings, U| T0 0 0 Cechˇ cohomology always vanishes — Lemma 2.3. X a topological space, a sheaf, and an open covering of X such F U that X . Then Hi( , ) = (0), i> 0. ∈U U F Proof of Lemma 2.3. Let X = U . For all s Zi( , ), define an (i 1)- ζ ∈ U ∈ U F − cochain by: t(α0,...,αi−1)= s(ζ, α0,...,αi−1) [OK since U U U = U U !] An easy calculation shows that ζ ∩ α0 ∩···∩ αi−1 α0 ∩···∩ αi−1 s = δt.  Hence the δ1-cohomology of the rows is (0) at the (p,q)-th spot, for all p > 0, q > 0. (C) Next there is a big diagram-chase —

p,q Lemma 2.4 (The easy lemma of the double complex). Let C , δ1, δ2 p,q≥0 be any 2 2 { } double complex (meaning δ1 = δ2 = 0 and δ1δ2 = δ2δ1). Assume that the δ2-cohomology: p,q p,q p,q+1 p,q−1 p,q H = Ker δ2 : C C / Image δ2 : C C δ2 −→ −→     222 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

is (0) for all p 0, q > 0. Then there is an isomorphism: ≥ p,0 p δ1-cohomology of H = ((d = δ1 + ( 1) δ2)-cohomology of total complex) δ2 − i.e., 

i,j x Cp,0 δ x = δ x = 0 x i+j=p C dx = 0 1 2 ∈ | . { ∈ p|−1,0 } ∼= δ1x x C with δ2x = 0 n dx Px Ci,j o { | ∈ } | ∈ i+j=p−1 n o Proof of Lemma 2.4. We give the proof in detailP for p = 2 in such a way that it is clear how to set up the proof in general. Start with x = (x ,x ,x ) Ci,j 2,0 1,1 0,2 ∈ i+j=2 such that dx = 0, i.e., P δ x = 0; δ x + δ x = 0; δ x δ x = 0; δ x = 0. 1 2,0 1 1,1 2 2,0 1 0,2 − 2 1,1 2 0,2 0O

δ2

_ δ1 x0,2 / y ±O δ2

_ δ1 x1,1 / z ±O δ2

_ δ1 x2,0 / 0 Now δ x =0 = x = δ x for some x . Alter x by the coboundary d(0, x ): 2 0,2 ⇒ 0,2 2 0,1 0,1 − 0,1 we find x x′ = (x′ ,x′ , 0) ( means cohomologous). ∼ 2,0 1,1 ∼ But then dx′ =0= δ x′ =0= x′ = δ x for some x . Alter x′ by the ⇒ 2 1,1 ⇒ 1,1 2 1,0 1,0 coboundary d(x1,0, 0): we find x′ x′′ = (x′′ , 0, 0) ∼ 2,0 ′′ ′′ ′′ ′′ 2 p,0 and dx =0 = δ1x and δ2x are 0. Thus x defines an element of H the complex H . ⇒ 2,0 2,0 2,0 δ1 δ2 This argument (generalized in the obvious way) shows that the map:   p,0 Φ: δ1-cohomology of H (d-cohomology of total complex) δ2 −→   is surjective. Now say x C2,0 satisfies δ x = δ x = 0. Say (x , 0, 0) = dx, 2,0 ∈ 1 2,0 2 2,0 2,0 x = (x ,x ) Ci,j, i.e., 1,0 0,1 ∈ i+j=1 x2P,0 = δ1x1,0; δ2x1,0 + δ1x0,1 = 0; δ2x0,1 = 0. − 0O

δ2

_ δ1 x0,1 / y ±O δ2

_ δ1 x1,0 / x2,0 Then δ x =0= x = δ x , for some x . Alter x by the coboundary dx : 2 0,1 ⇒ 0,1 2 0,0 0,0 − 0,0 we find x x′ = (x′ , 0) ∼ 0,0 2.THECASEOFSCHEMES:SERRE’STHEOREM 223

′ ′ ′ and dx = (x2,0, 0, 0). Then δ2x1,0 = 0 and δ1x1,0 = x2,0, i.e., x2,0 goes to 0 in the δ -cohomology of Hp,0. This gives injectivity of Φ.  1 δ2 If we combine (A), (B) and (C), applied both to the rows and columns of our double complex, we find isomorphisms:   Hn(total complex C( )) = Hn the complex Ker δ in C ,0 d ∼ δ1 2  = Hn the complex Ker δ in C0, . ∼ δ2 1
But
Ker δ : Cn,0 Cn,1 = (U U )= Cn( , ) 2 −→ ∼ F α0 ∩···∩ αn U F α ,...,α ∈S
0 Yn Ker δ : C0,n C1,n = (V V )= Cn( , ), 1 −→ ∼ F β0 ∩···∩ βn V F β ,...,β ∈T
0 Yn so in fact  Hn total complex C( ) = Hn( , ) d ∼ U F   = Hn( , ). ∼ V F It remains to check:

(D) The above isomorphism is the refinement map, i.e., if s(α0,...,αn) is an n-cocycle for , then s Cn,0 and refσ s C0,n are cohomologous in the total complex. In fact, U ∈ U,V ∈ define t C(n−1) by setting its (l,n 1 l)-th component equal to: ∈ − − t (α ,...,α , β ,...,β ) = ( 1)l res s(α ,...,α , σβ ,...,σβ ). l 0 l 0 n−1−l − 0 l 0 n−1−l Thus a straightforward calculation shows that dt = (refσ s) s. U,V − This completes the proof of Proposition 2.2.  Now return to the proof of Theorem 2.1 for quasi-coherent sheaves on schemes! The second step in its proof is the following explicit calculation: Proposition 2.5. Let Spec R be an affine scheme, = Spec R a finite distinguished U { fi }i∈I affine covering and M a quasi-coherent sheaf on X. Then Hi( , M) = (0), all i> 0. U ˇ Proof. Since M(Spec Rf ) = Mf and i∈I Spec Rfi = Spec R( fi), the complex of Cech f ∼ 0 fQi∈I0 cochains reduces to: f T Mf M(f ·f ) M(f ·f ·f ) . i −→ i0 i1 −→ i0 i1 i2 −→ · · · Yi∈I i0Y,i1∈I i0,iY1,i2∈I Using the fact that the covering is finite, we can write a k-cochain: m m(i ,...,i )= i0,...,ik , m M 0 k (f f )N i0,...,ik ∈ i0 · · · ik with fixed denominator. Then mi ,...,i mi ,i ,...,i m (δm)(i ,...,i )= 1 k+1 0 2 k+1 + + ( 1)k+1 i0,...,ik 0 k+1 (f f )N − (f f f )N · · · − (f f )N i1 · · · ik i0 i2 · · · ik+1 i0 · · · ik N N k+1 N fi mi1,...,ik+1 fi mi0,i2,...,ik+1 + + ( 1) fi mi0,...,ik = 0 − 1 · · · − k+1 . (f f )N i0 · · · ik+1 If δm = 0, then this expression is 0 in M , hence (fi0 ···fik+1 )

N ′ N N k+1 N (fi fi ) f mi ,...,i f mi ,i ,...,i + + ( 1) f mi ,...,i = 0 0 · · · k+1 i0 1 k+1 − i1 0 2 k+1 · · · − ik+1 0 k h i 224 VII.THECOHOMOLOGYOFCOHERENTSHEAVES in M if N ′ is sufficiently large. But rewriting the original cochain m with N replaced by N +N ′, we have ′ m ′ i0,...,ik ′ N m(i0,...,i )= ′ , m = (fi fi ) mi ,...,i k (f f )N+N i0,...,ik 0 · · · k 0 k i0 · · · ik so that

′ ′ ′ ( ) f N+N m′ f N+N m′ + + ( 1)k+1f N+N m′ =0 in M. ∗ i0 i1,...,ik+1 − i1 i0,i2,...,ik+1 · · · − ik+1 i0,...,ik Now since

′ Spec R = Spec Rfi = Spec R N+N , (fi ) ′ it follows that 1 (...,f N+N ,...), i.e.,[ we can write[ ∈ i ′ 1= g f N+N i · i Xi∈I for some g R. Now define a (k 1)-cochain n by the formula: i ∈ − n n(i ,...,i )= i0,...,ik 0 k−1 (f f )N+N ′ i0 · · · ik−1 ′ ni ,...,i = gl m . 0 k−1 · l,i0,...,ik=1 Xl∈I Then m = δn! In fact k j ni0 ,..., ij ,...,ik (δn)(i0,...,ik)= ( 1) − · N+N ′ j=0 (fi0 fij fik ) X · · · ·b · · k 1 b j N+N ′ ′ = ′ ( 1) f glm N+N ij l,i0,...,ij ,...,ik (fi0 fik ) − · · · · Xj=0 Xl∈I b k 1 j N+N ′ ′ = ′ gl ( 1) f m N+N ij l,i0,...,ij ,...,ik (fi0 fik ) − · · · Xl∈I Xj=0 b 1 ′ = g f N+N m′ (by ( )) N+N ′ l i i0,...,ik (fi0 fik ) ∗ · · · Xl∈I ′ m ′ i0,...,ik N+N = N+N ′ glfi (fi0 fik ) · · · Xl∈I = m(i0,...,ik). 

Corollary 2.6. Let X be an affine scheme, any affine covering of X and M a quasi- U coherent sheaf on X. Then Hi( , M) = (0), i> 0. U f Proof. Since the distinguishedf affines form a basis for the topology of X, and X is quasi- compact, we can find a finite distinguished affine covering of X refining . Consider the V U map ref : Hi( , M) Hi( , M). U,V U −→ V i i By Proposition 2.5, H ( , M) = (0) all i > 0, and H ( U , M U ) = (0) for all i > 0 and V f V| fS0 | S0 for all finite intersections U = U U (since each V U is a distinguished affine in S0 α0 ∩···∩ αp β ∩ S0 U too). Therefore by Propositionf 2.2, ref is an isomorphism,f hence Hi( , M) = (0) for all S0 U,V U i> 0.  f 2.THECASEOFSCHEMES:SERRE’STHEOREM 225

Theorem 2.1 now follows immediately from Proposition 2.2 and Corollary 2.6, in view of the fact that since X is separated, each US0 as a finite intersection of affines, is also affine as are the open sets V U that cover it. β ∩ S0 Theorem 2.1 implies: Corollary 2.7. For all schemes X, quasi-coherent and affine covering , the natural F U map: Hi( , ) Hi(X, ) U F −→ F is an isomorphism. The “easy lemma of the double complex” (Lemma 2.4) has lots of other applications in homological algebra. We sketch one that we can use later on. a) Let R be any commutative ring, let M (1), M (2) be R-modules, choose free resolutions (1) (1) (2) (2) F M and F M , i.e., exact sequences → → F (1) F (1) F (1) F (1) M (1) 0 −→ n −→ n−1 −→··· −→ 1 −→ 0 −→ −→ F (2) F (2) F (2) F (2) M (2) 0 −→ n −→ n−1 −→··· −→ 1 −→ 0 −→ −→ where all F (i) are free R-modules. Look at the double complex C = F (1) F (2), j i,j i ⊗R j 0 i, j with boundary maps ≤ d(1) : C C i,j −→ i−1,j d(2) : C C i,j −→ i,j−1 induced by the d’s in the two resolutions. Then Lemma 2.4 shows that (1) (2) H (total complex C ) = H (complex F M ) n , ∼ n ⊗R (1) (2) = H (complex M F ). ∼ n ⊗R Note that the arrows here are reversed compared to the situation in the text. For complexes in which d decreases the index, we take of homology on Hn instead of coho- mology on Hn. It is not hard to check that the above R-modules are independent of (1) (2) R (1) (2) the resolutions F , F . They are called Torn (M , M ). The construction could be globalized: if X is a scheme, (1), (2) are quasi-coherent sheaves, then there are F F canonical quasi-coherent sheaves or OX ( (1), (2)) such that for all affine open U X, T n F F ⊂ if U = Spec R

(i) = M (i), F then g or OX ( (1), (2)) = TorR(M (1), M (2)) . T n F F |U n I want to conclude this section with the classical explanation of the “meaning” of H1(X, ), OX via so-called “Cousin data”. Let me digress to give a little history:f in the 19th century Mittag- Leffler proved that for any discrete set of points αi C and any positive integer ni, there is a ∈ meromorphic function f(z) with poles of order ni at αi and no others. Cousin generalized this to n meromorphic functions f(z1,...,zn) on C in the following form: say Ui is an open covering n { } of C and fi is a meromorphic function on Ui such that fi fj is holomorphic on Ui Uj. Then − ∩ there exists a meromorphic function f such that f f is holomorphic on U . We can easily pose − i i an algebraic analog of this — a) Let X be a reduced and irreducible scheme. 226 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

b) Let R(X) = function field of X. c) Cousin data consists in an open covering Uα α∈S of X plus fα R(X) for each α such { } ∈ that f f Γ(U U , ), all α, β. α − β ∈ α ∩ β OX d) The Cousin problem for this data is to find f R(X) such that ∈ f f Γ(U , ), all α, − α ∈ α OX i.e., f and fα have the same “polar part” in Uα. For all Cousin data f , let g = f f Γ(U U , ). Then g is a 1-cocycle in { α} αβ α − β ∈ α ∩ β OX { αβ } for the covering U and by refinement, it defines an element of H1(X, ), which we call OX { α} OX ob( f )(= the “obstruction”). { α} Proposition 2.8. ob( f ) = 0 iff the Cousin problem has a solution. { α} Proof. If ob( f ) = 0, then there is a finer covering V and h Γ(V , ) such { α} { α}α∈T α ∈ α OX that if σ : T S is a refinement map, then → h h = res g = res(f f ) α − β σα,σβ σα − σβ (equality here being in the ring Γ(Vα Vβ, X )). But then in R(X), ∩ O h f = h f , α − σα β − σβ i.e., f h = F is independent of α. Then F has the same polar part as f in V . And σα − α σα α for any x U , take β so that x V too; then since f f , it follows that ∈ α ∈ β α − σβ ∈ Ox,X F f = (F f ) + (f f ) , i.e., F has the same polar part as f throughout U , so − α − σβ σβ − α ∈ Ox,X α α F solves the Cousin problem. Conversely, if such F exists, let h = f F ; then h h = g α α − α − β αβ and h Γ(U , ), i.e., g = δ( h ) is a 1-coboundary.  α ∈ α OX { αβ } { α} 3. Higher direct images and Leray’s spectral sequence One of the main tools that is used over and over again in computing cohomology is the higher direct image sheaf and the Leray spectral sequence. Let f : X Y be a continuous map → of topological spaces and let be a sheaf of abelian groups on X. For all i 0, consider the F ≥ presheaf on Y : a) U Hi(f −1(U), ), U Y open 7−→ F ∀ ⊂ b) if U U , then 1 ⊂ 2 res: Hi(f −1(U ), ) Hi(f −1(U ), ) 2 F −→ 1 F is the canonical map.

Definition 3.1. Rif ( ) = the sheafification of this presheaf, i.e., the universal sheaf which ∗ F receives homomorphisms: Hi(f −1(U), ) Rif (U), all U. F −→ ∗F Proposition 3.2. If X and Y are schemes, f : X Y is quasi-compact and is quasi- → F coherent -module, then Rif ( ) is quasi-coherent -module. Moreover, if U is affine or if OX ∗ F OY i = 0, then Hi(f −1(U), ) Rif ( )(U) F −→ ∗ F is an isomorphism. 3. HIGHER DIRECT IMAGES AND LERAY’S SPECTRAL SEQUENCE 227

Proof. In fact, by the sheaf axiom for , it follows immediately that the presheaf U F 7→ H0(f −1(U), ) = (f −1(U)) is a sheaf on Y . Therefore H0(f −1(U), ) R0f (U) is an F F F → ∗F isomorphism for all U. The rest of the proposition falls into the set-up of (I.5.9). As stated there, it suffices to verify that if U is affine, R = Γ(U, ) and g R, then we get an isomorphism: OX ∈ Hi(f −1(U), ) R ≈ Hi(f −1(U ), ). F ⊗R g −→ g F −1 But since f is quasi-compact, we may cover f (U) by a finite set of affines V1,...,VN = . −1 { } V Then f (Ug) is covered by

(V ) ∗ ,..., (V ) ∗ = −1 1 f (g) N f (g) V|f (Ug ) which is again an affine covering. Therefore  Hi(f −1(U), )= Hi(C ( , )) F V F i −1 i  H (f (U ), )= H (C ( −1 , )). g F V|f (Ug ) F The cochain complexes are: Ci( , )= (V V ) V F F α0 ∩···∩ αi 1≤α0,...,αY i≤N i C ( −1 , )= ((V ) ∗ (V ) ∗ ). V|f (Ug) F F α0 f g ∩···∩ αi f g 1≤α0,...,αY i≤N Since if S = Γ(V V , ): α0 ∩···∩ αi OX ((V ) ∗ (V ) ∗ ) = (V V ) S ∗ F α0 f g ∩···∩ αi f g ∼ F α0 ∩···∩ αi ⊗S f g = (V V ) R ∼ F α0 ∩···∩ αi ⊗R g it follows that i i C ( −1 , ) = C ( , ) R V|f (Ug ) F ∼ V F ⊗R g (since commutes with finite products). But now localizing commutes with kernels and ⊗    cokernels, so for any complex A of R-modules, Hi(A ) R = Hi(A R ). Thus ⊗R g ∼ ⊗R g Hi(f −1(U ), ) = Hi(f −1(U), ) R g F ∼ F ⊗R g as required.  Corollary 3.3. If f : X Y is an affine morphism (cf. Proposition-Definition I.7.3) and → a quasi-coherent -module, then F OX Rif = 0, i> 0. ∗F ∀ A natural question to ask now is whether the cohomology of on X can be reconstructed F by taking the cohomology on Y of the higher direct images Rif . The answer is: almost. The ∗F relationship between them is a spectral sequence. These are the biggest monsters that occur in homological algebra and have a tendency to strike terror into the heart of all eager students. I want to try to debunk their reputation of being so difficult4. Definition 3.4. A spectral sequence Epq = En consists in two pieces of data5: 2 ⇒ 4(Added in publication) Fancier notions of “derived categories and derived functors” have since become indispensable not only in algebraic geometry but also in analysis, mathematical physics, etc. Among accessible references are: Hartshorne [49], Kashiwara-Schapira [59], [60] and Gelfand-Manin [38]. 5 pq Sometimes one also has a spectral sequence that “begins” with an E1 . Then the first differential is pq pq p+1,q d1 : E1 −→ E1 pq p,q p−1,q and if you set E2 = (Ker d1 )/(Image d1 ), you get a spectral sequence as above. 228 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

pq E2 @E • • • @E d2 @E @E @E" Ep+2,q−1 • • @ 2 • @ d @ 3 @ @ @ @ @ @ Ep+3,q−2 • • • 2

etc. • • •

Figure VII.2

pq (A) A doubly infinite collection of abelian groups E2 , (p,q Z, p,q 0) called the initial pq ∈ ≥ terms plus filtrations on each E2 , which we write like this: Epq = Zpq Zpq Zpq Bpq Bpq Bpq = (0), 2 2 ⊃ 3 ⊃ 4 ⊃···⊃ 4 ⊃ 3 ⊃ 2 also, let pq pq Z∞ = Zr r \ pq pq B∞ = Br , r [ pq pq pq plus a set of homomorphisms dr that allow us to determine inductively Zr+1, Br+1 pq pq from the previous ones Zr , Br : dpq : Zpq Ep+r,q−r+1/Bp+r,q−r+1 r r −→ 2 r (cf. Figure VII.2). The d’s should have the properties pq pq p+r,q−r+1 pq pq i) Br Ker(dr ), Zr Image(dr ) so that dr induces a map ⊂ ⊃ Zpq/Bpq Zp+r,q−r+1/Bp+r,q−r+1. r r −→ r r pq pq This sub-quotient of E2 is called Er . ii) d2 = 0; more precisely, the composite Zpq/Bpq Zp+r,q−r+1/Bp+r,q−r+1 Zp+2r,q−2r+2/Bp+2r,q−2r+2 r r −→ r r −→ r r is 0. pq pq p+r,q−r+1 pq pq iii) Zr+1 = Ker(dr ); Br+1 = Image(dr ). This implies that Er+1 is the coho- pq mology of the complex formed by the Er ’s and the dr’s! (B) The so-called “abutment”: a simply infinite collection of abelian groups En plus a n p,n−p filtration on each E whose successive quotients are precisely the groups E∞ : En = F 0(En) F 1(En) F n(En) F n+1(En) = (0) ⊃ ⊃··· ······⊃ ⊃ 0,n 1,n−1 ··· n,0 =∼E∞ =∼E∞ =∼E∞ To illustrate what| is going{z on} here,| {z look} at| the{z terms} | of lowes{zt total degree.} One sees easily that one gets the following exact sequences: 00 0 a) E2 ∼= E . 3. HIGHER DIRECT IMAGES AND LERAY’S SPECTRAL SEQUENCE 229

d b) 0 E1,0 E1 E0,1 2 E2,0 E2. −→ 2 −→ −→ 2 −→ 2 −→ c) For all n, one gets “edge homomorphisms” En,0 En,0 En 2 −→ ∞ −→ and En E0,n E0,n : −→ ∞ −→ 2 i.e., n k EE kkk kkkk 0,n ku kk E2 F F n,0 E2

Theorem 3.5. 6 Given any quasi-compact morphism f : X Y and quasi-coherent sheaf → F on X, there is a canonical spectral sequence, called Leray’s spectral sequence, with initial terms Epq = Hp(Y, Rqf ) 2 ∗F and abutment En = Hn(X, ). F Proof. Choose open affine coverings = U of Y and = V of X and consider U { α}α∈S V { β}β∈T the double complex introduced in 2 for the two coverings f −1( ) and of X: § U V Cpq = (f −1U f −1U V V ). F α0 ∩···∩ αp ∩ β0 ∩···∩ βq α0,...,αYp∈S β0,...,βYq∈T Note that all the open sets here are affine because of Proposition II.4.5. Now the q-th row of our double complex is the product over all β0,...,βq T of the Cechˇ  ∈ complex C (f −1( ) V V , ), i.e., the Cechˇ complex for an affine open covering of U ∩ β0 ∩···∩ βq F an affine Vβ0 Vβq . Therefore all the rows are exact except at their first terms where their ∩···∩ q cohomology is (Vβ Vβ ), i.e., C ( , ). Hence by the easy lemma of the double β0,...,βq F 0 ∩···∩ q V F complex (Lemma 2.4), Q 1)  Hn(total complex) = Hn(C ( , )) ∼ V F = Hn(X, ). ∼ F But on the other hand, the p-th column of our double complex is the product over all α0,...,αp  ∈ S of the Cechˇ complex C ( f −1(U U ), ). The cohomology of this complex at V ∩ α0 ∩···∩ αp F the q-th spot is Hq(f −1(U U ), ) which is also the same as Rqf (U U ). α0 ∩···∩ αp F ∗F α0 ∩···∩ αp Therefore: q 2) [vertical δ2-cohomology of p-th column at (p,q)] = R f∗ (Uα Uα ). ∼ α0,...,αp∈S F 0 ∩···∩ p But now the horizontal maps δ : Cp,q Cp+1,q induce maps from the [δ -cohomology 1 → Q 2 at (p,q)-th spot] [δ2-cohomology at (p + 1,q)-th spot] and we see easily that →  q 3) [q-th row of vertical cohomology groups] = Cechˇ complex C ( , R f∗ ). There- as complex∼ U F fore finally: 4) [horizontal δ -cohomology at (p,q) of vertical δ -cohomology group] = Hp(Y, Rqf )! 1 2 ∼ ∗F Theorem 3.5 is now reduced to:

6Theorem 3.5 also holds for continuous maps of paracompact Hausdorff spaces and arbitrary sheaves F, but we will not use this. 230 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

p,q Lemma 3.6 (The hard lemma of the double complex). Let (C , δ1, δ2) be any double com- plex. Make no assumption on the δ2-cohomology, but consider instead its δ1-cohomology:   Ep,q = Hp (Hq (C , )). 2 δ1 δ2 p,q Then there is a spectral sequence starting at E2 and abutting at the cohomology of the total complex. Alternatively, one can “start” this spectral sequence at  Ep,q = Hq (Cp, ) = (cohomology in vertical direction) 1 δ2 7 with d1 being the maps induced by δ1 on δ2-cohomology . Also, since the rows and columns of a double complex play symmetric roles, one gets as a consequence a second spectral sequence with   Ep,q = Hp (Hq (C , )) 2 δ2 δ1 or  Ep,q = Hq (C ,p) = (cohomology in horizontal direction), 1 δ1 abutting also to the cohomology of the total complex.

A hard-nosed detailed proof of this is not very long but quite unreadable. I think the reader will find it easier if I sketch the idea of the proof far enough so that he/she can work out for pq himself/herself as many details as he/she wants. To begin with, we may describe E2 rather more explicitly as: x Cp,q δ x = 0 and δ x = δ y, some y Cp+1,q−1 Epq = { ∈ | 2 1 2 ∈ }. 2 δ (Cp,q−1)+ δ x Cp−1,q δ x = 0 2 1{ ∈ | 2 } The idea is — how hard is it to “extend” the δ2-cocycle x to a whole d-cocycle in the total complex: more precisely, to a set of elements x Cp,q δ x = 0 ∈ 2 y Cp+1,q−1 δ y = δ x 1 ∈ 2 1 1 y Cp+2,q−2 δ y = δ y 2 ∈ 2 2 1 1 etc. etc. so that d(x y1 y2 ) = 0 (the signs being mechanically chosen here taking into account ± ±p ±··· that d = δ1 + ( 1) δ2). See Figure VII.3. pq − pq pq Define Z∞ to be the subgroup of E2 for which such a sequence of yi’s exist; define Z3 to pq be the set of x’s such that such y1 and y2 exist; define Z4 to be the set of x’s such that such y1, y2 and y3 exist; etc. On the other hand, a δ2-cocycle x may be a d-coboundary in various ways — let w Cp−1,q, w Cp−2,q−1 Bpq = image in Epq of x Cp,q 1 ∈ 2 ∈ 3 2 ∈ δ w = x, δ w = δ w , δ w = 0  1 1 2 1 1 2 2 2 

w , w as above, w Cp−3,q−2 Bpq = image in Epq of x Cp,q 1 2 3 ∈ 4 2 ∈ δ w = x, δ w = δ w , δ w = δ w , δ w = 0  1 1 2 1 1 2 2 2 1 3 2 3 

etc.

(cf. Figure VII.4)

7More precisely, to construct the spectral sequence, one doesn’t need both gradings on L Cp,q and both p,q differentials; it is enough to have one grading (the grading by total degree), one filtration (Fk = Lp≥k C ) and the total differential: for details cf. MacLane [68, Chapter 11, §§3 and 6]. 3. HIGHER DIRECT IMAGES AND LERAY’S SPECTRAL SEQUENCE 231

0O δ2 δ _ 1 / z x O1 δ2 _ δ1 y / z 1 O2 δ2 _ δ1 y2 /

...... drx! x Ñ δ y 1 / z Ù r−1 r−O 1   δ2 _ δ1  yr−1 //.-,()*+?

Figure VII.3

0O δ2 _ wr / ur−1 δ1 O δ2 _ . wr−1 ..

......

. u .. O2

δ2 _ w2 / u1 δ1 O δ2 _ w1 / x δ1

Figure VII.4

pq pq p+r,q−r+1 p+r,q−r+1 p,q pq As for dr : Zr Er /Br , suppose x C defines an element of Zr , i.e., → ∈ y Cp+1,q−1,...,y Cp+r−1,q−r+1 such that δ y = δ y , i < r 1; δ y = δ x. Define ∃ 1 ∈ r−1 ∈ 2 i+1 1 i − 2 1 1 pq dr (x)= δ1yr−1.

p+r,q−r+1 p+r,q−r+1 p+r,q−r+1 This is an element of C killed by δ1 and δ2, hence it defines an element of Er /Br . At this point there are quite a few points to verify — that dr is well-defined so long as the im- age is taken modulo Br and that dr has the three properties of the definition. These are all mechanical and we omit them. 232 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

0 _ ?>=<89:;F k 0 . .. 0 x k,n−k . .. xn−1,1 _ xn,0

Figure VII.5

Finally, define the filtration on the cohomology of the total complex:

Ker d in Cp,q F k(En) =those elements of p+q=n  p,q  d p+qP=n−1 C which can be represented P by a d-cocycle  with components x Cp,q, x =0 if p < k pq ∈ pq (cf. Figure VII.5). The whole point of these definitions, which is now reasonable I hope, is the isomorphism: p n p+1 n p,n−p p,n−p F E /F E ∼= Z∞ /B∞ . The details are again omitted. 

An important remark is that the edge homomorphisms in the Leray spectral sequence: a) Hn(Y,f ) = En,0 En = Hn(X, ) ∗F ∼ 2 → ∼ F b) Hn(X, ) = En E0,n = H0(Y, Rnf ) F ∼ → 2 ∼ ∗F are just the maps induced by the functorial properties of cohomology (i.e., the set of maps f (U) (f −1(U)) means that there is a map of sheaves f with respect to f and ∗F → F ∗F → F this gives (a); and the maps Hn(X, ) Hn(f −1U, ) Rnf (U) for all U give (b)). This F → F → ∗F comes out if is a refinement of f −1( ) by the calculation used in the proof of Theorem 3.5. V U Proposition 3.7. Let be a quasi-coherent -module. If f : X Y is an affine mor- F OX → phism (cf. Proposition-Definition I.7.3), then ∼ Hp(X, ) Hp(Y,f ), p. F −→ ∗F ∀ Proof. Leray’s spectral sequence (Theorem 3.5) and Corollary 3.3. 

Corollary 3.8. Let be a quasi-coherent -module. If i: X Y is a closed immersion F OX → of schemes (cf. Definition 3.1), then ∼ Hp(X, ) Hp(Y, i ), p. F −→ ∗F ∀ Remark. If X is identified with its image i(X) in Y , i is nothing but the quasi-coherent ∗F -module obtained as the extension of the -module by (0) outside X. OY OX F 4. COMPUTING COHOMOLOGY (1): PUSH F INTO A HUGE ACYCLIC SHEAF 233

A second important application of the hard lemma (Lemma 3.6) is to hypercohomology and in particular to De Rham cohomology (cf. VIII.3 below). Let  be any complex of sheaves on § F a topological space X. Then if is an open covering, Hn( , ) is by definition the cohomology U U F of the total complex of the double complex Cq( , p), hence we get two spectral sequences U F abutting to it. The first is gotten by taking vertical cohomology (with respect to the superscript q): pq  E =Hq( , p)= En = Hn( , ) 1 U F ⇒ U F (with dpq the map induced on cohomology by d: p p+1). 1 F → F Passing to the limit over finer coverings, we get: pq  (3.9) E = Hq(X, p)= En = Hn(X, ). 1 F ⇒ F The second is gotten by taking horizontal cohomology (with respect to p) and then vertical p  cohomology. To express this conveniently, define presheaves pre( ) by H F p p+1  Ker( (U) (U)) p ( )(U)= F → F . Hpre F Image( p−1(U) p(U)) F → F The sheafification of these presheaves are just: p p+1  Ker( ) p( )= F → F H F Image( p−1 p) F → F p but pre will not generally be a sheaf already. The horizontal cohomology of the double complex q H p q p q p C ( , ) is just C ( , pre) and the vertical cohomology of this is H ( , pre), hence we get U F U H U H the second spectral sequence: pq   E = Hp( , q ( )) = En = Hn( , ). 2 U Hpre F ⇒ U F Passing to the limit over , this gives: U pq   (3.10) E = Hp(X, q ( )) = En = Hn(X, ). 2 Hpre F ⇒ F In good cases, e.g., X paracompact Hausdorff (cf. 1), the cohomology of a presheaf is the § cohomology of its sheafification, so we get finally: pq   (3.11) E = Hp(X, q( )) = En = Hn(X, ). 2 H F ⇒ F

4. Computing cohomology (1): Push into a huge acyclic sheaf F Although the apparatus of cohomology of quasi-coherent sheaves may seem at first acquain- tace rather formidable, it should always be remembered that it is really only fancy linear algebra. In many specific cases, it is no great problem to compute it. To stress the flexibility of the tools available for computing cohomology, we present in a fugal style four calculations each using a different method. A standard approach for cohomology is via a resolution of the type: 0 I I I −→ F −→ 0 −→ 1 −→ 2 −→······ where the Ik’s are injective, or “flasque” or “mou” or at least are acyclic. (See Godement [39] or Swan [99].) Sheaves of this type tend to be huge monsters, but there has been quite a bit of work done on injectives in the category of sheaves of -modules on a noetherian X (see OX Hartshorne [49, p. 120]). We use the method as follows: 234 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Lemma 4.1. If U X is affine and i: U X the inclusion map, then for all quasi-coherent ⊂ → on U, i is acyclic, i.e., Hp(X, i ) = (0), all p 1. F ∗F ∗F ≥ Proof. In fact, for V X affine, i−1(V )= U V is affine, so the presheaf V Hp(i−1V, ) ⊂ ∩ 7→ F is (0) on affines (p 1). Thus Rpi = (0) if p 1. Then Leray’s spectal sequence (Theorem ≥ ∗F ≥ 3.5) degenerates since Epq = Hp(X, Rqi ) = (0), q 1. 2 ∗F ≥ pq pq p+q Thus E2 ∼= E∞ ∼= E , and the edge homomorphism Hp(X, i ) Hp(U, ) ∗F −→ F is an isomorphism. Since Hp(U, ) = (0), p 1, the lemma is proven.  F ≥ If is quasi-coherent on X, and i: U X is the inclusion of an affine, there is a canonical F → map: φ: i ( ) F −→ ∗ F|U via res (V ) (U V ) = i ( )(V ), open V, F −→ F ∩ ∼ ∗ F|U ∀ which is an isomorphism on U. We can apply this to prove:

Proposition 4.2. Let X be a noetherian scheme and a quasi-coherent sheaf on X. Let F n = dim(Supp ), i.e., n is the maximum length of chains: F Z0 $ Z1 $ $ Zr Supp( ), Zi closed irreducible. · · · ⊂ F Then Hi(X, ) = (0) if i>n. F Proof. Use induction on n. If n = 0, then Supp is a finite set of closed points x ,...,x . F { 1 N } For all i, let U X be an affine neighborhood of X such that x / U , all j = i; let U be i ⊂ i j ∈ i 6 { β}β∈T an affine covering of X x1,...,xN . Then U1,...,UN Uβ is an affine covering of X such \{ } { }∪ { } i that for any two disjoint open sets U , U ′ in it, U U ′ Supp = . Thus C ( , ) = (0), α α α ∩ α ∩ F ∅ U F i 1, and hence Hi(X, ) = (0), i 1. ≥ F ≥ In general, decompose Supp into irreducible sets: F Supp = S S . F 1 ∪···∪ N Let U X be an affine open set such that i ⊂ U S = i ∩ i 6 ∅ U S = , all j = i. i ∩ j ∅ 6 Let i : U X be the inclusion map, and let k k → = i ( ). Fk k,∗ F|Uk As above we have a canonical map: N φ F −→ Fk Mk=1 given by: N N res (V ) (U V ) i ( ) (V ). k ∼= k,∗ Uk F −→ F ∩ " F| # Mk=1 Mk=1 Concerning φ, we have the following facts: 5. COMPUTING COHOMOLOGY (2): DIRECTLY VIA THE CECHCOMPLEXˇ 235

a) If i = j, U U Supp = , hence (U U ) = (0). Therefore if V U , 6 i ∩ j ∩ F ∅ F i ∩ j ⊂ k0 N (U V )= (U V )= (V ). F k ∩ F k0 ∩ F Mk=1 Therefore φ is an isomorphism of sheaves on each of the open sets Uk. b) If V S = , then V U Supp = so (V ) = (U V ) = (0). Thus ∩ k ∅ ∩ k ∩ F ∅ Fk F k ∩ Supp S . Fk ⊂ k c) Each is quasi-coherent by Proposition 3.2, hence = Ker φ and = Coker φ are Fk K1 K2 quasi-coherent. Putting all this together, if i = 1, 2 Supp (S S ) (open set where φ is an isomorphism) Ki ⊂ 1 ∪···∪ N \ N (S S U ). ⊂ k \ k ∩ k k[=1 Therefore dim Supp n: Hp(X, ) / Hp(X, ) / Hp(X, ) K1 F K3 by induction ( ) (0) (0) ∗ by induction by Lemma 4.1 p−1 p N p H (X, 2) / H (X, 3) / H (X, ). K K k=1 Fk This proves that Hp(X, ) = (0) if p>n. L  F 5. Computing cohomology (2): Directly via the Cechˇ complex

We illustrate this approach by calculating Hi(Pn , (m)) for any ring R. We need some more R O definitions first: a) Let R be a ring, f ,...,f R. Let M be an R-module. Introduce formal symbols 1 n ∈ ω1,...,ωn such that ω ω = ω ω , ω ω = 0. i ∧ j − j ∧ i i ∧ i Define an R-module: Kp(f ,...,f ; M)= M ω ω . 1 n · i1 ∧···∧ ip 1≤i1

2  Note that d = 0. This gives us the Koszul complex K (f1,...,fn; M): 0 / K0((f); M) / K1((f); M) / / Kn((f); M) / 0 · · · n M M ωi M ω ω i=1 · · 1 ∧···∧ n b) Now say R is a graded ring,L f R is homogeneous and M is a graded module. i ∈ di Then we assign ωi the degree di, so that fiωi is homogeneous of degree 0. Then  − K (f1,...,fn; M) is a complex of graded modules with degree preserving maps d. We  0 P let K (f1,...,fn; M) denote the degree 0 subcomplex, i.e., p 0 K (f1,...,fn; M) = Md +···+d ωi ωi . i1 ip · 1 ∧···∧ p i1direct limit of the system ∈ g ∼ multiplication by g multiplication by g multiplication by g N N N , −−−−−−−−−−−→ −−−−−−−−−−−→ −−−−−−−−−−−→· · · 5. COMPUTING COHOMOLOGY (2): DIRECTLY VIA THE CECHCOMPLEXˇ 237 it follows that (ν) (ν) 0 lim Mν(d +···d )ω ω = (Mf ···f ) . i1 ip i1 ∧···∧ ip ∼ i1 ip −→ν We leave it to the reader to check that δ and d correspond.  The complex Kp goes down to p = 0 while Cp−1 only goes down to p = 1. We can { } { } extend Cp−1 one further step so that it matches up with Kp as follows: { } { } ε 0 1 0 / M0 / C ( , M) / C ( , M) / alt U alt U · · · / lim (K0)0 / lim (K1f)0 / lim (K2f)0 / 0 ν ν ν −→ −→ −→ · · · where ε is the composite of the canonical maps: M Γ(Proj R, M) C0 ( , M). 0 −→ −→ alt U What we need next is a criterion for a Koszul complex to be exact: f f Proposition 5.2 (Koszul). Let R be a ring, f1,...,fn R and M an R-module. If fs is ∈  a non-zero-divisor in M/(f ,...,f ) M for 1 s t, then the complex K (f ,...,f ; M) is 1 s−1 · ≤ ≤ 1 n exact at Ks(f ,...,f ; M) for 0 s t 1. 1 n ≤ ≤ − Proof. To see how simple this is, it’s better to take the first non-trivial case and check it, rather than getting confused in a general inductive proof. Take t = 3 and check that d d Mω Mω ω Mω ω ω i −→ i1 ∧ i2 −→ i1 ∧ i2 ∧ i3 Mi iM1

c) If t 3, Hi(Proj R, M(d)) = (0), 1 i t 2, for all d. ≥ ≤ ≤ − This follows by combiningf Propositions 5.1 and 5.2, taking note that we must augment the Cechˇ complex C( , M(d)) by “0 M ” to get the lim of Koszul complexes (and also using U → d → the fact that a direct limit of exact sequences is exact).−→ f Corollary 5.4. Let S be a ring. Then a)

S-module of homogeneous polynomials l Γ(P , Pl (d)) f(X ,...,X ) of degree d −→ S O  0 l  is an isomorphism for all d Z, i l ∈ b) H (P , Pl (d)) = (0), for all d Z, 1 i l 1. S O ∈ ≤ ≤ −

Proof. Apply Proposition 5.3 to R = S[X0,...,Xl], n = l + 1, fi = Xi−1, 1 i l +1 and ν ≤ ≤ M = R. Then in fact multiplication by Xj is injective in the ring of truncated polynomials: R/(Xν ,...,Xν ) R 0 j−1 · so Proposition 5.3 applies with t = n. 

l l On the other hand, for any quasi-coherent on P , using the affine covering Ui = (P )X , F S S i 0 i l, we get non-zero alternating cochains Ci ( , ) only for 0 i l. Therefore: ≤ ≤ alt U F ≤ ≤ (5.5) Hi(Pl , ) = (0), i>l, all quasi-coherent . S F F

l l If we look more closely, we can describe the groups H (P , Pl (d)) too. Look first at the S O general situation R, (f1,...,fn), M:  Hn−1( , M)= Hn−1(C ( , M)) U alt U = Cn−1( , M)/δ(Cn−2) f alt U f alt n = M(U1 f Un) res M(U1 Ui,..., Un) ∩···∩ , ∩··· ∩ Xi=1 n f 0 f 0 b = M M . ( fj ) ( fj ) Q , Qj6=i   Xi=1   Thus in the special case: l l H (P , Pl (d)) =elements of degree d in the S[X0,...,Xl]-module S O ∼ l S[X0,...,X ] S[X0,...,X ] l ( Xj ) l ( Xj ) Q , Qj6=i (5.6) Xi=0 ∼=S-module of rational functions c Xα0 Xαl . α0,...,αl 0 · · · l αi≤−1 Xαi=d P l In particular H ( l (d)) = (0) if d > l 1. It is natural to ask to what extent this is a OP − − canonical description of Hl — for instance, if you change coordinates, how do you change the description of an element of Hl by a rational function. The theory of this goes back to Macaulay and his “inverse systems”, cf. Hartshorne [52, Chapter III]. 6. COMPUTING COHOMOLOGY (3): GENERATE F BY “KNOWN” SHEAVES 239

Koszul complexes have many applications to the local theory too. For instance in Chapter V, we presented smooth morphisms locally as:

X = Spec R[X1,...,Xn+r]/(f1,...,fr)

f  Y = Spec R and in Proposition V.3.19, we described the syzygies among the equations fi locally. We can strengthen Proposition V.3.19 as follows: let x X, y = f(x) so that ∈ = [X ,...,X ]p/(f ,...,f ) Ox,X Oy,Y 1 n+r 1 r for some prime ideal p. Then I claim:  K ((f), [X ,...,X ]p) 0 Oy,Y 1 n+r −→ Ox,X −→ is a resolution of as module over [X ,...,X ]p. This follows from the general fact: Ox,X Oy,Y 1 n+r Proposition 5.7. Let R be a regular local ring, M its maximal ideal and let f ,...,f M 1 r ∈ be independent in M/M 2. Then 0 K0((f), R) Kr((f), R) R/(f ,...,f ) 0 −→ −→ ··· −→ −→ 1 r −→ is exact.

Proof. Use Proposition 5.2.  Proposition 5.7 may also be applied to prove that if R is regular, f ,...,f M are inde- 1 n ∈ pendent in M/M 2, then: R Tori (R/(f1,...,fk), R/(fk+1,...,fn)) = (0), i> 0. (cf. discussion of Serre’s theory of intersection multiplicity, V.1.) § 6. Computing cohomology (3): Generate by “known” sheaves F There are usually no projective objects in categories of sheaves, but it is nontheless quite useful to examine resolutions of the type: 0 ··· −→E2 −→ E1 −→ E0 −→ F −→ where, for instance, the are locally free sheaves of -modules (on affine schemes, such Ei OX Ei are projective in the category of quasi-coherent sheaves). Let S be a noetherian ring. We proved in Theorem III.4.3 due to Serre that for every l coherent sheaf on P there is an integer n0 such that (n0) is generated by global sections. F S F This means that for some m0, equivalently, a) there is a surjection m0 Pl (n0) 0 O S −→ F −→ or b) there is a surjection m0 Pl ( n0) 0. O S − −→ F −→ Iterating, we get a resolution of by “known” sheaves: F m1 m0 Pl ( n1) Pl ( n0) 0. · · · −→ O S − −→ O S − −→ F −→ We are now in a position to prove Serre’s Main Theorem in his classic paper [87]: 240 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Theorem 6.1 (Fundamental theorem of F.A.C.). Let S be a noetherian ring, and a l F coherent sheaf on PS. Then i l 1) H (PS, (n)) is a finitely generated S-module for all i 0, n Z. F i l ≥ ∈ 2) n0 such that H (P , (n)) = (0) if i 1, n n0. ∃ S F ≥ ≥ 3) Every is of the form M for some finitely generated graded S[X0,...,Xl]-module M; F 0 l and if = M where M is finitely generated, then n1 such that Mn H (P , ) is F ∃ → S F an isomorphism if n nf1. f ≥ Proof. We prove (1) and (2) by descending induction on i. If i > l, then as we have seen Hi( (n)) = (0), all n (cf. Proposition 4.2). Suppose we know (1) and (2) for all and F F i > i 1. Given , put it in an exact sequence as before: 0 ≥ F n2 0 Pl ( n1) 0. −→ G −→ O S − −→ F −→ For every n Z, this gives us: ∈ n2 0 (n) Pl (n n1) (n) 0, −→ G −→ O S − −→ F −→ hence i0 n2 i0 i0+1 H ( Pl (n n1)) H ( (n)) H ( (n)). O S − −→ F −→ G i0+1 i0 By induction H ( (n)) is finitely generated for all n and (0) for n 0 and by 5, H ( Pl (n G ≫ § O S − n )) is finitely generated for all n and (0) for n 0: therefore the same holds for (n). 1 ≫ F The first half of (3) has been proven in Proposition III.4.4. Suppose = M. Let R = F S[X0,...,Xl] and let R( n ) R( m ) M 0 f − β −→ − α −→ −→ α Mβ M be a presentation of M by twists of the free rank one module R. Taking , this gives a presentation of : F f (6.2) Pl ( nβ) Pl ( mα) 0. O S − −→ O S − −→ F −→ α Mβ M Twisting by n and taking sections, we get a diagram: / / / (6.3) β Rn−nβ Rn−mα Mn 0

L ≈ ≈    0 β Γ( Pl (n nβ)) / α Γ( Pl (n mα)) / H ( (n)) / 0 O S − O S − F with top rowL exact, but the bottom rowL need not be so. But break up (6.2) into short exact sequences

0 / / α Pl ( mα) / / 0 G O S − F L 0 / / β Pl ( nβ) / / 0. H O S − G Choose n1 so that L H1( (n)) = H1( (n)) = (0), n n . G H ≥ 1 Then if n n ≥ 1 0 0 0 0 / H ( (n)) / α H ( Pl (n mα)) / H ( (n)) / 0 G O S − F 0 L 0 0 0 / H ( (n)) / α H ( Pl (n nβ)) / H ( (n)) / 0 H O S − G are exact, hence so is the bottom rowL of (6.3). This proves (3).  6. COMPUTING COHOMOLOGY (3): GENERATE F BY “KNOWN” SHEAVES 241

Corollary 6.4. Let f : X Y be a projective morphism (cf. Definition II.5.8) with Y a → noetherian scheme. Let be a relatively ample invertible sheaf on X. Then for all coherent L F on X: 1) Rif ( ) is coherent on Y . ∗ F 2) n such that Rif ( n) = (0) if i 1, n n . ∃ 0 ∗ F⊗L ≥ ≥ 0 3) n such that all the natural map ∃ 1 f ∗f ( n) n ∗ F⊗L −→F ⊗L is surjective if n n . ≥ 1 Proof. Since Y can be covered by a finite set of affines, to prove all of these it suffices to prove them over some fixed affine U = Spec R Y . Then choose n 1 and (s0,...,sk) −1 n −1 ⊂ k ′ ≥ k ∈ ,Γ(f (U), ) defining a closed immersion i: f (U) ֒ PR. Let X PR be the image of i ′ L ′ k →′ ⊂ ′ and let , be coherent sheaves on PR, (0) outside X and isomorphic on X to f −1(U) and F L ′ n F| −1 . By construction ′ (1) = ( ) . Then applying Serre’s theorem (Theorem 6.1): L|f (U) OX ∼ L 1) Rif ( ) = (Hi(X′, ′)) is coherent. ∗ F |U ∼ F 2) For any fixed m, f Rif ( m+νn) = (Hi(X′, ′ ( ′)m+νn)) ∗ F⊗L |U ∼ F ⊗ L = (Hi(X′, ( ′ ( ′)m)(ν))) ∼ F ⊗ L f = (0), if ν ν0. ≥ f Apply this for m = 0, 1,...,n 1 to get (2) of Corollary 6.4. − 3) For any fixed m, ∗ m+νn 0 ′ ′ ′ m+νn f f ( ) −1 = H (X , ( ) ) ′ ∗ F⊗L |f (U) ∼ F ⊗ L ⊗R OX 0 ′ ′ ′ m = H (X , ( ) (ν)) ′ ∼ F ⊗ L ⊗R OX and this maps onto ′ ( ′)m if ν ν . F ⊗ L ≥ 1 Apply this for m = 0, 1,...,n 1 to get (3) of Corollary 6.4.  − Combining this with Chow’s lemma (Theorem II.6.3) and the Leray spectral sequence (The- orem 3.5), we get:

Theorem 6.5 (Grothendieck’s coherency theorem). Let f : X Y be a proper morphism → with Y a noetherian scheme. If is a coherent -module, then Rif ( ) is a coherent - F OX ∗ F OY module for all i.

Proof. The result being local on Y , we need to prove that if Y = Spec S, then Hi(X, ) is F a finitely generated S-module. Since X is also a noetherian scheme, its closed subsets satisfy the descending chain condition and we may make a “noetherian induction”, i.e., assume the theorem holds for all coherent with Supp $ Supp . Also, if X is the ideal of functions f such G G F I ⊂ O ′ that multiplication by f is 0 in , we may replace X by the closed subscheme X , ′ = / . F OX OX I This has the effect that Supp = X. Now apply Chow’s lemma to construct F X′ π Ð ÐÐ ÐÐ X with π and f π projective ◦ f  Y 242 VII.THECOHOMOLOGYOFCOHERENTSHEAVES where res π : π−1(U ) U is an isomorphism for an open dense U X. Now consider the 0 → 0 0 ⊂ canonical map of sheaves α: π (π∗ ) defined by the collection of maps: F → ∗ F α(U): (U) π∗ (π−1U)= π (π∗)(U). F −→ F ∗ coherent implies π∗ coherent and since π is projective, π (π∗ ) is coherent by Corollary F F ∗ F 6.4. Look at the kernel, cokernel, and image: α 0 / / / π (π∗ ) / / 0. 1 G 8 ∗ F 2 K F GGG pp K G# ppp / 1 F: K O vv OOO vv OOO' 0 v 0 j Since α is an isomorphism on U0, Supp i X U0 $ X. Thus H ( i) are finitely generated K ⊂ \ K S-modules by induction. But now using the long exact sequences:

i−1 i i ∗ H ( 2) / H ( / 1) / H (π∗π ) O K F K F finitely generated  Hi( ) / Hi( ) / Hi( / ) K1 F F K1 it follows readily that if Hi(π π∗ ) is finitely generated, so is Hi( ). But now consider the ∗ F F Leray spectral sequence: Hp(Rqπ (π∗ )) = Epq = En = Hn(π∗ ) ∗ F 2 ⇒ F finitely generated S-module because X′ is projective over| {z Spec S}. q ∗ q ∗ If q 1, then R π∗(π ) U0 = (0); and since π is projective, R π∗(π ) is coherent by Corollary ≥ F | p q ∗ F 6.4. Therefore by noetherian induction, H (R π∗(π )) is finitely generated if q 1. In other F n pq ≥ words, we have a spectral sequence of S-modules with E (all n) and E2 (q 1) finitely p0 ≥  generated. It is a simple lemma that in such a case E2 must be finitely generated too.

7. Computing cohomology (4): Push into a coherent acyclic one F This is a variant on Method (1) taking advantage of what we have learned already — that l at least on PS there are plenty of coherent acyclic sheaves obtained by twists. It is the closest in spirit to the original Italian methods out of which cohomology grew. For simplicity we work n only on Pk (and its closed subschemes) for k an infinite field for the rest of 7. n § Let be coherent on P . Then if F (X0,...,Xn) is a homogeneous polynomial of degree d, F k multiplication by F defines a homomorphism: F (d). F −→ F If d is sufficiently large, Hi(Pn, (d)) = (0), i > 0, and the cohomology of can be deduced k F F from the kernel and cokernel of F as follows: K1 K2 0 / / / (d) / / 0 1 G 9 F 2 K F GGG s K G# sss / 1 F: K L vv LL vv LLL& 0 v 0 7. COMPUTING COHOMOLOGY (4): PUSH F INTO A COHERENT ACYCLIC ONE 243

i i i i+1 (7.1) / H ( 1) / H ( ) / H ( / 1) / H ( 1) / K F F K O K ≈ Hi−1( ), if i 2 K2 ≥ or H0( )/H0( (d)), if i = 1. K2 F This reduces properties of the cohomology of to those of and which have, in general, F K1 K2 lower dimensional support. In fact, one can easily arrange that F is injective, hence = (0) K1 too. In terms of Ass( ), defined in II.3, we can give the following criterion: F § n Proposition 7.2. Given a coherent on P , let Ass( ) = x1,...,xt . Then F : F k F { } F → (d) is injective if and only if F (xa) = 0, 1 a t (more precisely, if xa / V (Xna ), then the F d 6 ≤ ≤ ∈ function F/Xna is not 0 at xa). n d d Proof. Let Ua = P V (Xn ). If (F/X )(xa) = 0, then F/X = 0 on xa Ua. But k \ a na na { }∩ s (U ) with Supp(s) = x U , so (F/Xd )N s = 0 if N 0. Choose N so that ∃ ∈ F a { a}∩ a na · ≫ a (F/Xd )Na s =0 but (F/Xd )Na+1 s = 0. Then na · 6 na · F Na F s =0 in (d)(U ) · Xd · F a  na  so F is not injective. Conversely if F (x ) = 0 for all a and s (U) is not 0, then for some a, a 6 ∈ F s is not 0. But F/Xd is a unit in , so (F/Xd ) s = 0, so F s = 0.  xa ∈ Fxa na Oxa na · xa 6 · xa 6 Assuming then that F is injective, we get

≈ Hi( ) / Hi−1( ) if i 2 F K2 ≥ (7.1∗) ≈ H1( ) / H0( )/ Image H0( (d)). F K2 F It is at this point that we make contact with the Italian methods. Let X Pn be a projective ⊂ k variety, i.e., a reduced and irreducible closed subscheme. Let D be a Cartier divisor on X and X (D) the invertible sheaf of functions “with poles on D” (cf. III.6). Then X (D), extended O n § O by (0) outside X, is a coherent sheaf on P of Pn -modules (cf. Remark after Corollary 3.8) and k O its cohomology may be computed by (7.1∗). In fact, we may do even better and describe its cohomology by induction using only sheaves of the same type (D)! First, some notation — OX Definition 7.3. If X is an irreducible reduced scheme, Y X an irreducible reduced ⊂ subscheme and D is a Cartier divisor on X, then if Y * Supp D, define TrY D to be the Cartier divisor on Y whose local equations at y Y are just the restrictions to Y of its local equations ∈ at y X. Note that: ∈ (Tr D) = (D) . OY Y ∼ OX ⊗OX OY Now take a homogeneous polynomial F endowed with the following properties:

a) X * V (F ) and the effective Cartier divisor H = TrX (V (F )) is reduced and irreducible, b) no component Dj of Supp D is contained in V (F ). It can be shown that such an F exists (in fact, in the affine space of all F ’s, any F outside a proper union of subvarieties will have these properties). Take a second F ′ with the property c) H * V (F ′) 244 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

′ ′ and let H = TrX (V (F )). Start with the exact sequence 0 ( H) 0 −→ OX − −→ OX −→ OH −→ and tensor with (D + H′). We find OX 0 (D + H′ H) (D + H′) (Tr D + Tr H′) 0. −→ OX − −→ OX −→ OH H H −→ But the first sheaf is just (D) via: OX ′ multiply F/F ′ X (D) X (D + H H) O −−−−−−−−−→≈ O − and the second sheaf is just (D)(d) and the whole sequence is the same exact sequence as OX before: multiplication by F (7.4) 0 / X (D) / X (D)(d) / / 0 O O K2 multiplication multiplication ≈ ≈ by F/F ′ by F/F ′   ′ ′ ′ 0 / X (D + H H) / X (D + H ) / H (TrH D + TrH H ) / 0 O − natural O O inclusion

Thus (Tr D + Tr H′). This inductive precedure allowed the Italian School to K2 ≈ OH H H discuss the cohomology in another language without leaving the circle of ideas of linear systems. For instance H1( (D)) = Coker H0( (D + H′)) H0( (Tr D + Tr H′)) OX ∼ OX −→ OH H H space of linear conditions that must be imposed  ′ = on an f R(H) with poles on TrH D + TrH H before . ∼  ∈  it can be extended to an f R(X) with poles in D + H′  ∈  Classically one dealt with the projective space D +H′ of divisors V (s), s H0( (D +H′)), | |X ∈ OX (which is just the set of 1-dimensional subspaces of H0( (D + H′))), and provided dim X 2, OX ≥ we can look instead at: ′ subset of D + H X of divisors TrH ′ | | / TrH D + TrH H H E with H * Supp E | |  

E / TrH E. Then dim H1( (D)) =codimension of Image of Tr , called OX H the “deficiency”of Tr D + H′ . H | |X We go on now to discuss another application of method (4) — to the Hilbert polynomial. First of all, suppose X is any scheme proper over k and is a coherent sheaf on X. Then one F defines: dim X i i χ( )= ( 1) dimk H (X, ) (7.5) F − F Xi=0 = the Euler characteristic of , F which makes sense because of the Hi are finite-dimensional by Grothendieck’s coherency the- orem (Theorem 6.5). The importance of this particular combination of the dim Hi’s is that if 0 0 −→ F1 −→ F2 −→ F3 −→ 7. COMPUTING COHOMOLOGY (4): PUSH F INTO A COHERENT ACYCLIC ONE 245 is a short exact sequence of coherent sheaves, then it follows from the associated long exact cohomology sequence by a simple calculation that: (7.6) χ( )= χ( )+ χ( ). F2 F1 F3 This makes χ particularly easy to compute. In particular, we get:

Theorem 7.7. Let be a coherent sheaf on Pn. Then there exists a polynomial P (t) with F k deg P = dimSupp such that F χ( (ν)) = P (ν), all ν Z. F ∈ In particular, by Theorem 6.1, there exists an ν0 such that 0 dim H ( (ν)) = P (ν), if ν Z, ν ν0. F ∈ ≥ P (t) is called the Hilbert polynomial of . F Proof. This is a geometric form of Part I [76, (6.21)] and the proof is parallel: Let L(X) be a linear form such that L(x ) = 0 for any of the associated points x of . Then as above a 6 a F we get an exact sequence 0 L (1) 0 −→ F −→ F −→ G −→ for some coherent , with G Supp = Supp V (L) G F ∩ hence dimSupp = dimSupp 1. G F − Tensoring by Pn (l) we get exact sequences: O (7.8) 0 (l) (l + 1) (l) 0 −→ F −→ F −→ G −→ for every l Z, hence ∈ χ( (l + 1)) = χ( (l)) + χ( (l)). F F G Now we prove the theorem by induction: if dim Supp = 0, Supp is a finite set, so Supp = ≈ F F G ∅ and (l) (l + 1) for all l by (7.8). Therefore χ( (l)) = χ( ) = constant, a polynomial of F −→ F F F degree 0! In general, if s = dimSupp , then by induction χ( (l)) = Q(l), Q a polynomial of F G degree s 1. Then − χ( (l + 1)) χ( (l)) = Q(l) F − F hence as in Part I [76, (6.21)], χ( (l)) = P (l) for some polynomial P of degree s.  F This leads to the following point of view. Given , one often would like to compute F dim Γ( ): for = (D), this is the typical problem of the additive theory of rational func- k F F OX tions on X. But because of the formula (7.6), it is often easier to compute either χ( ) directly, F or dim Γ( (ν)) for ν 0, hence the Hilbert polynomial, hence χ( ) again. The Italians called k F ≫ F χ( ) the virtual dimension of Γ( ) and viewed it as dim Γ( ) (the main term) followed by an F F F alternating sum of “error terms” dim Hi( ), i 1. Thus one of the main reasons for computing F ≥ the higher cohomology groups is to find how far dim Γ( ) has diverged from χ( ). F F Recall that in Part I [76, (6.28)], we defined the arithmetic genus pa(X) of a projective variety X Pn with a given projective embedding to be ⊂ k p (X) = ( 1)r(P (0) 1) a − − where P (x) = Hilbert polynomial of X, r = dim X. It now follows: 246 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Corollary 7.9 (Zariski-Muhly).

p (X) = dim Hr( ) dim Hr−1( )+ + ( 1)r−1 dim H1( ) a OX − OX · · · − OX hence pa(X) is independent of the projective embedding of X.

Proof. By Theorem 7.7, P (0) = χ( ) so the formula follows using dim H0( ) = 1 OX OX (Corollary II.6.10). 

I’d like to give one somewhat deeper result analyzing the “point” vis-a-vis tensoring with (ν) at which the higher cohomology vanishes; and which shows how the vanishing of higher O cohomology groups alone can imply the existence of sections:

Theorem 7.10 (Generalized lemma of Castelnuovo and syzygy theorem of Hilbert). Let n F be a coherent sheaf on Pk . Then the following are equivalent: i) Hi( ( i)) = (0), 1 i n, F − ≤ ≤ ii) Hi( (m)) = (0), if m + i 0, i 1, F ≥ ≥ iii) there exists a “Spencer resolution”:

rn rn−1 r1 r0 0 Pn ( n) Pn ( n + 1) Pn ( 1) n 0. −→ O − −→ O − −→ · · · −→ O − −→ OP −→ F −→ If these hold, then the canonical map

0 0 0 H ( ) H ( Pn (l)) H ( (l)) F ⊗ O −→ F is surjective, l 0. ≥ Proof. We use induction on n: for n = 0, Pn = Spec k, = kn and the result is clear. k F So we may suppose we know the result on Pn−1. The implication (ii) = (i) is obvious and k ⇒ (iii) = (ii) follows easily from what we know of the cohomology off Pn (l), by splitting the ⇒ O resolution up into a set of short exact sequences:

r r 0 / Pn ( n) n / Pn ( n + 1) n−1 / n−1 / 0 O − O − F ______r 0 / / Pn ( 1) 1 / / 0 F2 O − F1 r0 0 / 1 / n / / 0. F OP F So assume (i). Choose a linear form L(X) such that L(x ) = 0 for any associated points x of a 6 a , getting sequences F ⊗L 0 (l 1) (l) (l) 0, all l Z −→ F − −→ F −→ G −→ ∈ where is a coherent sheaf on the hyperplane H = V (L). In fact is not only supported on H G G but is annihilated by the local equations L/Xj of H: hence is a sheaf of H -modules. Since n−1 G O H ∼= Pk , we are in a position to apply our induction hypothesis. The cohomology sequences give: Hi( ( i)) Hi( ( i)) Hi+1( ( i 1)) . −→ F − −→ G − −→ F − − −→ Applying this for i 1, we find that satisfies (i) also; applying it for i = 0, we find that ≥ G H0( ) H0( ) is surjective. Therefore by the theorem for , F → G G H0( ) H0( (l)) H0( (l)) G ⊗ OH −→ G 7. COMPUTING COHOMOLOGY (4): PUSH F INTO A COHERENT ACYCLIC ONE 247 is surjective. Consider the maps:

γ 0 0 0 H ( ) H ( Pn (l)) / H ( (l)) F ⊗ O F α β   H0( ) H0( (l)) / H0( (l)). G ⊗ OH G We prove next that γ is surjective for all l 0. By Proposition III.1.8, H0( (l)) is the space ≥ OH of homogeneous polynomials of degree l in the homogeneous coordinates on H: therefore each 0 is obtained by restricting to H a polynomial P (X0,...,Xn) of degree l and H ( Pn (l)) 0 0 O → H ( H (l)) is surjective. Therefore α is surjective. It follows that if s H ( (l)), then β(s)= O 0 0 0 ∈ F 0 uq vq, uq H ( ), vq H ( H (l)); hence lifgint uq to uq H ( ), vq to vq H ( Pn (l)), ⊗ ∈ G ∈ O 0 ∈ F ∈ O s uq vq lies in Ker β. But Ker β = Image of H ( (l 1)) under the map L: (l 1) P− ⊗ 0 F − 0 0 ⊗ F − → (l) and by induction on l, anything in H ( (l 1)) is in H ( ) H ( Pn (l 1)). Thus F P F − F ⊗ O − ′ ′ ′ 0 ′ 0 s u v = u v L, u H ( ), v H ( Pn (l 1)). − q ⊗ q q ⊗ q ⊗ q ∈ F q ∈ O −   Thus X X ′ ′ ′ 0 s = u v + u (v L), where v L H ( Pn (l)) q ⊗ q q ⊗ q ⊗ q ⊗ ∈ O as required. X X 0 n Next, note that this implies that is generated by H ( ). In fact, if x P , x / V (Xj), F F ∈ k ∈ and s , then Xl s (l) . For l 0, (l) is generated by H0( (l)). So ∈ Fx j · ∈ F x ≫ F F l 0 X s H ( (l)) ( Pn ) j · ∈ F · O x

0 0 H ( ) H ( Pn (l)) ( Pn ) F ⊗ O · O x i.e.,
l 0 0 X s = u v a , u H ( ), v H ( Pn (l)), a ( Pn ) j · q ⊗ q · q q ∈ F q ∈ O q ∈ O x hence X vq s = uq l aq . ⊗ Xj · X ∈(OPn )x We can now begin to construct a Spencer resolution:| {z let} s ,...,s be a basis of H0( ) and 1 r0 F define

r0 n 0 OP −→ F −→ by r0 (a ,...,a ) a s . 1 r0 7−→ q q q=0 X If is the kernel, then from the cohomology sequence it follows immediately that (1) satisfies F1 F1 Condition (i) of the theorem. Hence (1) is also generated by its sections and choosing a basis F1 t ,...,t of H0( (1)), we get the next step: 1 r1 F1 r1 n (1) 0 OP −→ F1 −→ r1 (a ,...,a ) a t 1 r1 7−→ q q q=1 X 248 VII.THECOHOMOLOGYOFCOHERENTSHEAVES hence r1 r0 Pn ( 1) / Pn / / 0. O − N < O F NN xx NNN& xx o7 F1 H ooo HH ooo HH# 0 o 0 Continuing in this way, we derive the whole Spencer resolution. It remains to check that after the last step: rn 0 Pn ( n) 0, −→ Fn+1 −→ O − −→ Fn −→ the sheaf is actually (0)! To prove this, we compute Hi( (l)) for 0 i n, 0 l n, Fn+1 Fn+1 ≤ ≤ ≤ ≤ using all the cohomology sequences ( ) associated to: ∗ m rm 0 Pn ( m) 0. −→ Fm+1 −→ O − −→ Fm −→ We get:

0 0 rn 0 a) H ( n+1(l)) = (0) (using injectivity of H ( ) H ( n(n)) when l = n) F by (∗)n F → F b) = (0) if l = n (using surjectivity of by (∗)n 0 rn 0  H ( ) H ( n(n))) 1  0 O → F H ( n+1)  ∼= H ( n(l)) = (0) if labelian category, define free abelian group on elements [X], one for each isomorphism class of objects in C, modulo relations 0   K (C)= [X2] = [X1] + [X3] for each short exact sequence:    0 X1 X2 X3 0   → → → →   in C.      7. COMPUTING COHOMOLOGY (4): PUSH F INTO A COHERENT ACYCLIC ONE 249

If X is any noetherian scheme, define K (X)= K0(Category of coherent sheaves of -modules on X) 0 OX K0(X)= K0(Category of locally free finite rank sheaves of -modules). OX Prove: a) a natural map K0(X) K (X). ∃ → 0 b) K0(X) is a contravariant functor in X, i.e., morphism f : X Y , we get ∀ → f ∗ : K0(Y ) K0(X) with the usual properties. → c) K0(X) is a commutative ring via [ ] [ ] = [ ] E1 · E2 E1 ⊗OX E2 0 and K0(X) is a K (X)-module via [ ] [ ] = [ ]. E · F E ⊗OX F d) K (X) is a covariant functor for proper morphisms f : X Y via 0 → ∞ f ([ ]) = ( 1)n[Rnf ]. ∗ F − ∗F n=0 X (2) Return to the case where k is an infinite field. a) Using the Spencer resolution, show that 0 n n K (P ) K0(P ) k −→ k is surjective and that they are both generated by the sheaves [ Pn (l)], l Z. O ∈ Hint: On any scheme X, if 0 0 −→ F −→ E1 −→ E0 −→ is exact, locally free and finitely generated, then is locally free and finitely Ei F generated, and locally on X, the sequence splits, i.e., 1 = 0.  E ∼ F⊕E b) Consider the Koszul complex K (X0,...,Xn; k[X0,...,Xn]). Take ] and hence 0 n show that [ Pn (l)] K (P ) satisfy O ∈ k n+1 ν n + 1 ( ) ( 1) [ Pn (ν + ν0)] = 0, ν0 Z, ∗ − ν O ∀ ∈ ν=0 X   0 n hence K (Pk ) is generated by [ Pn (ν)] for any set of ν’s of the form ν0 ν ν0+n. Oν ≤ ≤ Show that [ Lν ], 0 ν n, L = a fixed linear space of dimension ν, generate n O ≤ ≤ K0(Pk ). c) Let group of rational polynomials P (t) of degree n S = ≤ n taking integer values at integers   free abelian group on the polynomials = t .  P (t)= , 0 ν n  ν ν ≤ ≤   Prove that  [ ] Hilbert polynomial of F 7−→ F defines n K0(P ) Sn. k −→ 0 n ∼ n d) Combining (a), (b) and (c), show that K (P ) K0(P ). k −→ k 250 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

e) Using the result of Part I [76, 6C] show that if Z Pn is any subvariety, of § ⊂ k dimension r and g = p (Z H H ) ν a · 1 · · · r−ν = arithmetic genus of the ν-dimensional linear section of Z, (1 ν r) ≤ ≤ n d = deg Z, then in K0(Pk ):

[ ]= d [ n ] + (1 d g )[ r−1 ] + (g + g )[ r−2 ]+ OZ · OL − − 1 OL 1 2 OL n + ( 1) (g + g )[ 0 ]. · · · − r−1 r OL n (3) Because of (2), (d), K0(P ) inherits a ring structure. Using the sheaves or i defined in k T 2 as one of the applications of the “easy lemma of the double complex” (Lemma 2.4), § show that this ring structure is given by n ( ) [ ] [ ]= ( 1)i[ or ( , )]. ∗ F1 · F2 − T i F1 F2 Xi=0 In particular, check that or = (0) if i>n. (In fact, on any regular scheme X, it can T i be shown that or i = (0), i> dim X; and that ( ) defines a ring structure in K0(X)). T ∗ n Next apply this with 1 = X , 2 = X , X1, X2 subvarieties of P intersecting F O 1 F O 2 k properly and transversely at generic points of the components W ,...,W of X X 1 ν 1 ∩ 2 (cf. Part I [76, 5B]). Show by Ex. 2, 5D2??? , that if i 1, § § ≥ dim Supp( or ( , )) < dim X X . T i OX1 OX2 1 ∩ 2 Combining this with the results of (2), show Bezout’s Theorem: ν (deg X ) (deg X )= deg W . 1 · 2 i Xi=1 Hint: Show that [ r ] [ s ] = [ r+s−n ]. Show next that if i 1 OL · OL OL ≥ [ or ( , )] = combination of [ t ] for t< dim(X X ). T i OX1 OX2 OL 1 ∩ 2 8. Serre’s criterion for ampleness This section gives a cohomological criterion equivalent to ampleness for an invertible sheaf introduced in III.5. We apply it later to questions of positivity of intersections, formulated in § terms of the Euler characteristic. Theorem 8.1. Let X be a scheme over a noetherian ring A, embedded as a closed subscheme in a projective space over A, with canonical sheaf (1). Let be coherent on X. Then for OX F all i 0, Hi(X, ) is a finite A-module, and there exists an integer n such that for n n we ≥ F 0 ≥ 0 have Hi(X, (n)) = 0 for all i 1. F ≥ Proof. We have already seen in Corollary 3.8 that under a closed embedding X ֒ Pr the → A cohomology of over X is the same as the cohomology of viewed as a sheaf over projective F F r space. Consequently we may assume without loss of generality that X = PA, which we denote by P. i The explicit computation of cohomology H (P, P(n)) in Corollary 5.4 and (5.6) shows that O the theorem is true when = P(n) for all integers n. Now let be an arbitrary coherent F O F sheaf on P. We can represent in a short exact sequence (cf. 6) F § 0 0 −→ G −→ E −→ F −→ 8.SERRE’SCRITERIONFORAMPLENESS 251 where is a finite direct sum of sheaves P(d) for appropriate positive integers d, and is E O G defined to be the kernel of . We use the cohomology sequence, and write the cohomology E → F groups without P for simplicity:

Hi( ) Hi( ) Hi+1( ) −→ E −→ F −→ G −→ We apply descending induction. For i > r we have Hi( ) = 0 because P can be covered by F r + 1 open affine subsets, and the Cechˇ complex is 0 with respect to this covering in dimension r +1 (cf. (5.5)). If, by induction, Hi+1( ) is finite over A, then the finiteness of Hi( ) implies ≥ G E that Hi( ) is finite. F Furthermore, twisting by n, that is, taking tensor products with P(n), is an exact functor, O so the short exact sequence tensored with P(n) remains exact. This gives rise to the cohomology O exact sequence: Hi( (n)) Hi( (n)) Hi+1( (n)) → E → F → G → Again by induction, Hi+1( (n)) = 0 for n sufficiently large, and Hi( (n)) = 0 because of the G E i special nature of as a direct sum of sheaves P(d). This implies that H ( (n)) = 0 for n E O F sufficiently large, and concludes the proof of the theorem. 

Theorem 8.2 (Serre’s criterion). Let X be a scheme, proper over a noetherian ring A. Let be an invertible sheaf on X. Then is ample if and only if the following condition holds: For L L any coherent sheaf on X there is an integer n such that for all n n we have F 0 ≥ 0 Hi(X, n) = 0 for all i 1. F⊗L ≥ Proof. Suppose that is ample, so d is very ample for some d. We have seen (cf. Theorem L L III.5.4 and II.6) that X is projective over A. We apply Theorem 8.1 to the tensor products § , ,..., d−1 F F⊗L F⊗L and the very ample sheaf d = (1) to conclude the proof that the cohomology groups vanish L OX for i 1. ≥ Conversely, assume the condition on the cohomology groups. We want to prove that is L ample. It suffices to prove that for any coherent sheaf the tensor product n is generated F F⊗L by global sections for n sufficiently large. (cf. Definition III.5.1) By Definition III.2.1 it will suffice to prove that for every closed point P , the fibre k(P ) is generated by global sections. F⊗ Let be the ideal sheaf defining the closed point P as a closed subscheme. We have an exact IP sequence

0 P k(P ) 0. −→ I F −→F −→F⊗ −→ Since n is locally free, tensoring with n preserves exactness, and yields the exact sequence L L n n n 0 P k(P ) 0 −→ I F⊗L −→F ⊗L −→F ⊗ ⊗L −→ whence the cohomology exact sequence

H0( n) H0( k(P ) n) 0 F⊗L −→ F ⊗ ⊗L −→ because H1( n) = 0 by hypothesis. This proves that the fibre at P of n is generated IP F⊗L F⊗L by global sections, and concludes the proof of the theorem.  252 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

9. Functorial properties of ampleness This section gives a number of conditions relating ampleness on a scheme with ampleness on certain subschemes.

Proposition 9.1. Let X be a scheme of finite type over a noetherian ring and an invertible L sheaf, ample on X. For every closed subscheme Y , the restriction = is ample L|Y L⊗OX OY on Y .

Proof. Taking a power of we may assume without loss of generality that is very ample L L (cf. Theorem III.5.4), so (1) in a projective embedding of X. Then = (1) in that OX OX |Y OY same embedding. Thus the proposition is immediate. 

Let X be a scheme. For each open subset U we let il(U) be the ideal of nilpotnet elements N in (U). Then il is a sheaf of ideals, and the quotient sheaf / il defines a closed OX N OX N subscheme called the reduced scheme X . Its sheaf of rings has no nilpotent elements. If is red F a sheaf of -modules, then we let OX = / where = il . Fred F IF I N Alternatively, we can say that is the restriction of to X . Fred F red Proposition 9.2. Let X be a scheme, proper over a noetherian ring. Let be an invertible L sheaf on X. Then is ample on X if and only if is ample on X . L Lred red Proof. By Proposition 9.1, it suffices to prove one side of the equivalence, namely: if Lred is ample then is ample. Since X is noetherian, there exists an integer r such that if = il L N N is the sheaf of nilpotent elements, then r = 0. Hence we get a finite filtration N 2 r = 0. F⊃NF⊃N F⊃···⊃N F For each i = 1,...,r 1 we have the exact sequence − 0 i i−1 i−1 / i 0 −→ N F −→N F −→N F N F −→ whence the exact cohomology sequence Hp(X, i n) Hp(X, i−1 n) Hp(X, ( i−1 / i ) n). N F⊗L −→ N F⊗L −→ N F N F ⊗L For each i, i−1 / i is a coherent / -module, and thus is a sheaf on X . By hypothesis, N F N F OX N red and Theorem 8.2, we know that Hp(X, ( i−1 / i ) n) = 0 N F N F ⊗L for all n sufficiently large and all p 1. But i = 0 for i r. We use descending induction ≥ N F ≥ on i. We have Hp(X, i n) = 0 for all p> 0, i r, N F⊗L ≥ and n sufficiently large. Hence inductively, Hp(X, i n) = 0 for all p> 0 N F⊗L implies that Hp(X, i−1 n) = 0 for all p> 0 and n sufficiently large. This concludes the N F⊗L proof. 

Proposition 9.3. Let X be a proper scheme over a noetherian ring. Let be an invertible L sheaf on X. Then is ample if and only if is ample on each irreducible component X of L L|Xi i X. 9.FUNCTORIALPROPERTIESOFAMPLENESS 253

Proof. Since an irreducible component is a closed subscheme of X, Proposition 9.1 shows that it suffices here to prove one implication. So assume that is ample for all i. Let L|Xi Ii be the coherent sheaf of ideals defining Xi, and say i = 1,...,r. We use induction on r. We consider the exact sequence

0 / 0, −→ I1F −→ F −→ F I1F −→ giving rise to the exact cohomology sequence

Hp(X, n) Hp(X, n) Hp(X, ( / ) n). I1F⊗L −→ F⊗L −→ F I1F ⊗L Since is ample by hypothesis, it follows that L|X1 Hp(X, ( / ) n) = 0 F I1F ⊗L for all p > 0 and n n . Furthermore, is a sheaf with support in X X , so by ≥ 0 I1F 2 ∪···∪ r induction we have Hp(X, n) = 0 I1F⊗L for all p> 0 and n n . The exact sequence then gives ≥ 0 Hp(X, n) = 0 F⊗L for all p> 0 and n n , thus concluding the proof.  ≥ 0 Proposition 9.4. Let f : X Y be a finite (cf. Definition II.6.6) surjective morphism of → proper schemes over a noetherian ring. Let be an invertible sheaf on Y . Then is ample if L L and only if f ∗ is ample on X. L Proof. First note that f is affine (cf. Proposition-Definition I.7.3 and Definition II.6.6). Let be a coherent sheaf on X, so f is coherent on Y . For p 0 we get: F ∗F ≥ Hp(Y,f ( ) n)= Hp(Y,f ( (f ∗ )n)) ∗ F ⊗L ∗ F ⊗ L by the projection formula = Hp(X, (f ∗ )n) F ⊗ L by Proposition 3.7 9. If is ample, then the left hand side is 0 for n n and p > 0, so this L ≥ 0 proves that f ∗ is ample on X. L Conversely, assume f ∗ ample on X. We show that for any coherent -module , one has L OY G Hp(Y, n) = 0, p> 0 and n 0 G⊗L ∀ ≫ by noetherian induction on Supp( ). G By Propositions 9.2 and 9.3, we may assume X and Y to be integral. We follow Hartshorne [50, 4, Lemma 4.5, pp. 25–27] and first prove: § 9 Let f : X → Y be a morphism, F an OX -module and L an OY -module. The identity homomorphism ∗ ∗ ∗ f L → f L induces an OY -homomorphism L → f∗f L. Tensoring this with f∗F over OY and composing the result with a canonical homomorphism, one gets a canonical homomorphism

∗ ∗ f∗F ⊗OY L −→ f∗F ⊗OY f∗f L −→ f∗(F ⊗OX f L).

This can be easily shown to be an isomophism if L is a locally free OY -module of finite rank, giving rise to the “projection formula” ∼ ∗ f∗F ⊗OY L −→ f∗(F ⊗OX f L). 254 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Lemma 9.5. Let f : X Y be a finite surjective morphism of degree m of noetherian integral → schemes X and Y . Then for every coherent -module on Y , there exist a coherent - OY G OX module and an -homomorphism ξ : f ⊕m that is a generic isomorphism (i.e., ξ is F OY ∗F → G an isomorphism in a neighborhood of the generic point of Y ).

Proof of Lemma 9.5. By assumption, the function field R(X) is an algebraic extension of R(Y ) of degree m. Let U = Spec A X be an affine open set. Since R(X) is the quotient field ⊂ of A, we can choose s1,...,sm A such that s1,...,sm is a basis of R(X) as a vector space ∈ m { } over R(Y ). The X -submodule = X si of the constant X -module R(X) is coherent. O H i=1 O O Since s ,...,s H0(X, )= H0(Y,f ), we have an -homomorphism 1 m ∈ H P∗H OY m η : ⊕m = e f , e s (i = 1,...,m), OY OY i −→ ∗H i 7−→ i Xi=1 which is a generic isomorphism by the choice of s ,...,s . If a coherent -module is given, 1 m OY G η induces an -homomorphism OY ξ : ′ = om (f , ) om ( ⊕m, )= ⊕m, H H OY ∗H G −→ H OY OY G G which is a generic isomorphism. Since ′ is an f -module through the first factor of om H ∗OX H and f is finite, we have ′ = f for a coherent -module .  H ∗F OX F To continue the proof of Proposition 9.4, let be a coherent -module . Let be G OY G F a coherent -module as in Lemma 9.5, and let and be the kernel and cokernel of the OX K C -homomorphism ξ : f ( ) ⊕m. We have exact sequences OY ∗ F → G 0 f Image(ξ) 0 −→ K −→ ∗F −→ −→ 0 Image(ξ) ⊕m 0. −→ −→ G −→ C −→ and are coherent Y -modules, and Supp( ) $ Y and Supp( ) $ Y , since ξ is a generic K C O K C isomorphism. Hence by the induction hypothesis, we have

Hp(Y, n)= Hp(Y, n) = 0, p> 0 and n 0. K⊗L C⊗L ∀ ≫ By the cohomology long exact sequence, we have

p n ∼ p n ∼ p n ⊕m H (Y, (f∗ ) ) / H (Y, Image(ξ) ) / H (Y, ) F ⊗L ⊗L G⊗L Hp(X, (f ∗ )n) F ⊗ L for all p> 0 and n 0, the equality on the left hand side being again by the projection formula. ≫ Hp(X, (f ∗ )n) = 0 for all p> 0 and n 0, since f ∗ is assumed to be ample. Hence F ⊗ L ≫ L Hp(Y, n) = 0, p> 0 and n 0. G⊗L ∀ ≫ 

Proposition 9.6. Let X be a proper scheme over a noetherian ring A. Let be an invertible L sheaf on X, and assume that is generated by its global sections. Suppose that for every closed L integral curve C in X the restriction is ample. Then is ample on X. L|C L For the proof we need the following result given in Proposition VIII.1.7: Let C′ be a geometrically irreducible curve, proper and smooth over a field k. An invertible sheaf ′ on C′ is ample if and only if deg ′ > 0. L L 10. THE EULER CHARACTERISTIC 255

Proof. By Propositions 9.2 and 9.3 we may assume without loss of generality that X is integral. Since is generated by global sections, a finite number of these define a morphism L ϕ: X Pn −→ A ∗ such that = ϕ P(1). Then ϕ is a finite morphism. For otherwise, by Corollary V.6.5 some L O fiber of ϕ contains a closed integral curve C. Let ϕ(C)= P , a closed point of Pn . Let f : C′ C A → be a morphism obtained as follows: C′ is the normalization of C in a composite field k(P )R(C) obtained as a quotient of

k(P ) k R(C), ⊗ (P ) where k(P ) is the algebraic closure of k(P ). (C′ is regular by Proposition V.5.11, hence is proper ′ ∗ and smooth over k(P ).) Since C is ample, so is = f by Proposition 9.4. But then the ′ L| ′ ∗L ∗L∗ deg > 0 by Proposition VIII.1.7, while = f = f ϕ P(1). This contradicts the fact L L L O that ϕ(C) = P is a point. Hence ϕ is finite. Propositions 9.2, 9.3 and 9.4 now conclude the proof. 

10. The Euler characteristic Throughout this section, we let A be a local artinian ring. We let

X Spec(A) −→ be a projective morphism. We let be a coherent sheaf on X. F By Theorem 8.1, the cohomology groups Hi(X, ) are finite A-modules, and since A is F artinian, they have finite length. By (5.5) and Corollary 3.8, we also have Hi(X, ) =0 for i F sufficiently large. We define the Euler characteristic ∞ χ (X, )= χ ( )= ( 1)i length Hi(X, ). A F A F − F Xi=0 This is a generalization of what we introduced in (7.5) in the case A = k a field. As a general- ization of Theorem 7.7, we have:

Proposition 10.1. Let 0 ′ ′′ 0 −→ F −→ F −→ F −→ be a short exact sequence of coherent sheaves on X. Then

χ ( )= χ ( ′)+ χ ( ′′). A F A F A F Proof. This is immediate from the exact cohomology sequence

Hp(X, ′) Hp(X, ) Hp(X, ′′) −→ F −→ F −→ F −→ which has 0’s for p< 0 and p sufficiently large. cf. Lang [65, Chapter IV]. 

We now compute this Euler characteristic in an important special case.

r Proposition 10.2. Suppose P = PA. Then n + r (n + r)(n + r 1) (n + 1) χA( P(n)) = = − · · · for all n Z. O r r! ∈   256 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Proof. For n> 0, we can apply Corollary 5.4 to conclude that 0 χA( P(n)) = length H (P, P(n)), O O which is the number of monomials in T0,...,Tr of degree n, and is therefore equal to the binomial coefficient as stated. If n r 1, then similarly by (5.6), we have ≤− − r r χk( P(n)) = ( 1) length H (P, P(n)). O − O From the explicit determination of the cohomology in (5.6) if we put n = r d then the length r − − of H (P, P(n)) over A is equal to the number of r-tuples (q0,...,qr) of integers qj > 0 such O that q = r + d, which is equal to the number of r-tuples (q′ ,...,q′ ) of integers 0 such j 0 r ≥ that q′ = d 1. This is equal to P j − P d 1+ r n + r − = ( 1)r . r − r     i Finally, let r n 0. Then H (P, P(n)) = 0 for all i > 0 once more by Corollary 5.4 and − ≤ ≤ O (5.6). Also the binomial coefficient is 0. This proves the proposition.  Starting with the explicit case of projective space as in Proposition 10.2, we can now derive a general result, which is a generalization of Theorem 7.7 in the case A = k a field. Theorem 10.3. Let A be a local artinian ring. Let X be a projective scheme over Y = Spec(A). Let be an invertible sheaf on X, very ample over Y , and let be a coherent sheaf L F on X. Put (n)= n for n Z. F F⊗L ∈ i) There exists a unique polynomial P (T ) Q[T ] such that ∈ χA( (n)) = P (n) for all n Z. F ∈ ii) For n sufficiently large, χ ( (n)) = length H0(X, (n)). A F F iii) The leading coefficient of P (T ) is 0. ≥ Proof. By Theorem 8.1 we know that Hi( (n))=0 for i 1 and n large. F ≥ Hence χ ( (n)) is the length of H0( (n)) as asserted in (ii). In particular, χ ( (n)) is 0 A F F A F ≥ for n large, so the leading coefficient of P (T ) is 0 if such polynomial exists. Its uniqueness is ≥ obvious. To show existence, we reduce to the case of Proposition 10.2 by Jordan-H¨older techniques. Suppose we have an exact sequence 0 ′ ′′ 0. −→ F −→ F −→ F −→ Taking the tensor product with preserves exactness. It follows immediately that if (i) is true L for ′ and ′′, then (i) is true for . Let m be the maximal ideal of A. Then there is a finite F F F filtration m m2 ms = 0. F ⊃ F ⊃ F ⊃···⊃ F By the above remark, we are reduced to proving (i) when m = 0, because m annihilates each F factor sheaf mj /mj+1 . F F Suppose now that m = 0. Then can be viewed as a sheaf on the fibre X , where y is the F F y closed point of Y = Spec(A). The restriction of to X is ample by Proposition 9.1, and the L y cohomology groups of a sheaf on a closed subscheme are the same as those of that same sheaf viewed on the whole scheme. The twisting operation also commutes with passing to a closed 11. INTERSECTION NUMBERS 257 subscheme. This reduces the proof that χ ( (n)) is a polynomial to the case when A is a field A F k. Thus we are done by Theorem 7.7. 

When A = k is a field, the length is merely the dimension over k. For any coherent sheaf F on X we have by definition

d χ( )= ( 1)i dim Hi(X, ), F − k F Xi=0 where d = dim X. By Theorem 7.7, we know that

P (n)= χ( (n)) F is a polynomial of degree e where e = dimSupp . F

11. Intersection numbers Throughout this section we let X be a proper scheme over a field k. We let χ = χk.

Theorem 11.1 (Snapper). Let ,..., be invertible sheaves on X and let be a coherent L1 Lr F sheaf. Let d = dim Supp( ). Then there exists a polynomial P with rational coefficients, in r F variables, such that for all integers n1,...,nr we have

P (n ,...,n )= χ( n1 nr ). 1 r L1 ⊗···⊗Lr ⊗ F This polynomial P has total degree d. ≤ Proof. Suppose first that ,..., are very ample. Then the assertion follows by induc- L1 Lr tion on r and Theorem 7.7 (generalized in Theorem 10.3). Suppose X projective. Then there exists a very ample invertible sheaf such that ,..., are very ample (take any very L0 L0L1 L0Lr ample sheaf, raise it to a sufficiently high power and use Theorem III.5.10). Let

Q(n ,n ,...,n )= χ ( n0 ( )n1 ( )nr ) . 0 1 r L0 ⊗ L0L1 ⊗···⊗ L0Lr ⊗ F Then P (n ,...,n )= Q( n n ,n ,...,n ) 1 r − 1 −···− r 1 r and the theorem follows. If X is not projective, the proof is more complicated. We follow Kleiman [62]. The proof proceeds by induction on d = dim Supp( ). Since the assertion is trivial if d = 1, i.e., = (0), F − F we assume d 0. ≥ Replacing X by the closed subscheme pec ( / Ann( )) defined by the annihilator ideal S X OX F Ann( ), we may assume Supp( )= X. The induction hypothesis then means that the assertion F F is true for any coherent X -module with Supp( ) $ X, i.e., for torsion X -modules . O F F O F Let K be the abelian category of coherent -modules, and let K′ Ob(K) consist of those OX ⊂ ’s for which the assertion holds. K′ is obviously exact in the sense of Definition II.6.11. By F d´evissage (Theorem II.6.12), it suffices to show that K′ for any closed integral subscheme OY ∈ Y of X. In view of the induction hypothesis, we may assume Y = X, that is, X itself is integral. Then by Proposition III.6.2, there exists a Cartier divisor D on X such that = (D) and L1 OX that the intersections = ( D) as well as = (D) taken inside the function I OX − ∩ OX J OX ∩ OX field R(X) are coherent X -ideals not equal to X . Obviously, we have = X(D)= 1. O O J I⊗O I⊗L 258 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Tensoring the exact sequence 0 / 0 (resp. 0 / 0) →I→OX → OX I → →J →OX → OX J → with n1 (resp. n1−1), we have exact sequences L1 L1 n1 n1 n1 0 / / / ( X / ) / 0 I⊗L1 L1 L1 ⊗ O I

0 / J n1−1 / n1−1 / n1−1 ( / ) / 0. ⊗L1 L1 L1 ⊗ OX J Thus tensoring both sequences with n2 nr and taking the Euler characteristic, we have L2 ⊗···⊗Lr χ( n1 n2 nr ) χ( n1−1 n2 nr ) L1 ⊗L2 ⊗···⊗Lr − L1 ⊗L2 ⊗···⊗Lr =χ( n1 n2 nr ( / )) χ( n1−1 n2 nr ( / )). L1 ⊗L2 ⊗···⊗Lr ⊗ OX I − L1 ⊗L2 ⊗···⊗Lr ⊗ OX J The right hand side is a polynomial with rational coefficients in n1,...,nr of total degree < d since / and / are torsion -modules. Hence we are done, since χ( n2 nr ) OX I OX J OX L2 ⊗···⊗Lr is a polynomial in n ,...,n of total degree d by induction on r.  2 r ≤ We recall here the following result on integral valued polynomials.

Lemma 11.2. Let P (x1,...,xr) Q[x1,...,xr]= Q[x] be a polynomial with rational coeffi- ∈ cients, and integral valued on Zr. Then P admits an expression

x1 + i1 xr + ir P (x1,...,xr)= a(i1,...,ir) i1 · · · ir X     where a(i1,...,ir) Z, the sum is taken for i1,...,ir 0, ∈ ≥ x + i (x + i)(x + i 1) (x + 1) = − · · · if i> 0, i i!   and the binomial coefficients is 1 if i = 0, and 0 if i< 0. Proof. This is proved first for one variable by induction, and then for several variables by induction again. We leave this to the reader. 

Lemma 11.3. The coefficient of n n in χ( n1 nr ) is an integer. 1 · · · r L1 ⊗···⊗Lr ⊗ F Proof. Immediate from Lemma 11.2.  Let us define the intersection symbol: ( . . . . ) = coefficient of n n in the polynomial L1 Lr F 1 · · · r χ( n1 nr ). L1 ⊗···⊗Lr ⊗ F Lemma 11.4. (i) The function ( . . . . ) is multilinear in ,..., . L1 Lr F L1 Lr (ii) If c is the coefficient of nr in χ( n ), then L ⊗ F ( . . . . . )= r!c ( is repeated r times.) L L L F L Proof. Let , be invertible sheaves. Then L M ( n m n2 nr )= ann n + bmn n + L ⊗ M ⊗L2 ⊗···⊗Lr ⊗ F 2 · · · r 2 · · · r · · · with rational coefficients a, b. Putting n = 0 and m = 0 shows that a = ( . . . . . ) and b = ( . . . . . ). L L2 Lr F M L2 Lr F Let m = n = n1. It follows that (( ). . . . . ) = ( . . . . . ) + ( . . . . . ). L ⊗ M L2 Lr F L L2 Lr F M L2 Lr F Similarly, ( −1. . . . . )= ( . . . . . ). This proves the first assertion. L L2 Lr F − L L2 Lr F 12.THECRITERIONOFNAKAI-MOISHEZON 259

As to the second, let P (n) = ( n. ) and L F Q(n ,...,n ) = ( n1 . . . nr . ). 1 r L1 Lr F Let ∂ be the derivative, and ∂1,...,∂r be the partial derivatives. Then the second assertion follows from the relation ∂ ∂ Q(0,..., 0) = ∂rP (0). 1 · · · r 

The next lemma gives the additivity as a function of , in the sense of the Grothendieck F group.

Lemma 11.5. Let 0 ′ ′′ 0 −→ F −→ F −→ F −→ be an exact sequence of coherent sheaves. Then

( . . . . ) = ( . . . . ′) + ( . . . . ′′). L1 Lr F L1 Lr F L1 Lr F Proof. Immediate since the Euler characteristic satisfies the same type of relation. 

12. The criterion of Nakai-Moishezon Let X be a proper scheme over a field k. Let Y be a closed subscheme of X. Then Y is defined by a coherent sheaf of ideals , and IY = / OY OX IY is its structure sheaf. Let D1,...,Dr be divisors on X, by which we always mean Cartier divisors, so they correspond to invertible sheaves = (D ),..., = (D ). Suppose Y L1 OX 1 Lr OX r has dimension r. We define the intersection number (D ...D .Y ) = coefficient of n n in the polynomial 1 r 1 · · · r χ( n1 nr ). L1 ⊗···⊗Lr ⊗ OY (Dr.Y ) = (D...D.Y ), where D is repeated r times.

Lemma 12.1.

(i) The intersection number (D1 ...Dr.Y ) is an integer, and the function (D ,...,D ) (D ...D .Y ) 1 r 7−→ 1 r is multilinear symmetric. (ii) If a is the coefficient of nr in χ( n ), and = (D), then (Dr.Y )= r!a. L ⊗ OY L OX Proof. This is merely a repetition of Lemma 11.4 in the present context and notation. 

Remark. Suppose that Y is zero dimensional, so Y consists of a finite number of closed points. Then the higher cohomology groups are 0, and

(Y )= χ( ) = dim H0(Y, ) > 0, OY OY because H0(Y, ) is the vector space of global sections, and is not 0 since Y is affine. One can OY reduce the general intersection symbol to this case by means of the next lemma. 260 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Lemma 12.2. Let ,..., be invertible sheaves on X such that is very ample. Let D L1 Lr L1 1 be a divisor corresponding to 1 such that D1 does not contain any associated point of Y . Let ′ L O Y be the scheme intersection of Y and D1. Then ′ (D1 ...Dr.Y ) = (D2 ...Dr.Y ).

In particular, if D1,...,Dr are ample, then

(D1 ...Dr.Y ) > 0. Proof. If is the sheaf of ideals defining Y , and is the sheaf of ideals defining D , the IY I1 1 ( , ) defines Y D . By III.6 we know that is locally principal. The assumption in the IY I1 ∩ 1 § I1 lemma implies that we have an exact sequence ( ) 0 0. ∗ −→ I1 ⊗ OY −→ OY −→ OY ∩D1 −→ Indeed, let Spec(A) be an open affine subset of X containing a generic point of Y , and such that D1 is represented in A by the local equation f = 0, while Y is defined by the ideal I. Then the above sequence translates to 0 fA A/I A/I A/(I,f) 0 −→ ⊗ −→ −→ −→ which is exact on the left by our assumption on f. But = −1 by the definitions. Tensoring the sequence ( ) with I1 L1 ∗ n1 nr , L1 ⊗···⊗Lr taking the Euler characteristic, and using the additivity of the Euler characteristic, we get (D ...D .Y )n n (D ...D .Y )(n 1)n n + lower terms 1 r 1 · · · r − 1 r 1 − 2 · · · r = (D ...D .Y ′)n n + lower terms. 2 r 2 · · · r This proves the lemma. 

The intersection number (Dr.Y ) was taken with respect to the scheme X and it is sometimes necessary to include X in the notation, so we write (D ...D .Y ) or ( . . . .Y ) . 1 r X L1 Lr X On the other hand, let Z be a closed subscheme of X. Then we may induce the sheaves to Z to get ,..., . L|1Z L|rZ Lemma 12.3. Let Y Z X be inclusions of closed subschemes. Suppose Y has dimension ⊂ ⊂ r as before. Then ( . . . .Y ) = ( . . . .Y ) . L1 Lr X L|1Z L|rZ Z Proof. In the tensor products

n1 nr L1 ⊗···⊗Lr ⊗ OY we may tensor with each one of the factors without changing this tensor product. The OZ cohomology of a sheaf supported by a closed subscheme is the same as the cohomology of the sheaf in the scheme itself (cf. Corollary 3.8), so the assertion of the lemma is now clear. 

Theorem 12.4 (Criterion of Nakai-Moishezon). Let X be a proper scheme over a field k. Then a divisor D is ample on X if and only if (Dr.Y ) > 0 for all integral closed subschemes Y of dimension r, for all r dim X. ≤ 12.THECRITERIONOFNAKAI-MOISHEZON 261

Proof. Suppose D is ample. Replacing D by a positive multiple, we may assume without ∗ loss of generality that D is very ample. Let = (D), and let = f P(1) for a projective L O L O embedding f : X P over k. Abbreviate = P(1). Then the Euler characteristic → H O χ ( n1 nr ) k L ⊗···⊗L ⊗ OY is the same as the Euler characteristic χ ( n1 nr ) k H ⊗···⊗H ⊗ OY where Y is now viewed as a sheaf on P. This reduces the positivity to the case of projective O space, and D is a hyperplane, which is true by Lemma 12.2. The converse is more difficult and is the essense of the Nakai-Moishezon theorem. We assume that (Dr.Y ) > 0 for all integral closed subschemes Y of X of dimension r dim X and we want ≤ to prove that D is ample. By Propositions 9.2 and 9.3, we may assume that X is integral (reduced and irreducible), so X is a variety. For the rest of the proof we let = (D). L O By Lemma 12.3 and induction we may assume that is ample for every closed subscheme L|Z Z of X, Z = X. 6 Lemma 12.5. For n large, H0(X, n) = 0. L 6 Proof of Lemma 12.5. First we remark that χ( n) as n , for by Lemma 12.1 L →∞ →∞ (ii), χ( n)= and + lower terms L where d = dim X, and r!a = (Dd.X) > 0 by assumption. Next, we prove that Hi( n) Hi( n−1) for i 2 and n n . Since X is integral, we can L ≈ L ≥ ≥ 0 identify as a subsheaf of the sheaf of rational functions on X. We let L = −1 . I L ∩ OX Then is a coherent sheaf of ideals of , defining a closed subscheme Y = X. Furthermore I OX 6 is also a coherent sheaf of ideals, defining a closed subscheme Z = X. We have two exact I⊗L 6 sequences

0 / / X / Y / 0 I O O

0 / / X / Z / 0. I⊗L O O We tensor the first with n and the second with n−1. By induction, Hi( n )= Hi( n−1 )= L L L |Y L |Z 0 for i 1 and n n . Then the exact cohomology sequence gives isomorphisms for i 2 and ≥ ≥ 0 ≥ n n : ≥ 0 Hi( n) Hi( n) and Hi( n−1) Hi( n−1). I⊗L ≈ L I⊗L⊗L ≈ L This proves that Hi( n) Hi( n−1) for i 2. But then L ≈ L ≥ dim H0( n) χ( n) , L ≥ L →∞ thus proving the lemma. 

A global section of n then implies the existence of an effective divisor E nD, and since the L ∼ intersection number depends only on the linear equivalence class (namely, on the isomorphism class of the invertible sheaves), the hypothesis of the theorem implies that (Er.Y ) > 0 for all closed subschemes Y of X. It will suffice to prove that E is ample. This reduces the proof of the theorem to the case when D is effective, which we now assume. 262 VII.THECOHOMOLOGYOFCOHERENTSHEAVES

Lemma 12.6. Assume D effective. Then for sufficiently large n, n is generated by its global L sections. Proof of Lemma 12.6. We have = (D) where D is effective, so we have an exact L O sequence 0 −1 0. −→ L −→ OX −→ OD −→ Tensoring with n yields the exact sequence L 0 n−1 n n 0. −→ L −→ L −→ L |D −→ By induction, n is ample on D, so H1( n )=0 for n large. The cohomology sequence L |D L |D H0( n) H0( n ) H1( n−1) H1( n) H1( n ) L −→ L |D −→ L −→ L −→ L |D shows that H1( n−1) H1( n) is surjective for n large. Since the vector spaces H1( n) are L → L L finite dimensional, there exists n0 such that H1( n−1) H1( n) is an isomorphism for n n . L −→ L ≥ 0 Now the first part of the cohomology exact sequence shows that H0( n) H0( n ) is surjective for n n . L −→ L |D ≥ 0 Since n is ample on D, it is generated by global sections. By Nakayama, it follows that n L |D L is generated by global sections. This proves Lemma 12.6  We return to the proof of Theorem 12.4 proper. If dim X = 1, then (D) > 0, X is a curve, and every effective non-zero divisor on a curve is ample (cf. Proposition VIII.1.7 below). Suppose dim X 2. For every integral curve (subscheme of dimension one) C on X, we ≥ know by induction that n is ample on C. We can apply Proposition 9.6 to conclude the L |C proof.  CHAPTER VIII

Applications of cohomology

In this chapter, we hope to demonstrated the usefulness of the formidable tool that we developed in Chapter VII. We will deal with several topics that are tied together by certain common themes, although not in a linear sequence. We will start with possibly the most famous theorem in all algebraic geometry: the Riemann-Roch theorem for curves. This has always been the principal non-trivial result of an introduction to algebraic geometry and we would not dare to omit it. Besides being the key to the higher theory of curves, it also brings in differentials in an essential way — foreshadowing the central role played by the cohomology of differentials on all varieties. This theme, that of De Rham cohomology is discussed in 3. In order to be § able to prove strong result there, we must first discuss in 2 Serre’s cohomological approach to § Chow’s theorem, comparing analytic and algebraic coherent cohomologies. In 4 we discuss the § application, following Kodaira, Spencer and Grothendieck, of the cohomology of Θ, the sheaf of vector fields, to deformation of varieties. Finally, in 2, 3 and 4, we build up the tools to §§ be able at the end to give Grothendieck’s results on the partial computation of π1 of a curve in characteristic p.

1. The Riemann-Roch theorem As we discussed in VII.7, cohomology, disguised in classical language, grew out of the § attempt to develop formulas for the dimension of: space of 0 and non-zero rational functions f on X H0( (D)) = . OX with poles at most D, i.e., (f)+ D 0  ≥  (See also the remark in III.6.) § Put another way, the general problem is to describe the filtration of the function field R(X) given by the size of the poles. This one may call the fundamental problem of the additive theory of functions on X (as opposed to the multiplicative theory dealing with the group R(X)∗, and leading to Pic(X)). Results on dim H0( (D)) lead in turn to results on the projective OX embeddings of X and other rational maps of X to Pn, hence to many results on the geometry and classification of varieties X. The first and still the most complete result of this type is the Riemann-Roch theorem for curves. This may be stated as follows:

Theorem 1.1 (Riemann-Roth theorem). Let k be a field and let X be a curve, smooth and proper over k such that X is geometrically irreducible (also said to be absolutely irreducible, i.e., X Spec k is irreducible with k = algebraic closure of k). If n P (P X, closed ×Spec k i i i ∈ points) is a divisor on X, define P deg( niPi)= ni[k(Pi) : k]. Then for any divisor D on X: X X 1) dim H0( (D)) dim H1( (D)) = deg D g + 1, where g = dim H1( ) is the k OX − k OX − k OX genus of X, and

263 264 VIII.APPLICATIONSOFCOHOMOLOGY

2) (weak form) dim H1( (D)) = dim H0(Ω1 ( D)). k OX k X/k − The first part follows quickly from our general theory like this: Proof of 1). Note first that H0( ) consists only in constants in k. In fact H0( ) is OX OX a finite-dimensional k-algebra (cf. Proposition II.6.9), without nilpotents because X is reduced and without non-trivial idempotents because X is connected. Therefore H0( ) is a field L, OX finite over k. By the theorey of IV.2, X smooth over k = R(X) separable over k = L § ⇒ ⇒ separable over k; and X k k irreducible = k separable algebraically closed in R(X) = L × ⇒ ⇒ purely inseparable over k. Thus L = k, and (1) can be rephrased: χ( (D)) = deg D + χ( ). OX OX Therefore (1) follows from:

Lemma 1.2. If P is a closed point on X and is an invertible sheaf, then L χ( )= χ( ( P )) + [k(P ) : k]. L L − Proof of Lemma 1.2. Use the exact sequence:

0 ( P ) O k(P ) 0 −→ L − −→L −→L⊗ X −→ and the fact that invertible = O k(P ) = k(P ) (where: k(P ) = sheaf (0) outside P , L ⇒L⊗ X ∼ with stalk k(P ) at P ). Thus χ( )= χ( ( P )) + χ(k(P )) L L − and since H0(k(P ))= k(P ), H1(k(P )) = (0), the result follows.  

d To explain the rather mysterious second part, consider the first case k = C, D = i=1 Pi with the P distinct, so that deg D = d. Let z vanish to first order at P , so that z is i i ∈ OPi,X i P i a local analytic coordinate in a small (classical) neighborhood of P . Then if f H0( (D)), i ∈ OX we can expand f near each Pi as: ai f = + function regular at Pi, zi and we can map H0( (D)) / d OX C

f / (a1,...,ad) by assigning the coefficients of their poles to each f. Since only constants have no poles, this shows right away that dim H0( (D)) d + 1. OX ≤ Suppose on the other hand we start with a1,...,ad C and seek to construct f. From elementary ∈ complex variable theory we find obstructions to the existence of this f! Namely, regarding X as a compact Riemann surface ( = compact 1-dimensional complex manifold), we use the fact that if ω is a meromorphic differential on X, then the sum of the residues of ω at all 1 its poles is zero (an immediate consequence of Cauchy’s theorem). Now ΩX/C is the sheaf of algebraic differential forms on X and for any Zariski-open U X and ω Ω1 (U), ω defines ⊂ ∈ X/C a holomorphic differential form on U. (In fact if locally near x U, ∈ ω = a db , a , b j j j j ∈ Ox,X X 1. THE RIEMANN-ROCH THEOREM 265 then aj, bj are holomorphic functions near x too and ajdbj defines a holomorphic differential form: we will discuss this rather fine point more carefully in 3 below.) So if ω Γ(Ω1 ), then P § ∈ X/C write ω near Pi as:

ω = (bi(ω) + function zero at Pi) dzi, bi(ω) C. · ∈ If f exists with poles ai/zi at Pi, then fω is a meromorphic differential such that:

dzi fω = ai bi(ω) + (differential regular at Pi), · · zi hence res (fω)= a b (ω) Pi i · i hence d d 0= res (fω)= a b (ω). Pi i · i Xi=1 Xi=1 This is a linear condition on (a1,...,ad) that must be satisfied if f is to exist. Now Assertion (2) of Theorem 1.1 in its most transparent form is just the converse: if a b (ω) = 0 for every i · i ω Γ(Ω1 ), then f with polar parts a /z exists. How does this imply (2) as stated? Consider ∈ X/C i i P the pairing: Cd H0(Ω1 ) / × X/C C ((a ),ω) / a b (ω). i i · i 0 1 Clearly the null-space of this pairing on the H (ΩX/CP)-side is the space of ω’s zero at each Pi, i.e., 0 1 d 0 H (Ω ( Pi)). We have claimed that the null-space on the C -side is Image H ( X ( Pi)). X/C − O Thus we have a non-degenerate pairing: P P d 0 0 1 0 1 C / Image H ( X ( Pi)) H (Ω )/H (Ω ( Pi)) C. O × X/C X/C − −→ Taking dimensions, X   X  ( ) d dim H0( ( P ))+1 =dim H0(Ω1 ) dim H0(Ω ( P )). ∗ − OX i X/C − X/C − i Now it turns out that if d PXis a large enough positive divisor, H1( ( XP )) = (0) and i=1 i OX i H0(Ω1 ( P )) = (0) and this equation reads: X/C − i P P P d χ( ( P ))+1 = dim H0(Ω1 ), − OX i X/C and since by Part (1) of Theorem 1.1, χX( ( P )) = d g+1, it follows that g = dim H0(Ω1 ). OX i − X/C Putting this back in ( ), and using Part (1) of Theorem 1.1 again we get ∗ P g dim H0(Ω1 ( P )) = d + 1 χ( ( P )) dim H1( ( P )) − X/C − i − OX i − OX i X = g dim H1( X( P )) X − OX i hence Part (2) of Theorem 1.1. X A more careful study of the above residue pairing leads quite directly to a proof of Assertion (2) of Theorem 1.1 when k = C. Let us first generalize the residue pairing: if D1 and D2 are any two divisors on X such that D D is positive (D , D themselves arbitrary), then we get 2 − 1 1 2 a pairing:

0 1 X (D2)x/ X (D1)x H (Ω ( D1)) C O O × X/C − −→ x ! M 266 VIII.APPLICATIONSOFCOHOMOLOGY as follows: given (f ) representing a member of the left hand side (f (D ) ) and ω x x ∈ OX 2 x ∈ H0(Ω1 ( D )), pair these to res (f ω). Here f ω may have a pole of order > 1 at x, X/C − 1 x x x · x · but res still makes good sense: expand x P +∞ n fxω = cnt dt ! nX=−N where t has a simple zero at x, and set resx = c−1. Since 1 c = f ω −1 2πi x I (taken around a small loop around x), c−1 is independent of the choice of t. Note that if f ′ f + (D ) , then f ′ ω f ω Ω1 , hence res (f ′ ω) = res (f ω). If D = P , x ∈ x OX 1 x x · − x · ∈ x x x x x 2 i D = 0, we get the special case considered already. By the fact that the sum of the residues of 1 P any ω Ω1 is 0, the pairing factors as follows: ∈ R(X)/C 0 1 H (Ω ( D1)) x X (D2)x/ X (D1)x X/C − (residue pairing) O 0 O 0 1 C. Image H ( X (D2)) × H (Ω ( D )) −→ L O X/C − 2 It is trivial that this is non-degenerate on the right: i.e., if ω H0(Ω1 ( D )) H0(Ω1 ( D )), ∈ X/C − 1 \ X/C − 2 then for some (f ), res (f ω) = 0. But in fact: x x x 6 Theorem 1.3 (Riemann-Roch theorem (continued)). (2)-strong form: For every D1, D2 with D D positive, the residue pairing is non-degenerate on both sides. 2 − 1 Proof of Theorem 1.3. First, note that the left hand side can be interpreted via H1’s: namely the exact sequence: 0 (D ) (D ) (D )/ (D ) 0, −→ OX 1 −→ OX 2 −→ Ox 2 Ox 1 −→ x M where (D ) / (D ) is the skyscraper sheaf at x with stalk (D )/ (D ), induces an OX 2 x OX 1 x Ox 2 Ox 1 isomorphism

x(D2)/ x(D1) x O O = Ker H1( (D )) H1( (D )) . Image H0( (D )) ∼ OX 1 −→ OX 2 L OX 2 Now let D increase. Whenever D < D′ (i.e., D′ D a positive divisor), it follows that there 2 2 2 2 − 2 are natural maps: (D ) / (D ) (D′ ) / (D ) x OX 2 x OX 1 x ֒ x OX 2 x OX 1 x Image H0( (D )) injective→ Image H0( (D′ )) L OX 2 L OX 2 and H0(Ω1 ( D )) H0(Ω1 ( D )) X/C − 1 ։ X/C − 1 H0(Ω1 ( D )) surjective H0(Ω1 ( D′ )) X/C − 2 X/C − 2 compatible with the pairing. Passing to the limit, we get a pairing:

x∈X R(X)/ X (D1)x closed O 0 1 H (Ω ( D1)) C. R(XL) (embedded diagonally) × X/C − −→ It follows immediately that if this is non-degenerate on the left, so is the original pairing. Note here that the left hand side can be interpreted as an H1: namely the exact sequence:

0 X (D1) R(X) R(X)/ X (D1)x 0, −→ O −→ constant −→ O −→ sheaf x∈X closedM 1. THE RIEMANN-ROCH THEOREM 267 where R(X)/ X (D1)x is the skyscraper sheaf at x with stalk R(X)/ X (D1)x, induces an iso- O O morphism: x∈X R(X)/ X (D1)x closed O 1 = H ( X (D1)). L R(X) ∼ O Thus we are now trying to show that we have via residue a perfect pairing: 1 0 1 H ( X (D1)) H (Ω ( D1)) C. O × X/C − −→ This pairing is known as “Serre duality”. To continue, suppose

l : R(X)/ X (D1)x C O −→ x∈X closedM is any linear function. Then l = lx, where

Plx : R(X)/ X (D1)x C O −→ is a linear function. Now if tx has a simple zero at x, and nx = order of x in the divisor D1, then let −ν cν = lx(t ), all ν Z. x ∈ Note that c =0 if ν n . Then we can write l formally: ν ≤− x x l (f) = res (f ω ) x x · x where +∞ ν dtx ωx = cν tx · tx ν=−Xnx+1 is a formal differential at x; in fact ω Ω1 ( D ) . x ∈ X − 1 x This suggests defining, for the purposes of the proof only, pseudo-section of Ω1 ( D ) to be a X − 1 collection (ω ) , where ω Ω1 ( D ) are formal differentials and where x x∈X, closed x ∈ X − 1 x resx(f ωx) = 0, all f R(X). · ∈ x∈X closedX If we let H0(Ω1 ( D )) be the vector space of such pseudo-sections, then we see that X − 1 x∈X R(X)/ X (D1)x e closed O 0 1 H (Ω ( D1)) C L R(X) × X − −→ is indeed a perfect pairing, and we must merely checke that all pseudo-sections are true sections to establish the assertion. Now let D tend to as a divisor. If D′ < D , we get a diagram: 1 −∞ 1 1 H0(Ω1 ( D′ )) ⊂ H0(Ω1 ( D′ )) X − 1 X − 1 ∪ ∪ 0 1 0 1 H (Ω ( D1)) ⊂ He (Ω ( D )) X − X − 1 and clearly: e H0(Ω1 ( D′ )) H0(Ω1 ( D )) = H0(Ω1 ( D )). X − 1 ∩ X − 1 X − 1 Passing to the limit, we get: e Ω1 Ω1 R(X)/C ⊂ R(X)/C e 268 VIII.APPLICATIONSOFCOHOMOLOGY where set of meromorphic pseudo-differentials, i.e., 1 Ω = collections of ωx,X Ωx,X O R(X) such that . R(X)/C  ∈ ⊗ x   x resx(f ωx) = 0, all f R(X)  e · ∈ It suffices to prove that Ω1=PΩ1. But it turns out that if D′ is sufficiently negative, then D′ 1 − 1 is very positive and He1(Ω1 ( D′ )) = H0( (D′ )) = (0). X − 1 OX 1 Thus dim H0(Ω1 ( D′ )) = degΩ1 deg D′ g + 1 X − 1 X − 1 − dim H0(Ω1 ( D′ )) = dim H1( (D′ )) X − 1 OX 1 = deg D′ + g 1 e − 1 − hence

0 1 ′ 0 1 ′ 1 ′ dimC H (Ω ( D ))/H (Ω ( D )) = 2g 2 degΩ (independent of D ). X − 1 X − 1 − − X 1   1 1 1 1 1 Thus dimC Ωe /Ω < + . But Ω is an R(X)-vector space! So if Ω % Ω , R(X)/C R(X)/C ∞ R(X)/C 1 1 1 1 then dimC Ω /Ω =+ . Therefore Ω =Ω as required.  e ∞ e e All thise uses the assumption k = eC only in two ways: first in order to know that if we define the residue of a formal meromorphic differential via:

+∞ n res cnt dt = c−1, ! nX=−N then the residue remains unchanged if we take a new local coordinate t′ = a t + a t2 + , 1 2 · · · (a = 0). Secondly, if ω Ω1 , then we need the deep fact: 1 6 ∈ R(X)/k

resx ω = 0. x∈X closedX Given these facts, our proof works over any algebraically closed ground field k (and with a little more work, over any k at all). For a long time, only rather roundabout proofs of these facts were known in characteristic p (when characteristic = 0, there are simple algebraic proofs or one can reduce to the case k = C). Around the time this manuscript was being written Tate [100] discovered a very elementary and beautiful proof of these facts: we reproduce his proofs in an appendix to this section. Note that his “dualizing sheaf” is exactly the same as our “pseudo-differentials”. We finish the section with a few applications.

Corollary 1.4. If X is a geometrically irreducible curve, proper and smooth over a field k, then:

a) For all f R(X), deg(f) = 0; hence if X (D1) = X (D2), then deg D1 = deg D2. ∈ O ∼ O This means we can assign a degree to an invertible sheaf by requiring: L deg = deg D if = (D). L L ∼ OX b) If deg D< 0, then H0( (D)) = (0). OX 1. THE RIEMANN-ROCH THEOREM 269

Proof. Multiplication by f is an isomorphism ≈ ((f)), χ( ) = χ( ((f))), OX −→ OX OX OX so by Riemann-Roch (Theorem 1.1), deg(f) = 0. Secondly, if f H0( (D)), f = 0, then ∈ OX 6 D + (f) 0, so ≥ deg D = deg(D + (f)) 0. ≥ 

Corollary 1.5. If X is a geometrically irreducible curve, proper and smooth over a field k 1 of genus g (g = dim H ( X )), then: def O 0 1 1 1 a) dimk H (ΩX/k)= g, dimk H (ΩX/k) = 1, b) If K is a divisor such that Ω1 = (K) — a so-called canonical divisor — then X/k ∼ OX deg K = 2g 2. − Proof. Apply Riemann-Roch (Theorem 1.1) with D = K. 

Corollary 1.6. If X is a geometrically irreducible curve, proper and smooth over a field k of genus g, then deg D> 2g 2 implies: − a) H1( (D)) = (0) OX b) dim H0( (D)) = deg D g + 1. OX − Proof. If Ω1 = (K), then deg(K D) < 0, hence H0(Ω1 ( D)) = (0). Thus X/k ∼ OX − X/k − by Riemann-Roch (Theorem 1.1), H1( (D)) = (0) and dim H0( (D)) = χ( (D)) = OX OX OX deg D g + 1.  − Proposition 1.7. Added Let X be a geometrically irreducible curve proper and smooth over a field k. An invertible sheaf on X is ample if and only if deg > 0. L L Proof. We use Serre’s cohomological criterion (Theorem VII.8.2). Note that the coho- mology groups Hp for p > 1 of coherent -modules vanish since dim X = 1 (cf. Proposition OX VII.4.2). Thus we need to show that for any coherent -module one has H1(X, n) = 0, n 0 OX F F⊗L ≫ if and only if deg > 0. L Let r = rk , i.e., the dimension of the R(X)-vector space η (η = generic point). Then F F we claim that has a filtration F (0) = ⊂ F0 ⊂ F1 ⊂···⊂Fr−1 ⊂ Fr F by coherent -submodules such that OX = torsion -module F0 OX / = invertible -module for j = 1,...,r. Fj Fj−1 OX Indeed, for closed points x are discrete valuation rings since X is a regular curve. Thus Ox,X for the submodule ( ) of torsion elements in the finitely generated -module , the Fx tor Ox,X Fx quotient /( ) is a free -module. is the -submodule of with ( ) = ( ) Fx Fx tor Ox,X F0 OX F F0 x Fx tor for all closed points x and / is locally free of rank r. X is projective by Proposition V.5.11. F F0 Thus if we choose a very ample sheaf on X, then a sufficient twist / by it has a section. F F0 Untwising the result, we get an invertible subsheaf / . Let be the - M ⊂ F F0 F1 ⊂ F OX submodule containing such that / and that ( / )/ is the -submodule of F0 F1 F0 ⊃ M F1 F0 M OX torsions of ( / )/ . Obviously, / is an invertible submodule of / with / locally F F0 M F1 F0 F F0 F F1 free of rank r 1. The above claim thus follows by induction. − 270 VIII.APPLICATIONSOFCOHOMOLOGY

Since H1(X, n) = 0 for any n again by Proposition VII.4.2, the proposition follows if F0 ⊗L we show that for invertible one has H1(X, n) = 0, n 0 F F⊗L ≫ if and only if deg > 0. But this is immediate, since the cohomology group vanishes if deg( L F⊗ n) = deg + n deg > 2g 2 by Corollary 1.6, (a).  L F L − Remark. Added Using the filtration appearing in the proof above, we can generalize Theorem 1.1 (Riemann-Roch), (1) for a locally free sheaf of rank r as: E r dim H0(X, ) dim H1(X, ) = deg( )+ r(1 g). k E − k E E − Remark. Let X be a curve proper and smooth over an^ algebraically closed field k, and L an invertible sheaf on X. We can show: If deg 2g, then is generated by global sections. • L≥ L If deg 2g + 1, then is very ample (over k). • L≥ L Corollary 1.8. If X is a geometrically irreducible curve smooth and proper over a field k of genus 0, and X has at least one k-rational point x (e.g., if k is algebraically closed; or k a 1 finite field, cf. Proposition IV.3.5), then X ∼= Pk. Proof. Apply Riemann-Roch (Theorem 1.1) to (x). It follows that OX dim H0( (x)) 2, k OX ≥ hence f H0( (x)) which is not a constant. This f defines a morphism ∃ ∈ OX f ′ : X P1 −→ k such that (f ′)−1( )= x , with reduced structure. Then f ′ must be finite; and thus is a ∞ { } Ox,X finite 1 -module such that O∞,P

1 /m 1 / /(m 1 ) O∞,P ∞,P Ox,X ∞,P · Ox,X

k k(x)

k

′ is an isomorphism. Thus = 1 , hence f is birational, hence by Zariski’s Main Theorem Ox,X ∼ O∞,P ( V.6), f ′ is an isomorphism.  § Corollary 1.9. If X is a geometrically irreducible curve smooth and proper over a field k of genus 1, then Ω1 = . Moreover the map X/k ∼ OX set of k-rational invertible sheaves X(k)= / L points x X of degree 1 on X  ∈    x / (x) OX is an isomorphism, hence if x X(k) is a base point, X(k) is a group via x + y = z if and only 0 ∈ if (x) (y) = (z) (x ). OX ⊗ OX ∼ OX ⊗ OX 0 1. THE RIEMANN-ROCH THEOREM 271

Proof. Since H0(Ω1 ) = (0), Ω1 = (D) for some non-negative divisor D. But then X/k 6 X/k ∼ OX deg D = 2g 2 = 0, − so D = 0, i.e., Ω1 = . Next, if is an invertible sheaf of degree 1, then by Corollary 1.6, X/k ∼ OX L H1( ) = (0), hence by Riemann-Roch (Theorem 1.1), dim H0( ) = 1. This means there is a L k L unique non-negative divisor D such that = (D). Since deg D = 1, D = x where x X(k). L ∼ OX ∈ Finally, the invertible sheaves of degree 0 form a group under , hence so do the sheaves of ⊗ degree 1 if we multiply them by: ( , ) ( x ). L M 7−→ L ⊗ M ⊗ OX − 0 This proves that X(k) is a group.  In fact, it can be shown that X is a group scheme (in fact an abelian variety) over k (cf. VI.1) with origin x : especially there is a morphism § 0 µ: X X X ×Spec k −→ inducing the above addition on X(k): see Mumford [74, Chapter I, p. 36], and compare Part I [76, 7D]. § Corollary 1.10. If Θ = om(Ω1 , ) = ( K) is the tangent sheaf to X, then its X H X OX ∼ OX − cohomology is: g = 0 g = 1 g > 1 0 dim H (ΘX ) 3 1 0 dim H1(Θ ) 0 13g 3 X − 1 In fact, the three sections of Θ when X = Pk come from the infinitesimal section of the 1 3-dimensional group scheme PGL2,k acting on Pk; the one section of Θ when g = 1 comes from the infinitesimal action of X on itself, and the absence of sections when g > 1 is reflected in the fact that the group of automorphisms of such curves is finite. Thus three way division of curves, according as g = 0, g = 1, g > 1 is the algebraic side of the analytic division of Riemann surfaces according as whether they are a) the Gauss sphere, b) the plane modulo a discrete translation group or c) the unit disc modulo a freely acting Fuchsian group; and of the differential geometric division of compact surfaces according as they admit a metric with constant curvature K, with K > 0, K = 0, or K < 0. For further study of curves, an excellent reference is Serre [91, Chapters 2–5]. Classical references on curves are: Hensel-Landsberg [53], Coolidge [31], Severi [95] and Weyl [106]1. What happens in higher dimensions?2 The necessisity of the close analysis of all higher cohomolgy groups becomes much more apparent as the dimension increases. Part (1) of the curve Riemann-Roch theorem (Theorem 1.1) was generalized by Hirzebruch [56], and by Grothendieck (cf. [24])3 to a formula for computing χ( (D)) — for any smooth, projective variety X and divisor D — by a “universal polynomial” OX in terms of D and the Chern classes of X; this polynomial can be taken in a suitable cohomology ring of X, or else in the so-called Chow ring — a ring formed by cycles niZi (Zi subvarieties of X) modulo “rational equivalence” with product given by intersection. For this theory, see P Chevalley Seminar [29] and Samuel [84].

1(Added in publication) See also Iwasawa [57]. 2(Added in publication) There have been considerable developments on Kodaira dimension, Minimal model program, etc. 3(Added in publication) See also SGA6 [9] for further developments. 272 VIII.APPLICATIONSOFCOHOMOLOGY

Part (2) of the curve Riemann-Roch theorem (Theorem 1.1) was generalized by Serre and Grothendieck (see Serre [87], Altman-Kleiman [13] and Hartshorne [49]) to show, if X is a smooth complete varietie of dimension n, that a) a canonical isomorphism ǫ: Hn(X, Ωn ) ≈ k, and X/k −→ b) that — plus cup product induces a non-degenerate pairing Hi(X, (D)) Hn−i(X, Ωn ( D)) k OX × X − → for all divisors D and all i. Together however, these results do not give any formula in dim 2 involving H0’s alone. Thus ≥ geometric applications of Riemann-Roch requires a good deal more ingenuity (cf. for instance Shafarevitch et al. [96]). Three striking examples of cases where the higher cohomology groups can be dealt with so that a geometric conclusion is deduced from a cohomological hypothesis are: Theorem 1.11 (Criterion of Nakai-Moishezon). Let k be a field, X a scheme proper over k, and an invertible sheaf on X. Then L is ample, i.e., n 1 and reduced and irreducible L ≥ ∀ a closed immersion subvarieities Y X of    ⊂  . φ: X PN such that ⇐⇒ positive dimension,  ∗ → n   n   φ ( PN (1)) ∼=   χ( Y ) as n   O L   L ⊗ O →∞ →∞  (This is another form of Theorem VII.12.4. See also Kleiman [61]). Theorem 1.12 (Criterion of Kodaira). Let X be a compact complex analytic manifold and an invertible analytic sheaf on X. Then L can be defined by transition functions f L { αβ} X is a projective  for a covering Uα of X, where  2 { } ∞ variety and is fαβ = gα/gβ, gα positive real C on Uα and .  L  ⇐⇒  | 2 |  an ample sheaf on it  (∂ log gα/∂zi∂zj)(P )       positive definite Hermitian form at all P Uα   ∈    (For a proof, cf. Gunning-Rossi [48].) Theorem 1.13 (Vanishing theorem of Kodaira-Akizuki-Nakano). Let X be an n-dimensional complex projective variety, an ample invertible sheaf on X. Then L Hp(X, Ωq ) = (0), if p + q >n. X ⊗L (For a proof, cf. Akizuki-Nakano [11].)

Appendix: Residues of differentials on curves by John Tate Added

We reproduce here, in our notation, the very elementary and beautiful proof of Tate [100]. Here is the key to Tate’s proof: Let V be a vector space over a field k. A k-linear endomor- phism θ End (V ) is said to be finite potent if θnV is finite dimensional for a positive integer ∈ k n. For such a θ, the trace Tr (θ) k V ∈ is defined and has the following properties: (T1) If dim V < , then Tr (θ) is the ordinary trace. ∞ V APPENDIX: RESIDUES OF DIFFERENTIALS ON CURVES BY JOHN TATE 273

(T2) If W V is a subspace with θW W , then ⊂ ⊂

TrV (θ) = TrW (θ)+TrV/W (θ).

(T3) TrV (θ)=0 if θ is nilpotent. (T4) Suppose F End (V ) is a finite potent k-subspace, i.e., there exists a positive integer ⊂ k n such that θ θ θ V is finite dimensional for any θ ,...,θ F . Then 1 ◦ 2 ◦···◦ n 1 n ∈ Tr : F k is k-linear. V → (It does not seem to be known if

′ ′ TrV (θ + θ ) = TrV (θ)+TrV (θ )

holds in general under the condition θ, θ′ and θ + θ′ are finite potent.) (T5) Let φ: V ′ V and ψ : V V ′ be k-linear maps with φ ψ : V V finite potent. → → ◦ → Then ψ φ: V ′ V ′ is finite potent and ◦ →

Tr (φ ψ) = Tr ′ (ψ φ). V ◦ V ◦ n (T1), (T2) and (T3) characterize TrV (θ): Indeed, by assumption, W = θ V is finite dimen- sional for some n. Then TrV (θ) = TrW (θ). For the proof of (T4), we may assume F to be finite dimensional and compute the trace on the finite dimensional subspace W = F nV . As for (T5), φ and ψ induce mutually inverse isomorphisms between the subspaces W = n ′ n ′ (φ ψ) V and W = (ψ φ) V for n 0 under which (ψ φ) ′ and (φ ψ) correspond. ◦ ◦ ≫ ◦ |W ◦ |W Definition 1. Let A and B be k-subspaces of V . A is said to be “not much bigger than” B (denoted A B) if dim(A + B)/B < . • ≺ ∞ A is said to be “about the same size as” B (denoted A B) if A B and A B. • ∼ ≺ ≻ Proposition 2. Let A be a k-subspace of V . (1) E = θ End (V ) θA A is a k-subalgebra of End (V ). { ∈ k | ≺ } k (2) The subspaces E = θ End (V ) θV A 1 { ∈ k | ≺ } E = θ End (V ) θA (0) 2 { ∈ k | ≺ } E = E E = θ End (V ) θV A, θA (0) 0 1 ∩ 2 { ∈ k | ≺ ≺ } are two-sided ideals of E with E = E1 + E2, and E0 is finite potent so that there is a k-linear map Tr : E k. Moreover, E, E , E and E depend only on the V 0 → 1 2 0 -equivalence class of A. ∼ (3) Let φ, ψ End (V ). If either (i) φ E and ψ E, or (ii) φ E and ψ E , then ∈ k ∈ 0 ∈ ∈ 1 ∈ 2 [φ, ψ] := φ ψ ψ φ E ◦ − ◦ ∈ 0

with TrV ([φ, ψ]) = 0.

Proof. (1) is obvious. As for (2), express V as a direct sum V = A A′, and denote by ⊕ ε: V ։ A, ε′ : V ։ A′ the projections. Then id = ε + ε′ with ε E and ε E , so that V ∈ 1 ∈ 2 θ = θε + θε′ for all θ E. Obviously, θ θ V is finite dimensional for any θ ,θ E . (3) ∈ 1 ◦ 2 1 2 ∈ 0 follows easily from (T5).  274 VIII.APPLICATIONSOFCOHOMOLOGY

Theorem 3 (Abstract residue). Let K be a commutative k-algebra (with 1), V a k-vector space which is also a K-module, and A V a k-subspace such that fA A for all f K (hence ⊂ ≺ ∈ K acts on V through K E End (V ) with the image in E of each f K denoted by the → ⊂ k ∈ same letter f). Then there exists a unique k-linear map ResV :Ω1 k A K/k −→ such that for any pair f, g K, we have ∈ V ResA(fdg) = TrV ([f1, g1]) for f , g E such that 1 1 ∈ i) f f (mod E ), g g (mod E ) ≡ 1 2 ≡ 1 2 ii) either f E or g E . 1 ∈ 1 1 ∈ 1 The k-linear map is called the residue and satisfies the following properties: V V ′ ′ ′ ′ V V ′ (R1) Res = Res if V V A and KV = V . Moreover, Res = Res ′ if A A . A A ⊃ ⊃ A A ∼ (R2) (Continuity in f and g) We have fA + fgA + fg2A A = ResV (fdg) = 0. ⊂ ⇒ A Thus ResV (fdg) = 0 if fA A and gA A. In particular, ResV = 0 if A V is a A ⊂ ⊂ A ⊂ K-submodule. (R3) For g K and an integer n, we have ∈ n 0 ≥ ResV (gndg) = 0 if or A  n 2 and g invertible in K.  ≤− V In particular, ResA(dg) = 0.  (R4) If g K is invertible and h K with hA A, then ∈ ∈ ⊂ ResV (hg−1dg) = Tr (h) Tr (h). A A/(A∩gA) − gA/(A∩gA) In particular, if g K is invertible and gA A, then ∈ ⊂ V −1 ResA(g dg) = dimk(A/gA). (R5) Suppose B V is another subspace such that fB B for all f K. Then f(A+B) ⊂ ≺ ∈ ≺ A + B and f(A B) A B hold for all f K, and ∩ ≺ ∩ ∈ V V V V ResA +ResB = ResA+B +ResA∩B . (R6) Suppose K′ is a commutative K-algebra that is a free K-module of finite rank r. For a K-basis v ,...,v of K′, let { 1 r} r V ′ = K′ V A′ = v A. ⊗k ⊃ i ⊗ Xi=1 Then f ′A′ A′ holds for any f ′ K′, and the -equivalence class of A′ depends only ≺ ∈ ∼ on that of A and not on the choice of v ,...,v . Moreover, { 1 r} V ′ ′ V ′ ′ ′ Res ′ (f dg) = Res ((Tr ′ f )dg), f K , g K. A A K /K ∀ ∈ ∀ ∈ Proof of the existence of residue. By assumption, we have f, g E = E +E . Thus ∈ 1 2 f and g satisfying (i) and (ii) can be chosen. Then [f , g ] E by (ii), and [f , g ] [f, g] 1 1 1 1 ∈ 1 1 1 ≡ (mod E ) by (i) with [f, g] = 0 by the commutativity of K. Thus [f , g ] E E = E and 2 1 1 ∈ 1 ∩ 2 0 TrV ([f1, g1]) is defined. By Proposition 2, (3), it is unaltered if f1 or g1 is changed by an element APPENDIX: RESIDUES OF DIFFERENTIALS ON CURVES BY JOHN TATE 275 of E2 as long as the other is in E1. Moreover by (T4), TrV ([f1, g1]) is a k-bilinear function of f and g. Thus there is a k-linear map β : K K k such that β(f g) = Tr ([f , g ]). ⊗k −→ ⊗ V 1 1 We now show that β(f gh)= β(fg h)+ β(fh g), f,g,h K, ⊗ ⊗ ⊗ ∀ ∈ hence r factors through the canonical surjective homomorphism c: K K Ω1 , c(f g)= fdg. ⊗k −→ K/k ⊗ Indeed, for f,g,h K, choose suitable f , g , h E and let (fg) = f g , (fh) = f h and ∈ 1 1 1 ∈ 1 1 1 1 1 1 1 (gh)1 = g1h1. Then by the commutativity of K, we obviously have

[f1, g1h1] = [f1g1, h1] + [f1h1, g1]. 

We use the following lemma in proving the rest of Theorem 3:

Lemma 4. For f, g K, define subspaces B,C V by ∈ ⊂ B = A + gA C = B f −1(A) (fg)−1(A)= v B fv A and fgv A . ∩ ∩ { ∈ | ∈ ∈ } Then B/C is finite dimensional and V ResA(fdg) = TrB/C ([εf, g]), where ε: V ։ A is a k-linear projection.

Proof. B/C is finite dimensional, since B/ v B fv A and B/ v B fgv A are { ∈ | ∈ } { ∈ | ∈ } mapped injectively into the finite dimensional space (A+fA+fgA+fg2A)/A. Moreover, εf E ∈ 1 and εf f (mod E ), hence ResV (fdg) = Tr ([εf, g]). On the other hand, [εf, g]= εfg gεf ≡ 2 A V − maps V into B, and C into 0, since fg = gf. Thus the assertion follows by (T2), since  TrV = TrV/B + TrB/C + TrC . Proof of Theorem 3 continued. (R1) follows easily from Lemma 4, since B,C V ′. ⊂ As for (R2), we have B = C in Lemma 4. To prove (R3), choose g E such that g g (mod E ). If n 0, we have ResV (gndg)= 1 ∈ 1 1 ≡ 2 ≥ A Tr ([gn, g ]) = 0 since gn and g commute. If g is invertible, then g−2−ndg = (g−1)nd(g−1), V 1 1 1 1 − whose residue is 0 if n 0 by what we have just seen. ≥ −1 For the proof of (R4), let f = hg and apply Lemma 4. We have [εf, g]= εh ε1h, where −1 − ε1 = gεg is a projection of V onto gA. Since both A and gA are stable under h, we have ResV (fdg) = Tr (εh) Tr (gεg−1h) A (A+gA)/(A∩gA) − (A+gA)/(A∩gA) and we are done by computing the traces through A + gA A A gA and A + gA gA ⊃ ⊃ ∩ ⊃ ⊃ A gA, respectively. ∩ To prove (R5), choose projections εA : V ։ A, εB : V ։ B, εA+B : V ։ A + B, εA∩B : V ։ A B such that ∩ εA + εB = εA+B + εA∩B.

Then [εAf, g] and [εA+Bf, g] belong to F = θ E θV A + B,θ(A + B) A, θA (0) , { ∈ | ≺ ≺ ≺ } 276 VIII.APPLICATIONSOFCOHOMOLOGY which is finite potent, since θ θ θ V is finite dimensional for any θ ,θ ,θ F . Since 1 ◦ 2 ◦ 3 1 2 3 ∈ ε f E , ε f f (mod E ), ε f E and ε f f (mod E ), one has A ∈ 1 A ≡ 2 A+B ∈ 1 A+B ≡ 2 ResV (fdg) ResV (fdg) = Tr ([ε f, g]) Tr ([ε f, g]) A − A+B V A − V A+B = Tr ([(ε ε )f, g]) V A − A+B = Tr ([(ε ε )f, g]), V A∩B − B which, by a similar argument, equals ResV (fdg) ResV (fdg). A∩B − B As for (R6), a k-endomorphism ϕ of V ′ can be expressed as an r r matrix (ϕ ) of endo- × ij morphisms of V by the rule

ϕ( x v )= x ϕ v , for v V. j ⊗ j i ⊗ ij j j ∈ Xj Xij If F Endk(V ) is a finite potent subspace, then ϕ’s such that ϕij F for all i, j form a ⊂ ′ ′ ∈ ′ finite potent subspace F End (V ). We see that Tr ′ (ϕ) = Tr (ϕ ) for all ϕ F by ⊂ k V i V ii ∈ decomposing the matrix (ϕ ) into the sum of a diagonal matrix, a nilpotent triangular matrix ij P having zeros on and below the diagonal, and another nilpotent triangular matrix having zeros on and above the diagonal. For f ′ K′, write f ′x = x f with f K. Let ε: V ։ A be ∈ j i i ij ij ∈ a k-linear projection and put ε′( x v ) = x εv . Then ε′ : V ′ ։ A′ is a projection, i i ⊗ i i i ⊗P i and P P ′ ′ [f ε , g]ij = [fijε, g].  We are done since TrK′/K f = i fii. We are now ready to deal withP residues of differentials on curves. Let X be a regular irreducible curve proper over a field k, and denote by X0 the set of closed points of X. For each x X let ∈ 0 A = = m -adic completion of x Ox,X x,X Ox,X Kx = quotient field of Ax. b Define Res :Ω1 k x R(X)/k −→ by

Kx Resx(fdg) = Res (fdg), f,g R(X), Ax ∈ which makes sense since k(x) = Ax/mx,XAx is a finite dimensional k-vector space so that n Ax mx,XAx for any n Z and that for any non-zero f Kx we have fAx Ax since ∼ n ∈ ∈ ≺ fAx = mx,XAx for some n. Theorem 5. i) Suppose x X is k-rational so that A = k[[t]] and K = k((t)). For ∈ 0 x x f = a tν, g = b tµ K , ν µ ∈ x ν≫∞ µ≫∞ X X we have −1 ′ Resx(fdg)= coefficient of t in f(t)g (t)

= µaνbµ. ν+µ=0 X APPENDIX: RESIDUES OF DIFFERENTIALS ON CURVES BY JOHN TATE 277

ii) For any subset S X0, let (S)= x,X R(X). Then ⊂ O x∈S O ⊂ R Res (ω) = Res (TX)(ω), ω Ω1 . x O(S) ∀ ∈ R(X)/k xX∈S In particular Res (ω) = 0, ω Ω1 . x ∀ ∈ R(X)/k xX∈X0 iii) Let ϕ: X′ X be a finite surjective morphism of irreducible regular curves proper over → k. Then ′ ′ Resx′ (f dg) = Resx((TrR(X′)/R(X) f )dg) ′ −1 x ∈Xϕ (x) ′ ′ if f R(X ), g R(X) and x X0, while ∈ ∈ ∈ ′ ′ Resx′ (f dg) = Resx((TrK′ /K f )dg) x′ x ′ ′ ′ ′ ′ ′ if x X0 with ϕ(x ) = x, f Kx′ and g Kx. (Kx′ is the quotient field of the ∈ ′ ∈ ∈ m ′ ′ -adic completion A ′ of ′ ′ .) x ,X x Ox ,X Proof. (i) By the continuity (R2), we may assume that only finitely many of the aν and bµ are non-zero. Indeed, express f and g as

f = φ1(t)+ φ2(t)

g = ψ1(t)+ ψ2(t) in such a way that φ (t) and ψ (t) are Laurent polynomials and that φ (t), ψ (t) tnA for 1 1 2 2 ∈ x large enough n so that φ (t)ψ′ (t)+ φ (t)ψ′ (t)+ φ (t)ψ′ (t) A . 1 2 2 1 2 2 ∈ x Then fdg = f(t)g′(t)dt, and only the term in t−1 can give non-zero residue by (R3). By (R4) we have ResVx (t−1dt) = dim (x) = 1. Ax k k (Note that in positive characteristics it is not immediately obvious that the coefficient in question is independent of the choice of the uniformizing parameter t). For (ii), let

AS = Ax x∈S Y′ VS = Kx x∈S = Yf = (f ) f K , x S and f A for all but a finite number of x . { x | x ∈ x ∀ ∈ x ∈ x } Embedding R(X) diagonally into VS, we see that R(X) AS = (S). By (R5) we have ∩ O ResVS +ResVS = ResVS +ResVS . AS R(X) O(S) (R(X)+AS )

VS ResR(X) = 0 by (R2), since R(X) is an R(X)-module. We now show VS/(R(X)+ AS) to be finite dimensional, hence ResVS = 0 by (R1). It suffices to prove the finite dimensionality (R(X)+AS ) ։ when S = X0 because of the projection VX0 VS. Regarding R(X) as a constant sheaf on X, we have an exact sequence

0 X R(X) R(X)/ X = Kx/Ax 0, −→ O −→ constant −→ O −→ sheaf xM∈X0 278 VIII.APPLICATIONSOFCOHOMOLOGY where Kx/Ax is the skyscraper sheaf at x with stalk Kx/Ax. The associated cohomology long exact sequence induces an isomorphism ∼ 1 VX /(R(X)+ AX ) H (X, X ), 0 0 −→ O the right hand side of which is finite dimensional since X is proper over k. To complete the proof of (ii), it remains to show

VS Res (ω)= Resx(ω), ω = fdg. AS ∀ xX∈S Let S′ S be a finite subset containing all poles of f and g. We write ⊂ VS = VS\S′ Kx × ′ xY∈S AS = AS\S′ Ax. × ′ xY∈S By (R5), V ′ VS S\S Res (fdg) = Res (fdg)+ Resx(fdg). AS AS\S′ ′ xX∈S VS\S′ ′ ′ Res (fdg) = 0 and Resx(fdg) =0 for x S S by the choice of S . The last assertion in AS\S′ ∈ \ (ii) follows, since (X )= = H0(X, ) O 0 Ox,X OX x\∈X0 VX0 is finite dimensional over k so that (X0) (0) and Res = 0 by (R1). O ∼ O(X0) To prove (iii), regard the function field R(X′) of X′ as a finite algebraic extension of R(X). ′ ′ Then (iii) follows from (R6), since the integral closure of x (resp. Ax) in R(X ) (resp. K ′ ) is O x a finite module over (resp. A ).  Ox x

Recall that X0 is the set of closed points of an irreducible regular curve X proper over k. Each x X determines a prime divisor on X, which we denote by [x]. Thus a divisor D on X ∈ 0 is of the form

D = nx[x], with nx = 0 for all but a finite number of x. xX∈X0 We denote ordx D = nx. Let ′ V = VX0 = Kx x∈X0 Y A = AX0 = Ax. xY∈X0 For a divisor D on X, let V (D)= f = (f ) V ord f ord D, x X . { x ∈ | x x ≥− x ∀ ∈ 0} Then by an argument similar to that in the proof of Theorem 5, (ii), we get 1 H (X, X (D)) = V/(R(X)+ V (D)). O ∼ Let JR = λ Homk(V, k) λ(R(X)+ V (D)) = 0, D divisor (X)/k { ∈ | ∃ } = lim Hom (H1(X, (D)), k), k OX −→D APPENDIX: RESIDUES OF DIFFERENTIALS ON CURVES BY JOHN TATE 279 which is nothing but the space of meromorphic “pseudo-differentials” appearing in 1. § JR(X)/k is a vector space over R(X) by the action

(gλ)(f)= λ(gf), g R(X), f = (fx) V, ∀ ∈ ∀ ∈ since obviously (gλ)(R(X)) = 0, while (gλ)(V ((g)+ D)) = 0. As in 1, let us assume X to be smooth and proper over k and geometrically irreducible. § Then R(X) is a regular transcendental extension of transcendence degree one so that the module 1 Ω is a one-dimensional vector space over R(X). Moreover, for any x X0, the stalk R(X)/k ∈ Ω1 is a free -submodule of Ω1 of rank one. If t is a local parameter at x so that X/k,x Ox,X R(X)/k x m = t , then each ω Ω1 can be expressed as x,X xOx,X ∈ R(X)/k ω = hdtx, for some h R(X). ∈ Let us denote ordx(ω) = ordx(h), which is independent of the choice of the local parameter tx. We then denote (ω)= ordx(ω)[x], xX∈X0 which is easily seen to be a divisor on X. For any divisor D, one has H0(X, Ω1 ( D)) = ω Ω1 (ω) D X/k − { ∈ R(X)/k | ≥ } and Ω1 = lim H0(X, Ω1 ( D)). R(X)/k X/k − −→D The abstract residue gives rise to an R(X)-linear map σ :Ω1 J R(X)/k −→ R(X)/k defined by σ(ω)(f)= Res (f ω), ω Ω1 , f = (f ) V. x x ∀ ∈ R(X)/k ∀ x ∈ xX∈X0 This makes sense, since σ(ω)(R(X)) = 0 by Theorem 5, (ii), while σ(ω)(V (D)) =0 for D = (ω) by (R2). For any divisor D, we see easily that σ induces a k-linear map 0 1 1 σD : H (X, Ω ( D)) Homk(V/(R(X)+ V (D)), k) = Homk(H (X, X (D)), k). X/k − −→ O Theorem 6 (Serre duality). σ :Ω1 J R(X)/k −→ R(X)k is an isomorphism, which induces an isomorphism 0 1 ∼ 1 σD : H (X, Ω ( D)) Homk(V/(R(X)+ V (D)), k) = Homk(H (X, X (D)), k), X/k − −→ O for any divisor D. Consequently, (2)-strong form of the Riemann-Roch theorem (Theorem 1.3) holds, giving rinse to a non-degenerate bilinear pairing H0(X, Ω1 ( D)) H1(X, (D)) k. X/k − × OX −→ In particular, Part (2) of Theorem 1.1 holds. To show that σ and σ are isomorphisms, we follow Serre [91, Chapter II, 6 and 8]. D §§ Lemma 7. dim J 1. R(X) R(X)/k ≤ 280 VIII.APPLICATIONSOFCOHOMOLOGY

′ Proof. Suppose λ, λ JR were R(X)-linearly independent. Hence we have an injec- ∈ (X)/k tive homomorphism ′ R(X) R(X) (g, h) gλ + hλ JR . ⊕ ∋ 7−→ ∈ (X)/k There certainly exists D such that λ(V (D)) = 0 and λ′(V (D)) = 0. Fix x X and let P = [x]. ∈ 0 For a positive integer n and g, h R(X) with (g) + nP 0 and (h) + nP 0, we have ∈ ≥ ≥ (gλ + hλ′)(V (D nP )) = 0. Thus we have an injective homomorphism − H0(X, (nP )) H0(X, (nP )) (g, h) gλ + hλ′ Hom (H1(X, (D nP )), k). OX ⊕ OX ∋ 7−→ ∈ k OX − Hence we have ( ) dim H1(X, (D nP )) 2dim H0(X, (nP )). ∗ OX − ≥ k OX The right hand side of ( ) is greater than or equal to 2(n deg P g + 1) by Theorem 1.1, (1). ∗ − On the other hand, again by Theorem 1.1, (1), the left hand side of ( ) is equal to ∗ deg(D nP )+ g 1+dim H0(X, (D nP )) − − − k OX − = n deg P + (g 1 deg D)+dim H0(X, (D nP )). − − k OX − However, one has deg(D nP ) < 0 for n 0, hence H0(X, (D nP )) = 0 by Corollary 1.4, − ≫ OX − (b). Thus ( ) obviously leads to a contradiction for n 0.  ∗ ≫ Lemma 8. Under the R(X)-linear map σ :Ω1 J , R(X)/k −→ R(X)/k ω Ω1 belongs to H0(X, Ω1 ( D)) if σ(ω)(V (D)) = 0. ∈ R(X)/k X/k − Proof. Otherwise, there exists y X such that ord (ω) < ord D. Let n = ord (ω) + 1, ∈ 0 y y y hence n ord D. Define f = (f ) V by ≤ y x ∈ f =0 if x = y x 6 f = 1/tn (t being a local parameter at y).  y y y Obviously, Res (f ω)=0 for x = y, while x x 6 ord (f ω) = ord ((1/tn)ω)= n + ord (ω)= 1, y y y y − y − hence σ(ω)(f) = Res (f ω) = 0 by (R4). Since n ord (D), one has f V (D), a contradiction y y 6 ≤ y ∈ to the assumption σ(ω)(V (D)) = 0.  Proof of Theorem 6. σ is injective, for if σ(ω) = 0, then ω H0(X, Ω1 ( D)) for all ∈ X/k − D by Lemma 8, hence ω = 0. σ is surjective, since σ is a non-zero R(X)-linear map with dimR 1 by Lemma 7. (X) ≤ Moreover, σ is surjective, for if λ J satisfies λ(V (D)) = 0, then there exists ω Ω1 D ∈ R(X)/k ∈ R(X)/k with σ(ω)= λ. We see that ω H0(X, Ω1 ( D)) by Lemma 8.  ∈ X/k − 2. Comparison of algebraic with analytic cohomology In almost all of this section, we work only with complex projective space and its non-singular n n n subvarieties. We abbreviate PC to P and recall that the set of closed points of P has two topologies: the Zariski topology and the much finer classical (or ordinary) topology. By (Pn in the classical topology) we mean the set of closed points of Pn in the classical topology and by n n (P in the Zariski topology) we mean the scheme PC as usual. Note that there is a continuous map ǫ: (Pn in classical topology) (Pn in the Zariski topology). −→ We shall consider sheaves on the space on the left. The following class is very important. 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 281

n Definition 2.1. The holomorphic or analytic structure sheaf Pn,an on (P in the classical O topology) is the sheaf:

Pn,an(U) = ring of analytic functions f : U C. O → n If U P is an open set, then a sheaf of Pn,an-modules on U is called a coherent analytic ⊂ F O sheaf if for all x U, there is a (classical) open neighborhood U U of x and an exact sequence ∈ x ⊂ of sheaves of Pn -modules on U : O ,an x n1 n0 n n 0. OP ,an|Ux −→ OP ,an|Ux −→ F|Ux −→ For basic results on coherent analytic sheaves, we refer to Gunning-Rossi [48]. Among the standard results given there are:

(2.2) If φ: is an Pn -module homomorphism of coherent analytic sheaves on some F → G O ,an U, then Ker φ, Image φ and Coker φ are coherent; thus the coherent analytic sheaves on U form an abelian category. n (2.3) If U PC is a polycylinder in some affine piece and is coherent on U, then i ⊂ F 0 H (U, ) = (0), i> 0, and is generated as Pn -module by H (U, ). F|U F|U O ,an F|U (2.4) If X U is a closed analytic subset, then the sheaf X of analytic functions vanishing ⊂ n I on X is coherent. If is coherent, then x P x = (0) is a closed analytic subset. F { ∈ | F 6 } Now if is an algebraic coherent sheaf on Pn, one can define canonically an associated analytic F C coherent sheaf as follows: for all classical open U, let: Fan

(U) = submodule of ( Pn ) n Fan O ,an x ⊗OP ,x Fx xY∈U consisting of families s satisfying the following condition: { x} for all x U, classical neighborhood U of x and a Zariski ∈ 1 neighborhood U of x, f Pn (U ) and t (U ) such that 2 i ∈ O ,an 1 i ∈ F 2 s = f t , x U U . x i ⊗ i ∈ 1 ∩ 2 This looks a bit cumbersomeP but, in fact, it is the natural way to define f ∗ for any morphism F f of ringed spaces, and sheaf of modules on the image space. In the present situation, one F has = ǫ∗ . An elementary calculation gives the stalks of : Fan F Fan

( ) = ( Pn ) n . Fan x O ,an x ⊗OP ,x Fx Also, is obviously a functor, i.e., any Pn -homomorphism φ: induces F 7−→ Fan O F1 → F2 φ : . We now invoke the basic fact: an F1,an → F2,an Lemma 2.5 (Serre). C X1,...,Xn , the ring of convergent power series, is a flat C[X1,...,Xn]- { } module.

Proof. In fact, the completion of a noetherian local ring is a faithfully flat -module O O O (Atiyah-MacDonald [19, (10.14) and Exercise 7, Chapter 10]), hence C[[X1,...,Xn]] is faith- fully flat over C X1,...,Xn and overb C[X1,...,Xn] . Hence M N P over { } (X1,...,Xn) ∀ → → C[X1,...,Xn], M N P exact = M C[[X]] N C[[X]] P C[[X]] exact → → ⇒ ⊗ → ⊗ → ⊗ = M C X N C X P C X exact. ⇒ ⊗ { }→ ⊗ { }→ ⊗ { } 

Corollary 2.6. is an exact functor from the category of all Pn -modules to the F 7−→ Fan O category of all Pn -modules. O ,an 282 VIII.APPLICATIONSOFCOHOMOLOGY

Proof. If is exact, then by Proposition IV.4.3 F→G→H

Fan,x −→ Gan,x −→ Han,x is exact for all x, hence is exact.  Fan → Gan →Han Corollary 2.7. If is a coherent algebraic sheaf, then is a coherent analytic sheaf. F Fan Proof. If is coherent, then on any affine U, there is an exact sequence F n1 n0 n n 0 OP |U −→ OP |U −→ F|U −→ hence an exact sequence:

n1 n0 0. OPn,an|U −→ OPn,an|U −→ Fan|U −→ 

Note that covering the identity map

ǫ: (Pn in the classical topology) (Pn in the Zariski topology) −→ there is a map ǫ∗ : of sheaves. This induces a canonical map on cohomology Hi( ) F → Fan F → Hi( ). Fan Our goal is now the following fundamental theorem:

Theorem 2.8 (Serre). (Fundamental “GAGA”4 comparison theorem) i) For every coherent algebraic , and every i, F i n i n H (P in the Zariski topology, ) = H (P in the classical topology, an). F ∼ F ii) The categories of coherent algebraic and coherent analytic sheaves are equivalent, i.e., every coherent analytic ′ is isomorphic to , some , and F Fan F

Hom n ( , ) = Hom n ( , ). OP F G ∼ OP ,an Fan Gan We will omit the details of the first and most fundamental step in the proof (for these we refer the reader to Gunning-Rossi [48, Chapter VIII A]). This is the finiteness assertion: given i n a coherent analytic , then dimC H (P , ) < + , for all i. The proof goes as follows: F F ∞ a) For all C > 1, 0 i n, let ≤ ≤ n Xj Ui,C = x P x / V (Xi) and (x) < C, 0 j n . ∈ ∈ X ≤ ≤  i  n n n b) Then Ui,C = P so we have an open covering C = U0,C , , Un,C of P . i=0 U { · · · } c) Note that each intersection U U can be mapped bihilomorphically onto S i1,C ∩···∩ ik,C a closed analytic subset Z of a high-dimensional polycylinder D by means of the set of functions X /X , 0 j n, 1 l k. Therefore, every coherent analytic on j il ≤ ≤ ≤ ≤ F U U corresponds to a sheaf ′ on Z and, extending it to D Z by (0) outside i1,C ∩··· ik,C F \ Z, a coherent analytic ′ on D. Then F Hi(U U , ) = Hi(Z, ′) = Hi(D, ′) = (0), i> 0. i1,C ∩··· ik,C F ∼ F ∼ F ∼

4Short for “g´eom´etrie analytique et g´eom´etrie alg´ebrique” 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 283

d) Therefore by Proposition VII.2.2 it follows that i n i H (P , ) = H ( C , ) F ∼ U F and that the refinement maps (for C>C′ > 1): i i i ref ′ : C ( , ) C ( ′ , ) C,C UC F −→ UC F induce an isomorphism on cohomology. e) The key step is to show that the space of sections: (U U ) F i1,C ∩··· ik,C is a topological vector space, in fact, a Fr´echet space in a natural way; and that all restriction maps such as (U U ) (U U ) F i2,C ∩··· ik,C −→ F i1,C ∩··· ik,C are continuous, and that restriction to a relatively compact open subset, as in ′ (U U ) (U ′ U ′ ) (C>C ) F i1,C ∩··· ik,C −→ F i1,C ∩··· ik,C is compact. This last is a generalization of Montel’s theorem that holomorphic functions on holomorphic functions on res: disc z

i i−1 ref +δ i Z ( , ) C ( ′ ) / Z ( ′ , ) UC F ⊕ UC UC F / i (a, b) refC,C′ a + δb is surjective. A standard fact in the theory of Fr´echet spaces is that if α, β : V V 1 −→ 2 are two continuous maps of Fr´echet spaces, with α surjective and β compact, then α+β has closed image of finite codimension. Apply this with α = ref+δ, β = ref and we i − find that Coker(δ)= H ( ′ , ) is finite-dimensional. UC F The second step in the proof is the vanishing theorem — if is coherent analytic, then F i n n for i > 0, m 0, H (P , (m)) = (0). (Here (m) = OPn P ,an(m) as usual.) We ≫ F F def F ⊗ ,an O prove this by induction on n, the complex dimension of the ambient projective space, since it is obvious for n = 0. As in VII.7, we use the L: (m) (m + 1), where L = c X is a § ⊗ F → F i i linear form. This induces exact sequences: P (2.9) 0 / L(m) / (m) / (m + 1) / L(m) / 0 G F F7 H LL oo LL& ooo ′ L(m) 8F Q pp QQQ ppp QQQ 0 p ( 0 n where both L and L are annihilated by L/Xi on P V (Xi). Therefore they are coherent G H n−1 \ analytic sheaves on V (L) = P , and the induction assumption applies to them, i.e., m0(L) ∼ ∃ such that i n H (P , L(m)) ∼= (0) i n G if m m0(L), 1 i n. H (P , L(m)) = (0) ≥ ≤ ≤ H ∼ 284 VIII.APPLICATIONSOFCOHOMOLOGY

The cohomology sequence of (2.9) then gives us: i n ∼ i n L: H (P , (m)) H (P , (m + 1)), m m0(L). ⊗ F −→ F ≥ i n In particular, dim H (P , (m)) = Ni, independent of m for m m0(L). Now fix one linear F ≥ form L and consider the maps: F : Hi( (m (L))) Hi( (m (L)+ d)) ⊗ F 0 −→ F 0 for all homogeneous F of degree d. If Rd is the vector space of such F ’s, then choosing fixed bases of the above cohomology groups, we have a linear map: R / vector space of (N N )-matrices d i × i F / matrix for F . ⊗ ∞ ∞ Let Id be the kernel. It is clear that I = d=1 Id is an ideal in R = d=0 Rd and that 2 dim Rd/Id Ni . Thus the degree of the Hilbert polynomial of R/I is 0, hence the subscheme n ≤ P L V (I) P with structure sheaf Pn /I is 0-dimensional. If V (I) = x1,...,xt , it follows that ⊂ O { } the only associated prime ideals of I can be either e m = ideal of forms F with F (x ) = 0, 1 i t xi i ≤ ≤ or

(X0,...,Xn). By the primary decomposition theorem, it follows that t I (X ,...,X )d0 mdi ⊃ 0 n ∩ xi i\=1 for some d0,...,dt. Now fix linear forms Li such that Li(xi) = 0. Let t F = Lmax(d0,m0(Li)−m0(L)) Ldi . · i Yi=1 On the one hand, we see that F I, hence F on Hi( (m (L))) is 0. But on the other hand, ∈ ⊗ F 0 if m = max(d ,m (L ) m (L)), then F factors: 1 0 0 i − 0 ⊗ i ⊗L ⊗L i H ( (m0(L))) H ( (m0(L)+ m1)) F −−→≈ · · · −−→≈ F ⊗L1 ⊗Lt i . . . H ( (m0(L)+ m1 + di)) −−−→≈ −−→≈ F which is an isomorphism. It follows that Hi( (m (L))) = (0) as required.X F 0 The third step is to show that if is coherent analytic, then (ν) is generated by its sections n F F if ν 0. For each x P , let mx = sheaf of functions zero at x, and consider the exact sequence: ≫ ∈ 0 / m (ν) / (ν) / (ν)/m (ν) / 0 x · F F F x · F (m )(ν). x · F There exists ν such that if ν ν , then H1(m (ν)) = (0), hence H0( (ν)) H0( (ν)/m x ≥ x x ·F F → F x · (ν)) is onto. Let be the cokernel: F G 0 H ( (ν )) C Pn (ν ) 0. F x ⊗ O ,an −→ F x −→ G −→ Then is coherent analytic and G /m = (ν) / m (ν) + Image H0( (ν) = (0). Gx x · Gx ∼ F x x · F x F
2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 285

Therefore by Nakayama’s lemma, x = (0) and by coherency, a neighborhood Ux of x in which G 0 ∃ (0). It follows that (νx) is generated by H ( (νx)) in Ux and hence (ν) is generated by G0 ≡ F Fn F H ( (ν)) in Ux for ν νx too! By compactness P is covered by finitely many of these Ux’s, F ≥ say U ,...,U . Then if ν max(ν ), (ν) is generated everywhere by H0( (ν)). x1 xt ≥ xi F F The fourth step is to show that 0 H ( Pn (m)) = vector space of homogeneous forms of O ,an degree m in X0,...,Xn just as in the algebraic case. We do this by induction first on n, since it is clear for n = 0; and then by a second induction on m, since it is also clear for m = 0, i.e., by the maximum principle the only global analytic functions on the compact space Pn are constants. The induction step uses the exact sequence:

⊗Xn 0 Pn (m 1) Pn (m) n−1 (m) 0 −→ O ,an − −−−→O ,an −→ OP ,an −→ which gives:

Polynomials in Polynomials in Polynomials in Xn 0 / X ,...,X / X ,...,X / X ,...,X / 0  0 n   0 n   0 n−1  of degree m 1 of degree m of degree m  −               0 ⊗Xn 0 0 0 / H ( Pn (m 1)) / H ( Pn (m)) / H ( Pn−1 (m)). O ,an − O ,an O ,an “Chasing” this diagram shows the required assertion for Pn (m). O ,an The fifth step is that every coherent analytic ′ is isomorphic to , some coherent algebraic F Fan . By the third step there is a surjection: F n0 ′(m ) 0 OPn,an −→ F 0 −→ for suitable n0 and m0, hence a surjection: n ′ Pn ( m ) 0 0. O ,an − 0 −→ F −→ Applying the same reasoning to the kernel, we get a presentation: ′ n φ n ′ Pn ( m ) 1 Pn ( m ) 0 0. O ,an − 1 −→ O ,an − 0 −→ F −→ ′ ′ Now φ is given by an (n n )-matrix of sections φ of Pn (m m ), hence by an (n n )- 0 × 1 ij O ,an 1 − 0 0 × 1 matrix F of polynomials of degree m m . Thus the F defines φ, with cokernel : ij 1 − 0 ij F n φ n Pn ( m ) 1 Pn ( m ) 0 0. O − 1 −→ O − 0 −→ F −→ By exactness of the functor , it follows that ′ = . Using the same set-up, we can G 7→ Gan F ∼ Fan also conclude that H0( (m)) = H0( (m)) for m 0. In fact, twist enough so that the H1 F ∼ Fan ≫ of the kernel and image of both φ and φ′ are all (0): then the usual sequences show that the two rows below are exact: 0 n 0 n 0 H ( Pn (m m ) 1 ) / H ( Pn (m m ) 0 ) / H ( (m)) / 0 O − 1 O − 0 F    0 n1 0 n0 0 H ( Pn,an(m m1) ) / H ( Pn,an(m m0) ) / H ( (m)) / 0. O − O − Fan Thus H0( (m)) H0( (m)) is an isomorphism. F → Fan The sixth step is to compare the cohomologies of (m) and (m) for all m. We know that F Fan for m 0, all their cohomology groups are isomorphic and we may assume by induction on n ≫ that we know the result for sheaves on Pn−1. We use a second induction on m, i.e., assuming 286 VIII.APPLICATIONSOFCOHOMOLOGY the result for Hi( (m + 1)), all i, deduce it for Hi( (m)), all i. Use the diagram (2.9) above F F for any linear form L. We get

Hi−1( (m + 1)) / Hi−1( (m)) / Hi( ′ (m)) / Hi( (m + 1)) / Hi( (m)) F HL FL F HL      i−1 i−1 i ′ i i H ( (m + 1)) / H ( L,an(m)) / H ( (m)) / H ( (m + 1)) / H ( L,an(m)) Fan H FL,an Fan H and

Hi−1( ′ (m)) / Hi( (m)) / Hi( (m)) / Hi( ′ (m)) / Hi+1( (m)) FL GL F FL GL      i−1 ′ i i i ′ i+1 H ( (m)) / H ( L,an(m)) / H ( (m)) / H ( (m)) / H ( L,an(m)). FL,an G Fan FL,an G By the 5-lemma, the result for Hi( (m + 1)) and Hi−1( (m + 1)) implies it for Hi( ′ (m)). F F FL And the result for Hi( ′ (m)) and Hi−1( ′ (m)) implies it for Hi( (m)). FL FL F The seventh step is to compare Hom( , ) and Hom( , ). Presenting as before, we F G Fan Gan F get:

n0 ◦φ n1 Hom( , ) = Ker Hom( Pn ( m ) , ) Hom( Pn ( m ) , ) F G ∼ O − 0 G −→ O − 1 G

h ⊗Fij i = Ker H0( (m )n0 ) H0( (m )n1 ) ∼ G 0 −−−→ G 1   ⊗Fij = Ker H0( (m )n0 ) H0( (m )n1 ) ∼ Gan 0 −−−→ Gan 1   n0 ◦φ n1 = Ker Hom( Pn ( m ) , ) Hom( Pn ( m ) , ) ∼ O ,an − 0 Gan −→ O ,an − 1 Gan = Hom(h , ). i ∼ Fan Gan 

Corollary 2.10. A new proof of Chow’s theorem ( Part I [76, (4.6)]): If X Pn is a closed ⊂ analytic subset, then X is a closed algebraic subset.

n Proof. If X P is a closed analytic subset, then X Pn,an is a coherent analytic sheaf, ⊂ I ⊂ O so = for some coherent algebraic Pn . So X = Supp Pn / = Supp Pn / is IX Jan J ⊂ O O ,an IX O J a closed algebraic subset. 

Corollary 2.11. If X1 and X2 are two complete verieties over C, then every holomorphic map f : X X is algebraic, i.e., a morphism. 1 → 2 Proof. Apply Chow’s lemma (Theorem II.6.3) to find proper birational π : X′ X and 1 1 → 1 π : X′ X with X′ projective. Let Γ X X be the graph of f. Then (π π )−1Γ 2 2 → 2 i ⊂ 1 × 2 1 × 2 ⊂ X′ X′ is a closed analytic subset of projective space, hence is algebraic by Chow’s theorem. 1 × 2 Since π π is proper, Γ = (π π )[(π π )−1Γ] is also a closed algebraic set. In order to 1 × 2 1 × 2 1 × 2 see that it is the graph of a morphism, we must check that p : Γ X is an isomorphism. This 1 → 1 follows from:

Lemma 2.12. Let f : X Y be a bijective morphism of varieties. If f is an analytic → isomorphism, then f is an algebraic isomorphism.

Proof of Lemma 2.12. Note that f is certainly birational since #f −1(y) = 1 for all y Y . ∈ Let x X, y = f(x). We must show that f ∗ : is surjective. The local rings of ∈ OY,y → OX,x 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 287 analytic functions on X and Y at x and y and the formal completions of these rings are related by: f ∗ / OY,y OX,x  f ∗  ( ) an / ( ) OY,an y OX,an x   f ∗ / . OY,y OX,x Now f ∗ is an isomorphism by assumption. If a , write a = b/c, b, c using the fact an b ∈ OX,xb ∈ OY,y that f is birational. Then f ∗ isomorphism = a′ ( ) with b = c a′ an ⇒∃ ∈ OY,an y · = b c . ⇒ ∈ OY,y ∩ · OY,y But for any ideal a , a = a , so b c , i.e., a .  ⊂ OY,y OY,y ∩ · OY,y ∈ · ObY,y ∈ OY,y  b Corollary 2.13 (Projective case of Riemann’s Existence Theorem). Let X be a complex projective variety5. Let Y be a compact topological space and π : Y (X in the classical topology) e −→ a covering map (since Y is compact, this amounts merely to requiring that π is a local homeo- e morphism). Then there is ae unique scheme Y and ´etale proper morphism π : Y X such that → there exists a homeomorphisme ρ: e ≈ / (Y in the classical topology) Y QQ ρ QQQ gg QQQ ggggg QQQ ggggg e QQ( gs ggg (X in the classical topology).

Proof. Given Y , note first that since π is a local homeomorphism we can put a unique analytic structure on it making π into a local analytic isomorphism. Let = π∗( Y ): this is a B −1 O sheaf of X,an-algebras.e Now every x X hase a neighborhood U such that π (U)e= disjoint O ∈ ∼ union of l copies of U; hence e = l as a sheaf of algebras. In particular,e is a B|U ∼ i=1 OX,an B coherent analytic sheaf of -modules. Recall that we can identify sheavese of -modules OX,an L OX,an with sheaves of Pn -modules, (0) outside X and killed by multiplication by . Therefore by O ,an IX the fundamental GAGA Theorem 2.8, = B for some algebraic coherent sheaf of -modules B ∼ an OX B. Multiplication in defines an -module homomorphism B OX,an µ: , B⊗OX,an B −→B hence by the GAGA Theorem 2.8 again this is induced by some -module homomorphism: OX ν : B B B. ⊗OX −→ The associative law for µ implies it for ν and so this makes B into a sheaf of -algebras. OX The unit in similarly gives a unit in B. We now define Y = pec (B), with π : Y X the B S X → canonical map (proper since B is coherent by Proposition II.6.5). How are Y and Y related? We have e 5The theorem is true in fact for any variety X and finite-sheeted covering π∗ : Y ∗ → X, but this is harder, cf. SGA4 [7, Theorem 4.3, Expos´e11], where Artin deduces the general case from Grauert-Remmert [40]; or SGA1 [4, Expos´eXII, Th´eor`eme 5.1, p. 332], where Grothendieck deduces it from Hironaka’s resolution theorems [55]. 288 VIII.APPLICATIONSOFCOHOMOLOGY

i) a continuous map ζ : Y (closed points of X) → ii) a map backwards covering ζ: e ζ∗ : B (sheaf of continuous C-valued functions on Y ) −→ such that x Y and f B , ∀ ∈ ∀ ∈ ζ(x) e ζ∗f(x)= f(ζ(x)), e defined as the composite B(U) (U) (π−1U) [continuous functions on π−1U]. −→ B −→ OY,an −→ e These induce a continuous map: e e η : Y (closed points of Y ) −→ by e ∗ f Bζ(x) ζ f(x) = 0 η(x) = point corresponding to maximal ideal { ∈ | } m B X,ζ(x) · ζ(x) via the correspondence π−1(ζ(x)) ⇆ maximal ideals in B /m B . ζ(x) ζ(x) · ζ(x) Now η has the property

2.14. f (V ), the composite map ∀ ∈ OY η f η−1(V ) (closed points of V ) C −→ −→ is a continuous function on η−1(V ) (in the classical topology).

But a basis for open sets in the classical topology on Y is given by finite intersections of the sets: V Zariski open, f (V ), let ∈ OY W = x V x closed and f(x) <ǫ . f,ǫ { ∈ | | | } −1 Because of (2.14), η (Wf,ǫ) is open in Y , i.e., η is a continuous map from Y to (Y in the classical topology). Now in fact η is bijective too. In fact, if U X is a classical open so that ⊂ π−1(U) = (disjoint union of n copies of U)e and = l , then for alle x U, B|U i=1 OX,an|U ∈ l L x/mx x = C B ·B ∼ Mi=1 and the correspondence between points of π−1(x) and maximal ideals of /m given by Bx x ·Bx y f f(y) = 0 is bijective. On the other hand, since = B ( ) , it follows that 7→ { | } Bx ∼ x ⊗Ox,x OX,an x /m = B /m B . Thus η is a continuous bijective map from a compact space Y to (Y Bx x ·Bx ∼ x x · x in the classical topology). Thus η is a homeomorphism. Finally is a free ( ) -module, Bx OX,an x hence it follows that B is a free -module: Hence π : Y X is a flat morphism. Ande the x OX,x → scheme-theoretic fibre is: π−1(x) = Spec B /m B x x · x = Spec /m ∼ Bx x ·Bx l ∼= Spec C = l reduced points. Mi=1 Thus π is ´etale. 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 289

As for the uniqueness of Y , it is a consequence of the stronger result: say

Y1 Y2 CC { π1 C! {} {π2 X are two ´etale proper morphisms. Then any map continuous in the classical topology:

f (Y in the classical topology) (Y in the classical topology) 1 −→ 2 with π f = π is a morphism. To see this, note that π are local analytic isomorphisms, hence 2 ◦ 1 i f is analytic, hence by Corollary 2.11, f is a morphism. 

This Corollary 2.13 implies profound connections between topology and field theory. To alg explain these, we must first define the algebraic fundamental group π1 (X) for any normal noetherian scheme X6. We have seen in V.6 that morphisms: §

π : Y X −→ Y normal irreducible (a) π proper, surjective, π−1(x) finite for all x, π generically smooth are uniquely determined by the function field extension R(Y ) R(X), which is necessarily ⊃ separable; and that conversely, given any finite separable K R(X), we obtain such a π by ⊃ setting Y = the normalization of X in K. In particular, suppose we start with a morphism:

π : Y X −→ (b) Y connected π proper and ´etale.

Then Y is smooth over a normal X, hence is normal by Proposition V.5.5. Being connected, Y is also irreducible. Thus Y = normalization of X in R(Y ). Now choose a specific separable algebraic closure R(X) of R(X) and let

G = Gal(R(X)/R(X)), the Galois group ∼= lim Aut(K/R(X)) ←−K where R(X) K R(X), K normal over R(X) with [K : R(X)] < + . ⊂ ⊂ ∞

6Normality is not necessary and noetherian can be weakened. For a discussion of the results below in more general case, see SGA1 [4, Expos´es V and XII]. 290 VIII.APPLICATIONSOFCOHOMOLOGY

As usual, G, being an inverse limit of finite groups, has a natural structure of compact, totally disconnected topological group. One checks easily7 that there is an intermediate field:

R(X) R^(X) R(X) ⊂ ⊂ such that for all K R(X), finite over R(X): ⊂ the normalization YK of X K R^(X). in K is ´etale over X ⇐⇒ ⊂  Note that because of its defining property, R^(X) is invariant under all automorphisms of R(X), i.e., it is normal over R(X) and its Galois group over R(X) is a quotient G/N of G. By Galois theory, the closed subgroups of finite index in G are in one-to-one correncepondence with the subfields K R(X) finite over R(X). So the closed subgroups of finite index in G/N are in ⊂ one-to-one correspondence with the subfields K R^(X) finite over R(X), hence with the set ⊂ of schemes YK ´etale over X. It is therefore reasonable to call G/N the algebraic fundamental alg group of X, or π1 (X): alg ^ (2.15) π1 (X) = Gal(R(X)/R(X)).

Next in the complex projective case again choose a universal covering space Ω of X in the classical topology. Then the topological fundamental group is:

top π1 (X) = group of homeomorphisms of Ω over X, and its subgroups of finite index are in one-to-one correspondence with the compact covering spaces Y dominated by Ω: π Ω Y X, e −→ −→e which give, by algebraization (Corollary 2.13), connected normal complete varieties Y , ´etale over e X. This must simply force a connection between the two groups and, in fact, it implies this:

7This follows from two simple facts:

a) K1 ⊂ K2, YK2 ´etale over X =⇒ YK1 ´etale over X,

b) YK1 and YK2 ´etale over X =⇒ YK1·K2 ´etale over X. To prove (a), note that we have a diagram

YK2 −→ YK1 −→ X.

Now YK2 ´etale over X =⇒ ΩYK2 /X = (0) =⇒ ΩYK2 /YK1 = (0) =⇒ YK2 ´etale over YK1 by Criterion 4.1 for smoothness in §V.4. In particular, YK2 is flat over YK1 , hence if y2 ∈ YK2 has images y1 and x in YK1 and X, then Oy2 is flat over Oy1 , hence mx ·Oy2 ∩Oy1 = mx ·Oy1 . Thus ∼ Oy1 /mx ·Oy1 ⊂Oy2 /mx ·Oy2 = product of separable field extensions of k(x) hence Oy1 /mx ·Oy1 is also a product of separable field extensions of k(x). This shows ΩYK1 /X ⊗ k(y1) = (0), hence by Nakayama’s lemma, ΩYK1 /X = (0) near y1, hence by Criterion 4.1 for smoothness in §V.4, YK1 is ´etale over X at y1.

To prove (b), note that YK1 ×X YK2 will be ´etale over X, hence normal. We get a morphism

φ YK1·K2 −→ YK1 ×X YK2 and if Z = component of YK1 ×X YK2 containing Image φ, then φ: YK1·K2 → Z is birational. Since Z is normal and the fibres of φ are finite, φ is an isomorphism by Zariski’s Main Theorem in §V.6. 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 291

n Theorem 2.16. Let X be a normal subvariety of PC and let πtop(X) = lim πtop(X)/H, over all H πtop(X) of finite index 1 1 ⊂ 1 ←−H top b = “pro-finite completion” of π1 (X). top alg Then π1 (X) and π1 (X) are isomorphic as topological groups, the isomorphism being canonical up to an inner automorphism. b Proof. Choose a sequence H of normal subgroups of πtop of finite index, with H { ν } 1 ν+1 ⊂ H , such that for any H of finite index, H H for some ν. Let πtop/H = G and let H ν ν ⊂ 1 ν ν ν define Y X. Then ν → πtop = lim G e 1 ∼ ν ←−ν and bGν ∼= group of homeomorphisms of Yν over X. Algebraize Y to a scheme Y ´etale over X by Corollary 2.13. Then the map Y Y comes ν ν e ν+1 → ν from a morphism Yν+1 Yν and we get a tower of function field extensions: e → e e R(Yν+1) R(Yν) R(X). ··· ←− ←− ←−··· ←− Note that AutR(X)(R(Yν )) ∼= AutX (Yν) ∼= AutX (Yν ) ∼= Gν and since #Gν = degree of the covering (Yν X) = [R(Yν ) : R(X)], this shows that R(Yν) is → e a normal extension of (X). The fact that Y Y is ´etale over X shows that (Y ) is R ν ∼= R(Yν ) R ν e isomorphic to a subfield of R^(X). Now choose an R(X)-isomorphism: ∞ φ: R(Yν) R^(X). −→ ν[=1 It is easy to see that φ is surjective by going backwards from an ´etale Y X to a topological K → covering Y X and dominating this by Ω. So we get the sought for isomorphism: K → πtop = lim G e 1 ∼ ν ←−ν b ∼= lim Gal(R(Yν )/R(X)) ←−ν ^ ∼= Gal(R(X)/R(X)) alg ∼= π1 . The only choice here is of φ and varying φ changes the above isomorphism by an inner auto- morphism. 

As a final topic I would like to discuss Grothendieck’s formal analog of Serre’s fundamental theorem. His result is this: Let R = noetherian ring, complete in the topology defined by the powers of an ideal I. Let X Spec R be a proper morphism. −→ Consider the schemes: X = X Spec R/In+1 n ×Spec R 292 VIII.APPLICATIONSOFCOHOMOLOGY i.e., X ⊂ ⊂ X ⊂ ⊂ X 0 · · · n ······    Spec R/I ⊂ ⊂ Spec R/In+1 ⊂ ⊂ Spec R · · · ······ Define: a formal coherent sheaf on X is a set of coherent sheaves on X plus isomor- F Fn n phisms: = . Fn−1 ∼ Fn ⊗OXn OXn−1 Note that every coherent on X induces a formal by letting F Ffor = . Ffor,n ∼ F ⊗OX OXn Then: Theorem 2.17 (Grothendieck). (Fundamental “GFGA”8 comparison theorem) i) For every coherent algebraic on X and every i, F Hi(X, ) = lim Hi(X , ) F ∼ n Fn ←−n where = . Fn F ⊗OX OXn ii) The categories of formal and algebraic coherent sheaves are equivalent, i.e., every formal ′ is isomorphic to , some , and F Ffor F Hom ( , ) = Formal Hom ( , ). OX F G ∼ OX Ffor Gfor The result for H0 is essentially due to Zariski, whose famous [108] proving this and apply- ing it to prove the connectedness theorem (see (V.6.3) Fundamental theorem of “holomorphic functions”) started this whole development. A complete proof of Theorem 2.17 can be found in EGA [1, Chapter 3, 4 and 5].9 Here we will prove only the special case: §§ R complete local, I = maximal ideal, k = R/I X projective over Spec R m (which suffices for most applications). If X is projective over Spec R, we can embed X in PR m for some m, and extend all sheaves from X to PR by (0): thus it suffices to prove Theorem 2.17 m for X = PR . Before beginning the proof, we need elementary results on the category of coherent formal sheaves. For details, we refer the reader to EGA [1, Chapter 0, 7 and Chapter 1, 10]; however § § none of these facts are very difficult and the reader should be able to supply proofs.

n+1 2.18. If A is a noetherian ring, complete in its I-adic topology and Un = Spec A/I , then there is an equivalence of categories between: a) sets of coherent sheaves on U plus isomorphism Fn n = Fn−1 ∼ Fn ⊗OUn OUn−1 b) finitely generated A-modules M given by: M = lim Γ(U , ) n Fn ←−n = M/I^n+1M. Fn 8Short for “g´eom´etrie formelle et g´eom´etrie alg´ebrique”. 9(Added in publication) Illusie’s account in FAG [3, Chapter 8] “provides an introduction, explaining the proofs of the key theorems, discussing typical applications, and updating when necessary.” 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 293

In particular, Category (a) is abelian: [but kernel is not the usual sheaf-theoretic kernel because M M does not imply M /In+1M M /In+1M !]. 1 ⊂ 2 1 1 ⊂ 2 2 2.19. Given A as above, and f A, then ∈ A = lim A /In A = lim(A/InA) f f · f f ←−n ←−n is flat over A.

Corollary 2.20. The category of coherent formal sheaves on a scheme X, proper {Fn} over Spec R (R as above) is abelian with Coker[ ]= Coker( ) {Fn} −→ {Gn} { Fn −→ Gn }n=0,1,... but Ker[ ]= {Fn} −→ {Gn} {Hn} where for each affine U X: ⊂ (U)= (U)/In+1 (U) Hn H H (U) = Ker lim (U) lim (U) . H Fn −→ Gn "←−n ←−n # Proof of Corollary 2.20. Applying (2.18) with A = lim (U) we construct kernels n OXn of for each affine U as described above.←− Use (2.19) to check that on each {Fn|U } → {Gn|U } distinguished open U U, the restriction of the kernel on U is the kernel on U .  f ⊂ f 2.21. If A is any ring and 0 K L M 0 −→ n −→ n −→ n −→ are exact sequences of A-modules for each n 0 fitting into an inverse system ≥ 0 / Kn+1 / Ln+1 / Mn+1 / 0    0 / Kn / Ln / Mn / 0 and if for each n, the decreasing set of submodules Image(K K ) of K is stationary for n+k → n n k large enough, then 0 lim K lim L lim M 0 −→ n −→ n −→ n −→ ←−n ←−n ←−n is exact.

Proof of Theorem 2.17. We now begin the proof of GFGA. To start off, say = n m F {F } is a coherent formal sheaf on PR . Introduce ∞ gr R = In/In+1 : a finitely generated graded k-algebra Mn=0 S = Spec(gr R) : an affine scheme of finite type over k ∞ n m gr = I n : a quasi-coherent sheaf on P . F · F k n=0 M ∞ n n+1 Note that gr is in fact a sheaf of ( I /I ) Pm -modules and since F n=0 ⊗ O k InL/In+1 In ⊗k F0 −→ · Fn 294 VIII.APPLICATIONSOFCOHOMOLOGY

∞ n n+1 is surjective, gr is finitely generated as a sheaf of ( I /I ) Pm -modules. In other F n=0 ⊗ O k words, we can form a coherent sheaf gr on F L ∞ n n+1 m pec g I /I Pm = P . S S ⊗ O k S n=0 ! ! M Moreover, ∞ q m q m n H (P , gr )= H (P ,I n). S F k · F n=0 M This same holds after twisting by theg standard invertible sheaf (l), hence: F O ∞ q m q m n H (P , gr (l)) = H (P ,I n(l)). S F k · F n=0 M But since gr R is a noetherian ring,g the left hand side is (0) if l l (for some l ) and q 1. ≥ 0 0 ≥ Thus: q m n H (P ,I n(l)) = (0), if q 1, n 0, l l0. k · F ≥ ≥ ≥ Now look at the exact sequences: 0 In (l) (l) (l) 0. −→ · Fn −→ Fn −→ Fn−1 −→ It follows from the cohomology sequences by induction on n that: q m H (P , n(l)) = (0), if q 1, R F ≥ 0 m 0 m and H (P , n(l)) H (P , n−1(l)) surjective for all n 0, l l0. R F −→ R F ≥ ≥ The next step (like the third step of the GAGA Theorem 2.8) is that for some l (l) is 1 {Fn } generated by its sections for all l l : i.e., there is a set of surjections: ≥ 1 N n+1 N (2.22) Pm /I Pm n(l) 0 O R · O R −→ F −→ commuting with restriction from n + 1 to n. To see this, take l l so that (l) is generated 1 ≥ 0 F0 by its sections for l l . This means there is a surjection: ≥ 1 N Pm 0(l) 0. O k −→ F −→ By (2.21), this lifts successively to compatible surjections as in the third step of the GAGA Theorem 2.8. In other words, we have a surjection of formal coherent sheaves: N (2.23) Pm ( l)for n . O R − −→ {F }

Next, as in the fourth step of the GAGA Theorem 2.8, we prove 0 n+1 lim H ( Pm (l)/I Pm (l)) = (R-module of homogeneous forms of degree l) O R · O R ∼ ←−n 0 = H ( Pm (l)). ∼ O R n+1 m This is obvious since Pm (l)/I Pm (l) is just the structure sheaf of P , where Rn = O R · O R Rn R/In+1 R. Then the fifth step follows GAGA in Theorem 2.8 precisely: given , we take · {Fn} the kernel of Corollary 2.20 and repeat the construction, obtaining a presentation:

N1 φ N0 Pm ( l1)for Pm ( l0)for n 0. O R − −→ O R − −→ {F } −→ By the fourth step, φ is given by a matrix of homogeneous forms, hence we can form the algebraic coherent sheaf: N1 N0 = Coker φ: Pm ( l1) Pm ( l0) F O R − −→ O R − h i 2. COMPARISON OF ALGEBRAIC WITH ANALYTIC COHOMOLOGY 295 and it follows immediately that = /In+1 , i.e., = . Fn ∼ F · F {Fn} ∼ Ffor The rest of the proof follows that of GAGA in Theorem 2.8 precisely with Hq( ) replaced Fan by lim Hq( /In+1 ), once one checks that n F · F ←− lim Hq( /In ) F 7−→ F · F ←−n is a “cohomological δ-functor” of coherent algebraic sheaves , i.e., if 0 0 is F →F→G→H→ exact, then one has a long exact sequence

0 lim H0( /In ) lim H0( /In ) lim H0( /In ) −→ F · F −→ G · G −→ H ·H ←−n ←−n ←−n δ lim H1( /In ) . −→ F · F −→······ ←−n But this follows by looking at the exact sequences:

0 /( In ) /In /In 0. −→ F F ∩ · G −→ G ·G −→H ·H −→ By (2.21), the cohomology groups lim Hq( /( In )) F F ∩ · G ←−n lim Hq( /In ) G · G ←−n lim Hq( /In ) H ·H ←−n fit into a long exact sequence (since for each n, the n-th terms of these limits are finitely generated (R/In R)-modules, hence are of finite length). But by the Artin-Rees lemma (Zariski-Samuel · [109, vol. II, Chapter VIII, 2, Theorem 4′, p. 255]), the sequence of subsheaves In of § F∩ · G F is cofinal with the sequence of subsheaves In : in fact l such that for all n l: · F ∃ ≥ In In = In−l ( Il ) In−l . ·F⊂F∩ · G · F ∩ · G ⊂ · F Therefore lim Hq( /( In )) = lim Hq( /In ). F F ∩ · G F · F ←−n ←−n 

Corollary 2.24. Every formal closed subscheme Yfor of X (i.e., the set of closed subschemes Y X such that Y = Y X ) is induced by a unique closed subscheme Y of X (i.e., n ⊂ n n−1 n ×Xn n−1 Y = Y X ). n ×X n Corollary 2.25. Every formal ´etale covering π : Y X (i.e., a set of coverings π : Y for → n n → X plus isomorphisms Y = Y X ) is induced by a unique ´etale covering π : Y X n n−1 ∼ n ×Xn n−1 → (i.e., Y = Y X ). n ∼ ×X n

In fact, it turns out that an ´etale covering π0 : Y0 X0 already defines uniquely the whole alg alg→ 10 formal covering, so that it follows that π1 (X0) ∼= π1 (X): See Corollary 5.9 below. Another remarkable fact is that the GAGA and GFGA comparison theorems are closer than it would

10(Added in publication) See §5 for other applications of GFGA in connection with deformations (e.g., Theorem 5.5 on algebraization). See also Illusie’s account in FAG [3, Chapter 8]. 296 VIII.APPLICATIONSOFCOHOMOLOGY seem at first. In fact, if R is a complete discrete valuation ring with absolute value , note m | | that for AR :

0 m n+1 lim H (A n+1 , Am ) = lim(R/I )[X1,...,Xm] (R/I ) O ←−n ←−n α ∼= ring of “convergent power series” cαX where c R and c 0 as α X . α ∈ | α|→ | |→∞ This is the basis of a connection between the above formal geometry and a so-called “rigid” or “global” analytic geometry over the quotient field K of R. For an introduction to this, see Tate [101] and ????? .

Exercise . (1) Let X be a normal irreducible noetherian scheme and let L R(X) be a separable ⊃ Galois extension such that the normalization Y of X in L is ´etale over X. Let π : Y L L → X be the canonical morphism. Let G = Gal(L/R(X)). Then G acts on YL over X: show that for all y Y , if x = π(y), then: ∈ L a) G acts transitively on π−1(x). b) If Gy G is the subgroup leaving y fixed, then Gy acts naturally on k(y) leaving ⊂ k(x) fixed. c) k(y) is Galois over k(x) and, via the action in (b),

≈ Gy Gal(k(y)/k(x)). −→ n times

[Hint: Let n = [L: R(X)]. Using the fact that L R L = L L and that ⊗ (X) ∼ ×···× Y Y is normal, prove that Y Y = disjoint union of n copies of Y . Prove L ×X L L ×X L z }| { L that if G acts on Y Y non-trivially on the first factor but trivially on the second, L ×X L then it permutes these components simply transitively.] (2) Note that the first part of the GFGA theorem (Theorem 2.17) would be trivial if the following were true: X a scheme over Spec A a quasi-coherent sheaf of -modules F OX B an A-algebra.

Then for all i, the canonical map

Hi(X, ) B Hi(X Spec B, B) F ⊗A −→ ×Spec A F ⊗A is an isomorphism. Show that if B is flat over A, this is correct. (3) Using (2), deduce the more elementary form of GFGA:

f : Z X proper, X noetherian −→ a coherent sheaf of -modules. F OX Then for all i, and for all x X, ∈ lim Rif ( ) / mn Rif ( ) = lim Hi(f −1(x), /mn ). ∗ F x x · ∗ F x ∼ F x · F ←−n ←−n
3. DE RHAM COHOMOLOGY 297

3. De Rham cohomology

As in 2 we wish to work in this section only with varieties X over C. For any such X, we § have the topological space (X in the classical topology) and for any group G, we can consider the “constant sheaf GX ” on this: functions f : U G, constant on each G (U)= → X connected component of U.   It is a standard fact from algebraic topology (cf. for instance, Spanier [98, Chapter 6, 9]; or § Warner [104]) that if a topological space Y is nice enough — e.g., if it is a finite simplicial i complex — then the sheaf cohomology H (Y, GY ) and the singular cohomology computed by G- valued cochains on all singular simplices of Y as in Part I [76, 5C] are canonically isomorphic. § One may call these the classical cohomology groups of Y . I would like in this part to indicate the basic connection between these groups for G = C, and the coherent sheaf cohomology studied above. This connection is given by the ideas of De Rham already mentioned in Part I [76, 5C]. § We begin with a completely general definition: let f : X Y be a morphism of schemes. → We have defined the K¨ahler differentials ΩX/Y in Chapter V. We now go further and set:

k k ΩX/Y = (ΩX/Y ), i.e., the sheafification of the pre-sheaf def ^ k U of the (U)-module Ω (U). 7→ OX X/Y One checks by the methods used^ above (e.g., Ex. ???? ) that this is quasi-coherent and that

k Ωk (U)= over (U) of Ω (U) for U affine. X/Y OX X/Y In effect, this means that for U affine^ in X lying over V affine in Y : Ωk (U) =free (U)-module on generators dg dg , X/Y OX 1 ∧···∧ k (g (U)), modulo i ∈ OX a) d(g + g′ ) dg = dg dg + dg′ dg 1 1 ∧···∧ k 1 ∧···∧ k 1 ∧···∧ k b) d(g g′ ) dg = g dg′ dg + g′ dg dg 1 1 ∧···∧ k 1 1 ∧···∧ k 1 1 ∧···∧ k c) dg dg = sgn(ǫ) dg dg (ǫ =permutation) ǫ1 ∧···∧ ǫk · 1 ∧···∧ k d) dg dg dg =0 if g = g 1 ∧ 2 ∧···∧ k 1 2 d) dg dg =0 if g (V ). 1 ∧···∧ k 1 ∈ OY The derivation d: Ω extends to maps: OX → X/Y d:Ωk Ωk+1 (not -linear) X/Y −→ X/Y OX given on affine U by: d(fdg dg )= df dg dg , f,g (U). 1 ∧···∧ k ∧ 1 ∧···∧ k i ∈ OX (Check that this is compatible with relations (a)–(e) on Ωk and Ωk+1, hence d is well-defined.) It follows immediately from the definition that d2 = 0, i.e.,  Ω : 0 d Ω1 d Ω2 d X/Y −→ OX −→ X/Y −→ X/Y −→ · · · i  is a complex. Therefore as in VII.3 we may define the hypercohomology H (X, ΩX/Y ) of this § i complex, which is known as the De Rham cohomology HDR(X/Y ) of X over Y . Grothendieck 298 VIII.APPLICATIONSOFCOHOMOLOGY

[41], putting together more subtly earlier ideas of Serre, Atiyah and Hodge with Hironaka’s resolution theorems [55] has proven the very beautiful:

Theorem 3.1 (De Rham comparison theorem). If X is a variety smooth (but not necessarily proper) over C, then there is a canonical isomorphism: i i HDR(X/C) ∼= H ((X in the classical topology), CX ). We will only prove this for projective X referring the reader to Grothendieck’s elegant paper [41] for the general case. Combining Theorem 3.1 with the spectral sequence of hypercohomology gives:

Corollary 3.2. There is a spectral sequence with

pq q p E1 = H (X, ΩX/C) pq and d being induced by d:Ωp Ωp+1 abutting to Hν((X in the classical topology), C). In 1 → particular, if X is affine, then closed ν-forms { } = Hν((X in the classical topology), C). exact ν-forms ∼ { } To prove the theorem in the projective case, we must simply combine the GAGA comparison theorem (Theorem 2.8) with the so-called Poincar´elemma on analytic differentials. First we recall the basic facts about analytic differentials. If X is an n-dimensional complex manifold, then the tangent bundle TX has a structure of a rank n complex analytic vector bundle over X, i.e.,

TX = (P, D) P X, D : ( X,an)P C a derivation over C centered at P ∼ { | ∈ O → } (cf. Part I [76, SS1A, 5C, 6B]). Thus if U X is an open set with analytic coordinates z ,...,z , ⊂ 1 n then the inverse image of U in TX has coordinates (P, D) (z (P ),...,z (P ), D(z ),...,D(z )) 7−→ 1 n 1 n p under which it is analytically isomorphic to U Cn. We then define the sheaves Ω of × X,an holomorphic p-forms by: p Ωp (U)= holomorphic sections over U of the complex vector bundle (T ∗ ) . X,an { X } (Here E∗ = Hom(E, C) is the dual bundle.) Locally such a section ω is written^ as usual by an expression ω = c dz dz , c (U), i1,...,ip i1 ∧···∧ ip i1,...,ip ∈ OX,an 1≤i1

d:Ωp Ωp+1 X,an −→ X,an given by p+1 ∂c k+1 i1,...,ik,...,ip+1 dω = ( 1) b dzi1 dzip+1 . − ∂zik ∧···∧ 1≤i1<···Xcommutative algebra in which d is a 7→ ∧ p X,an derivation. L 3. DE RHAM COHOMOLOGY 299

Lemma 3.3 (Poincar´e’s lemma). The sequence of sheaves: d d d d 0 Ω1 Ω2 Ωn 0 −→ OX,an −→ X,an −→ X,an −→ · · · −→ X,an −→ is exact, except at the 0-th place where Ker(d: Ω1 ) is the sheaf of constant functions OX,an → X,an CX . For an elementary proof of this see Hartshorne [51, Remark after Proposition (7.1), p. 54]. (See also Wells [105, Chapter II, 2, Example 2.13, p. 49] as well as the proof of the Delbeault § Lemma in Gunning-Rossi [48, Chapter I, D, 3.Theorem, p. 27].) Now if X is a variety smooth § over C, an essential point to check is that the general functor an of 2 takes the K¨ahler p F 7→ Fp § p-forms ΩX/C to the above-defined sheaf of holomorphic p-forms ΩX,an. This is virtually a tautology but to tie things together properly, we can proceed like this. For the sake of this p p argument, we write ΩX,alg for K¨ahler differentials on the scheme X, parallel to ΩX,an defined above: a) For all U X affine, Ωp (U) is the universal skew-commutative (U)-algebra ⊂ p X,alg OX with derivation (i.e., the free algebra on elements df, f X (U), modulo the standard Lp ∈ O identities); since p ΩX,an(U) is another skew-commutative algebra with derivation over (U) (via the inclusion (U) (U)), there is a unique collection of OX L OX ⊂ OX,an maps: Ωp (U) Ωp (U) X,alg −→ X,an commuting with and d. ∧ b) From a general sheaf theory argument, such a collection of maps factors through a map of sheaves of -modules (on X in the classical topology): OX,an p p Ω ΩX,an. X,alg an −→   c) If z ,...,z m induce a basis of m /m2 , then we have: 1 n ∈ X,x X,x X,x n (Ω ) = dz X,alg x ∼ OX,x · i Mi=1 hence p Ω = X,x dzi1 dzip , X,alg x ∼ O · ∧···∧   1≤ii

b) a map of the spectral sequences abutting to these too:

pq / pq algebraic E1 analytic E1

q p q p H (X in the Zariski topology, ΩX/C) H (X in the classical topology, ΩX,an).

pq But by the GAGA comparison theorem (Theorem 2.8), the map on E1 ’s is an isomorphism. Now quite generally, if Epq = Eν 2 ⇒ Epq = Eν 2 ⇒ are two spectral sequences, and e e φp,q : Epq Epq 2 −→ 2 φν : Eν Eν −→ e are homomorphisms “compatible with the spectral sequences”, i.e., commuting with the d’s, tak- e l ν l ν pq p p+q p+1 p+q ing F (E ) into F (E ) and commuting with the isomorphisms of E∞ with F (E )/F (E ), then it follows immediately that e φp,q isomorphisms, all p,q = φν isomorphisms, all ν. ⇒ In our case, this means that the map in (a) is an isomorphism. ν  Now compute H (X in the classical topology, ΩX,an) by the second spectral sequence of hy- percohomology (cf. (VII.3.11)). Since X in its classical topology is paracompact Hausdorff, we get (cf. VII.1) § Ker d:Ωq Ωq+1 an an  Hp X in the classical topology, sheaf → = Hν(X, Ω ).   q−1 q  ⇒ X,an Image d:Ωan Ωan →    By Poincar´e’s lemma (Lemma 3.3), all but one of these sheaves are (0) and the spectral sequence degenerates to an isomorphism:

ν ν  H (X in the classical topology, CX ) ∼= H (X in the classical topology, ΩX,an). This proves Theorem 3.1 in the projective case. In the projective case and more generally for any complete variety X, the spectral sequence of Corollary 3.2:

pq p E = Hq(X, Ω )= Hν(X in the classical topology, C) 1 X/C ⇒ simplifies quite remarkably. In fact the Theory of Hodge implies:

pq Fact. I: All dr ’s are 0.

This implies that

q p pq p p+q p+1 p+q p+q H (X, ΩX/C) ∼= E∞ ∼= p-th graded piece: F (H )/F (H ) of H (X, C). Note that Hp+q(X, C) = Hp+q(X, Z) C, hence there is a natural complex conjugation x x ∼ ⊗ 7→ on Hp+q(X, C).

Fact. II: In Hp+q(X, C), F q+1(Hp+q) is a complement to the subspace F p(Hp+q). 4. CHARACTERISTIC p PHENOMENA 301

This implies that Hp+q splits canonically into a direct sum: Hν (X, C)= Hp,q p+q=ν p,qM≥0 such that a) Hq,p = Hp,q. p p+q p′,q′ b) F (H )= p′≥p H . Combining both facts,L Hp,q Hq(X, Ωp ) ∼= X/C hence (3.4) Hν(X, ) Hq(X, Ωp ). C ∼= X/C p+q=ν M

Fact. III: If we calculate Hν(X, C) by C∞ differential forms, then set of cohomology classes representable by forms ω  of type (p,q), i.e., in local coordinates z1,...,zn,  Hp,q = . ∼  ω = ci1,...,ip,j1,...,jq dzi1 dzip dzj1 dzjq   ∧···∧ ∧ ∧···∧   1≤i1<··· 0, then the De Rham groups ν  H (X, ΩX/k) are finite-dimensional k-vector spaces which usually behave quite like their counterparts in char- acteristic 0 and are “reasonable” candidates for the cohomology of X with coefficients in k12. 1 0 1 For instance, if X is a complete non-singular curve of genus g, then dim H ( X ) = dim H (Ω )= 1  O g, dim H (ΩX ) = 2g in all characteristics. However the De Rham groups have a much richer structure in characteristic p even in the case of curves. The simplest examples of this are the cohomology operations: F : Hn(X, ) Hn(X, ), OX −→ OX X = any scheme in which p 0 · OX ≡ given by p F ( ai ,...,i )= a { 0 n } { i0,...,in } 11(Added in publication) There have been considerable advances, since the original manuscript was written. See the footnote at the end of this section. 12 There are some cases where their dimension is larger than the expected n-th Betti number Bn and there are also cases where the spectral sequence q p n  H (Ω )=⇒ H (Ω ) does not degenerate: cf. ???, ???, ??? . This is apparently connected with the presence of p-torsion on X. And if X is affine instead of complete, these groups are not even finite-dimensional. 302 VIII.APPLICATIONSOFCOHOMOLOGY on the cocycle level. Note that if X is a scheme over k, char k = p, so that Hν(X, ) is a OX k-vector space, then F is not k-linear; in fact F (α x)= αp F (x), α k, x Hν( ). Such · · ∀ ∈ ∈ OX a map we call p-linear. Expanded in terms of a basis of Hν( ), F is given by a matrix which OX is called the ν-th Hasse-Witt matirix of X. p-linear maps do not have eigenvalues; instead they have the following canonical form:

Lemma 4.1. Let k be an algebraically closed field of characteristic p, let V be a finite- dimensional vector space over k and let T : V V be a p-linear transformation. Then V has a → unique decomposition: V = V V s ⊕ n where a) T (V ) V and T is nilpotent on V . n ⊂ n n b) T (V ) V and V has a basis e ,...,e such that T (e )= e . Furthermore, s ⊂ s s 1 n i i e Vs T e = e = miei mi Z/pZ . { ∈ | } { | ∈ } Proof. Let V = ∞ Image T ν and V X= ∞ Ker T ν. Since dim V < + , V = s ν=1 n ν=1 ∞ s Image T ν, V = Ker T ν for ν 0. Now if ν 0: n T ≫ ≫ S x V V = T νx = 0 and x = T νy ∈ s ∩ n ⇒ = T 2νy = 0 ⇒ = T νy = 0 ⇒ = x = 0 ⇒ and since dim V = dimKer T ν + dim Image T ν, it follows that V = V V . Then T is ∼ s ⊕ n |Vn nilpotent and T is bijective. Now choose x V and take ν minimal such that there is a |Vs ∈ s relation T νx = a x + a T (x)+ + a T ν−1(x). 0 1 · · · ν−1 If a0 = 0, then T ν−1x = a′ x + + a′ T ν−2(x) 1 · · · ν−1 and ν would not be minimal. Now try to solve the equation: T (λ x + + λ T ν−1(x)) = λ x + + λ T ν−1(x). 0 · · · ν−1 0 · · · ν−1 This leads to λp a = λ ν−1 · 0 0 λp + λp a = λ 0 ν−1 · 1 1 ······ λp + λp a = λ . ν−2 ν−1 · ν−1 ν−1 By substitution, we get: ν ν−1 ν−1 ν−2 λp ap + λp ap + + λp a λ = 0 ν−1 · 0 ν−1 · 1 · · · ν−1 · ν−1 − ν−1 which has a non-zero solution. Solving backwards, we find λν−2,...,λ0 as required, hence an x V with Tx = x. Now take a maximal independent set of solutions e ,...,e to the equation ∈ s 1 l Tx = x. If W = k e , then T : W W is bijective, hence T : V /W V /W is also bijective. · i → s → s If W $ Vs, the argument above then shows x Vs/W such that T x = x. Lifting x to x Vs, P ∃ ∈ ∈ we find

Tx = x + λiei. X 4. CHARACTERISTIC p PHENOMENA 303

Let µ k satisfy µp µ = λ . Then e = x µ e also lifts x but it satisfies T e = e . i ∈ i − i i l+1 − i i l+1 l+1 This proves that e span V .  i s P We can apply this decomposition in particular to H1(X, ) and we find the following OX interpretations of the eigenvectors: Theorem 4.2. Let X be a complete variety over an algebraically closed field k of character- istic p. Consider F acting on H1(X, ). Then: OX a) There is a one-to-one correspondence between α H1( ) F α = α and pairs { ∈ OX | } (π, φ): w Y φ π  X π ´etale, proper, π φ = π, φp = 1 , such that x X closed, #π−1(x) = p and φ ◦ Y ∀ ∈ permutes these points cyclically: we call this, for short, a p-cyclic ´etale convering13. b) If X is non-singular, there is an isomorphism: 1 0 1 a H ( X ) F α = 0 = ω H (Ω ) ω = df, some f R(X) . { ∈ O | } ∼ { ∈ X | ∈ } Proof. (a) Given α with F α = α, represent α by a cocycle f . Then F α is represented { ij} by f p and since this is cohomologous to α: { ij} f = f p + g g ij ij i − j g (U ). i ∈ OX i But then define a sheaf of -algebras by: A OX = [t ]/(f ) A|Ui OX i i f (t )= tp t + g i i i − i i and by the glueing: ti = tj + fij over U U . Let Y = pec ( ). Since (df /dt )(t )= 1, Y is ´etale over X. Since is integral i ∩ j α S X A i i i − α A and finitely generated over , Y is proper over X (cf. Corollary II.6.7). Define φ : Y Y OX α α α → α by ∗ φα(ti)= ti + 1. For all closed points x U , let a be one solution of tp t + g (x)= 0. Then π−1(x) consists of ∈ i i − i i the p points t = a, a + 1,...,a + p 1 which are permuted cyclically by φ . Finally, and this i − α is where we use the completeness of X, note that Yα depends only on α: if f ′ = f + h h ij ij i − j and f ′ = (f ′ )p + g′ g′ ij ij i − j is another solution to the above requirements, then g′ g′ = f ′ (f ′ )p i − j ij − ij = f f p + h h (h h )p ij − ij i − j − i − j = (g + h hp) (g + h hp), i i − i − j j − j hence g′ = g + h hp + ξ, ξ Γ( ). i i i − i ∈ OX 13Compare the statement of (a) and the proof with provisionally, Exercise (5) after §III.6, which treats p-cyclic coverings for p 6= char k. 304 VIII.APPLICATIONSOFCOHOMOLOGY

Thus ξ k, hence ξ = η ηp for some η k and we get an isomorphism ∈ − ∈ ≈ [t ]/(f p t + g ) / [t′ ]/((t′ )p t′ + g′) OX i i − i i OX i i − i i t / t′ (h + η). i i − i

We leave it to the reader to check that (Yα, φα) ∼= (Yβ, φβ) only if α = β. Conversely, suppose π : Y X and φ: Y Y are a p-cyclic ´etale covering. By Proposition → → II.6.5, Y = pec , a coherent sheaf of -algebras. S X A A OX Now π ´etale = π flat = is a flat -module. Now a finitely presented flat module ⇒ ⇒ Ax Ox,X over a local ring is free (cf. Bourbaki [26, Chapter II, 3.2]), hence is a free -module. § Ax Ox,X In fact

x/mx,X x ∼= Γ( π−1(x)) ∼= k(y) A ·A O −1 y∈πM(x) so is free of rank p, and the function 1 , since it is not in m , may be taken as Ax ∈ Ax x,X ·Ax a part of a basis. Moreover, φ induces an automorphism φ∗ : in terms of which we can A→A characterize the subsheaf : OX ⊂A (U)= f (U) φ∗f = f . OX { ∈A | } In fact for all closed points x X, we get an inclusion: ∈  /m / /m Ox,X x,X Ax x,X ·Ax

k(x) Γ( π−1(x)) O

y∈π−1(x) k(y)

∗L and clearly k(x) is characterized as the set of φ -invariant functions in y∈π−1(x) k(y). So if U is affine and f (U) is φ∗-invariant, then ∈A L ( ) f [ (U)+ m (U)]. ∗ ∈ OX x,X ·A x∈U closed\ But if U is small enough, has a free basis: A|U p = e A|U OX |U ⊕ OX |U · i Xi=2 and if we expand f = f + p f e , then ( ) means that f (x) = 0, 2 i p, x U 1 i=2 i · i ∗ i ≤ ≤ ∀ ∈ closed. By the Nullstellensatz, f = 0, hence f (U). Let x X be a closed point and let P i ∈ OX ∈ π−1(x)= y,φy,...,φp−1y . We can find a function e such that e (y) = 1, e (φiy) = 0, { } x ∈Ax x x 1 i p 1. Then ≤ ≤ − Tr e = p−1(φi)∗e satisfies φ∗(Tr e ) = Tr e and has value 1 at all points of π−1(x), • x i=0 x x x hence is invertible in . P Ax Set • p−1 1 i ∗ fx = − i (φ ) ex. Tr ex · · Xi=0 4. CHARACTERISTIC p PHENOMENA 305

∗ i p A small calculation shows that φ (f ) = f +1 and f (φ y) = i. Let g = f fx . Then x x x x x − φ∗g = g , hence g . Define a homomorphism x x x ∈ Ox,X p λx : x,X [tx]/(tx tx + gx) / O − Ax tx / fx.

i Note that since fx has distinct values at all points φ y, fx generates p−1 i x/mx,X x = k(φ y), A ·A ∼ Mi=0 hence by Nakayama’s lemma, λ is surjective. But as λ is a homomorphism of free - x x Ox,X modules of rank p, it must be injective too. Now λx extends to an isomorphism in some neighborhood of x and covering X by such neighborhoods, we conclude that X has a covering U such that { i} = [t ]/(tp t + g ), g (U ). A|Ui ∼ OX |Ui i i − i i i ∈ OX i Over U U , φ∗(t t )= t t , hence t = t + f , f (U U ). Then i ∩ j i − j i − j i j ij ij ∈ OX i ∩ j f f p = (t t ) (t t )p ij − ij i − j − i − j = g g , i − j so α = f is a cohomology class in such that F α = α. This completes the proof of (a). { ij} OX (b) Given α with F α = 0, represent α by a cocycle f . Then { ij} f p = g g ij i − j g (U ) i ∈ OX i hence dg = dg on U U . Therefore the dg ’s define a global section ω of Ω1 of the form df, i j i ∩ j i α X f R(X). If ∈ f ′ = f + h h ij ij i − j (f ′ )p = g′ g′ ij i − j is another solution to the above requirements, then g′ g′ = (f ′ )p i − j ij = f p + hp hp ij i − j = (g + hp) (g + jp) i i − j j hence g′ = g + hp + ξ, ξ Γ( )= k. i i i ∈ OX Thus dg′ = dg and ω depends only on α. Conversely, if we are given ω Γ(Ω1 ), ω = df, i i α ∈ X f R(X), the first step is to show that for all x X, ω = dfx for some fx x,X too. We use ∈ ∈ ∈ O the following important lemma:

Lemma 4.3. Let X be a non-singular n-dimensional variety over an algebraically closed field k, and assume z ,...,z Γ( ) such that ∃ 1 n ∈ OX n Ω1 = dz . X/k ∼ OX · i Mi=1 Consider as a sheaf of p -modules: is a free p -module with basis consisting of OX OX OX OX monomials n zαi , 0 α p 1. i=1 i ≤ ≤ − Q 306 VIII.APPLICATIONSOFCOHOMOLOGY

Another way to view this is to consider the pair Y = (X, p ) consisting of the topological OX space X and the sheaf of rings p : this is a scheme too, in fact it is isomorphic to X as scheme OX — but not as scheme over k — via:

≈ identity : X / Y ≈ p-th power: / p = . OX OX OY Thus Y is in fact an irreducible regular scheme, ant it is of finite type over k, i.e., a non-singular k-variety. Now ≈ identity: X / X p inclusion: Y = / O OX OX induces a k-morphism π : X Y −→ which is easily seen to be bijective and proper. Thus π is a coherent -module, and we ∗OX OY are asserting that it is free with basis n zαi , 0 α p 1. i=1 i ≤ i ≤ − Proof of Lemma 4.3. To checkQ that zαi generate π , it suffices to prove that for all i ∗OX closed points x Y , zαi generate (π ) /m (π ) over k. But identifying with ∈ i ∗OX Qx x,Y · ∗OX x Ox,Y p , m = f p f m : write this m[p] . Then Ox,X x,Y { | Q∈ x,X } x,X [p] (π ) /m (π ) = /m . ∗OX x x,Y · ∗OX x ∼ Ox,X x,X · Ox,X Let a = z (x) and y = z a . Then y ,...,y generate m and = k[[y ,...,y ]] by i i i i − i 1 n x,X Ox,X ∼ 1 n Proposition V.3.8. Thus b (π ) /m (π ) = k[[y ,...,y ]]/(yp,...,yp) ∗OX x x,Y · ∗OX x ∼ 1 n 1 n and the latter has a basis given by the monomials yαi , 0 α p 1, hence by zαi , b b i ≤ i ≤ − i 0 α p 1. ≤ i ≤ − Q Q But now suppose there was a relation over U X: ⊂ cp zα = 0, c (T ) not all zero. α · α ∈ OX α=(α1,...,αn) 0≤αXi≤p−1 Then for some closed point x U, c (x) = 0 for some α, hence there would be relation over k: ∈ α 6 c (x)p zα = 0 α · α=(α1,...,αn) 0≤αXi≤p−1 in (π ) /m (π ) . But the above proof showed that the zα were k-independent in ∗OX x x,Y · ∗OX x (π∗ X )x/mx,Y (π∗ X )x.  O b · O b 1 bTo return to theb proof of Theorem 4.2, let x X, f R(X) and suppose df (Ω )x. ∈ ∈ ∈ X/k Write f = g/hp, g, h , and by Lemma 4.3 expand: ∈ Ox,X g = cp zα, z ,...,z a generator of m . α { 1 n} x,X α=(α1,...,αn) 0≤αXi≤p−1 4. CHARACTERISTIC p PHENOMENA 307

Then

n p cα α1 αl−1 αn df =  αlz z z  dzl h 1 · · · l · · · n l=1 α=(α ,...,α ) X  X1 n     0≤αi≤p−1    hence

cp α zα1 zαl−1 zαn = hp b , b . α l 1 · · · l · · · n · l l ∈ Ox,X α=(α1,...,αn) 0≤αXi≤p−1 0<αl

α p p p Expanding b by Lemma 4.3, and equating coefficients of z , it follows that cα h if l ∈ · Ox,X αl > 0. Since this is true for all l = 1,...,n, it follows:

g = cp + hp f , f . (0,...,0) · x x ∈ Ox,X Therefore

p df = d(g/h )= dfx.

Now we can find a covering Ui of X and fi X (Ui) such that ω = dfi. Then in Ui Uj, p{ } ∈ O ∩ d(fi fj) = 0, hence fi fj = gij, gij (Ui Uj) (prove this either by Lemma 4.3 again, or by − − p ∈1 O ∩ p p field theory since d: R(X)/R(X) Ω is injective and x R(X) = x by the normality → X/k O ∩ O of X). Then g defines α H1( ) such that F α = 0. This completes the proof of Theorem { ij } ∈ OX 4.2. 

The astonishing thing about (b) is that any f [R(X) k] must have poles and in charac- ∈ \ teristic 0,

f / = df / (Ω1 ) . ∈ Ox,X ⇒ ∈ X/k x In fact, if f has an l-fold pole along an irreducible divisor D, then df has an (l + 1)-fold pole along D. But in characteristic p, if p l then the expected pole of df may sometimes disappear! | Nonetheless, this is relatively rare phenomenon even in characteristic p. For instance, in char = 2, consider a hyperelliptic curve C. This is defined to be the 6 normalization of P1 in a quadratic field extension k(X, f(X)). Explicitly, if we take f(X) to be a polynomial with no multiple roots and assume its degree is odd: say 2n + 1, then C is p covered by two affine pieces:

C = Spec k[X,Y ]/(Y 2 f(X)) 1 − C = Spec k[X, Y ]/(Y 2 g(X)) 2 − where e e e e

X = 1/X Y = Y/Xn+1 e g(X) = (X)2n+2 f(1/X). e · e e e 308 VIII.APPLICATIONSOFCOHOMOLOGY

Then consider ω = dX/Y : On C : 2YdY = f ′(X) dX, so 1 · ω = dX/Y = 2dY/f ′(X) and since Y,f ′(X) have no common zeroes, ω has no poles. On C : 2Y dY = g′(X) dX, and one checks 2 · ω = (X)n−1dX/Y = 2(X)n−1dY /g′(X) and since e e e− e − Y , g′(X) have no common zeroes, ω has no poles. e e e e e e But now say e e f(X)= h(X)p + X. Then f ′(X) = 1, so ω = d(2Y ) is exact! The area of characteristic p De Rham theory is far from being completely understood. For further developments, see Serre [90, p.24] (from which our theorem has been taken), Grothendieck [42] and Monsky [71, p.451]14.

5. Deformation theory We want to study here some questions of a completely new type: given an artin local ring R, with maximal ideal M, residue field k = R/M and some other ideal I such that I M = (0), · we get Spec R Spec R/I Spec k. ⊃ ⊃ Then

a) Suppose X1 is a scheme smooth and of finite type over R/I. How many schemes X2 are there, smooth and of finite type over R, such that X = X Spec R/I? 1 ∼ 2 ×Spec R X2 ⊃ X1   Spec R ⊃ Spec R/I

Such an X2 we call a deformation of X1 over R. b) Suppose X2, Y2 are two schemes smooth and of finite type over R, and let X1 = X Spec R/I, Y = Y Spec R/I. Suppose f : X Y is an R/I- 2 ×Spec R 1 2 ×Spec R 1 1 → 1 morphism. How many R-morphisms f : X Y are there lifting f ? 2 2 → 2 1 In fact the methods that we use to study these questions can be extended to the case where the X’s and Y ’s are merely flat over R or R/I (this is another reason why flat is such an important concept). (For this, see ????? .) We can state the results in the smooth case as follows: In case (a), let X = X Spec k. As in V.3, let 0 1 ×Spec R/I § 1 ΘX = om(Ω , X ) 0 H X0/k O 0 = sheaf of k-derivations from to ∼ OX0 OX0 be the tangent sheaf to X0. Then

ai) In order that at lease one X2 exist, it is necessary and sufficient that a canonically defined obstruction α H2(X , Θ ) I vanishes. (α will be denoted by obstr(X ) ∈ 0 X0 ⊗k 1 below.)

14(Add in publication) There have been considerable developments since the manuscript was written. See, for instance, Chambert-Lior [47], Ast´erisque volumes [45], [46] on “p-adic cohomology” related to “crystalline cohomology” initiated by Grothendieck [42]. 5. DEFORMATION THEORY 309

a ) If one X exists, consider the set of pairs (X , φ), with X as above and φ: X ≈ ii 2 2 2 1 −→ X Spec R/I an isomorphism, modulo the equivalence relation 2 ×Spec R ′ ′ ≈ ′ (X2, φ) (X2, φ ) if an R-isomorphism X2 X2 ∼ ∃ −→ψ such that

ψ×1 R/I / ′ X2 Spec R Spec R/I X2 Spec R Spec R/I × UUU ii× UUUU iiii UUU* it iii X1 commutes. Denote this set Def(X1/R): then Def(X1/R) is a principal homogeneous space over the group H1(X , Θ ) I: i.e., the group acts freely and transitively on the set. 0 X0 ⊗k aiii) Given two smooth schemes X1 and Y1 over R/I and a morphism over R/I:

f X Y 1 −→ 1 the obstructions to deforming X1 and Y1 are connected by having the same image in H2(X ,f ∗Θ ) I: 0 Y0 ⊗k 2 obstr(X1) H (X0, ΘX0 ) k I ∈ ⊗ [[[ df0 [[[[[[[[- H2(X ,f ∗Θ ) I 1 0 0 Y0 k cccccccc ⊗ ccc f ∗ obstr(Y ) H2(Y , Θ ) I 0 1 ∈ 0 Y0 ⊗k where f = f k : X Y and df : Θ f ∗Θ is the differential of f . 0 ⊗R/I 0 → 0 0 X0 → 0 Y0 0 In case (b), let X = X Spec k, Y = Y Spec k and let f induce f : X 0 1 ×Spec R/I 0 1 ×Spec R/I 1 0 0 → Y0. We have:

bi) In order that at least one lifting f2 exist, it is necessary and sufficient that a canonically defined obstruction α H1(X ,f ∗Θ ) I vanishes. ∈ 0 0 Y0 ⊗k bii) If one lifting f2 exists, denote the set of all lifts by Lift(f1/R). Then Lift(f1/R) is a 0 ∗ principal homogeneous space over the group H (X0,f0 ΘY0 ) k I. 1 ⊗ biii) The action of H (X0, ΘX0 ) k I on Def(X1/R) is a special case of the obstructions ′ ⊗ in (i): namely, if X2, X2 are two deformations of X1 over R, then the element of 1 H (X0, ΘX0 ) k I by which they differ is the obstruction to lifting 1X1 : X1 X1 to a ⊗ ′ → morphism from X2 to X2. biv) Given three schemes and two morphisms:

f g X 1 Y 1 Z , 1 −→ 1 −→ 1 the obstructions to lifting compose as follows: if α = (obstruction for f ) H1(X ,f ∗Θ ) 1 ∈ 0 0 Y0 β = (obstruction for g ) H1(Y , g∗Θ ) 1 ∈ 0 0 Z0 γ = (obstruction of g f ) H1(X , (g f )∗Θ ) 1 ◦ 1 ∈ 0 0 ◦ 0 Z0 then ∗ γ = dg0(α)+ f0 (β) where dg : Θ g∗Θ is the differential of g . 0 Y0 → 0 Z0 0 310 VIII.APPLICATIONSOFCOHOMOLOGY

Note, in particular, what these say in the affine case15 Affine a) If X is affine, ! deformation X of X smooth over R. 0 ∃ 2 1 Affine b) If X and Y are affine, then every f lifts to some f : X Y and if X = Spec A , 0 0 1 2 2 → 2 0 0 Y0 = Spec B0 then these liftings are a principal homogeneous space under: Γ(X ,f ∗Θ ) I = Der (B , A ) I. 0 0 Y0 ⊗k ∼ k 0 0 ⊗k If one is interested only in the existence of a lifting in (b), then the smoothness of X2 is irrelevant and one can prove:

Lifting Property for smooth morphisms: : If X2, Y2 are of finite type over R, Y2 smooth and X affine, then any f : X Y lifts to an f : X Y . 2 1 1 → 1 2 2 → 2 Variants of this lifing property have been used by Grothendiek to characterize smooth mor- phisms (cf. “formal smoothness” in Criterion V.4.10, EGA [1, Chapter IV, 17] and SGA1 [4, § Expos´eIII]). Our method of proof will be to analyze the deformation problem in an even more local case and then to analyze the patching problem via Cechˇ cocycles. In fact if Z is smooth over Spec R′, then we know that locally Z is isomorphic to U where

′ U = (Spec R [X1,...,Xn+l]/(f1,...,fl))g  Spec R′

′ where in R [X1,...,Xn+l] ∂f det i h = g, some h R′[X]. 1≤i,j≤l ∂X · ∈  n+j  Let’s call such U special smooth affine schemes over R′.

Step. I: If X is a special smooth affine over R/I, then a deformation X of X over R 1 ∃ 2 1 which is again a special smooth affine.

Proof. Write X1 = (Spec(R/I)[X]/(f))g as above, with det h = g. Simply choose any ′ ′ · ′ polynomials fi, h with coefficients in R which reduce mod I to fi, h. Let X2 = (Spec R[X]/(f ))g′ , where g′ = det′ h′.  · Step. II: If X2 is any affine over R (not even necessarily smooth) and Y2 is a special smooth affine over R, then any f : X Y lifts to an f : X Y . 1 1 → 1 2 2 → 2

Proof. If X2 = Spec A2 and Y2 = (Spec R[X]/(f))g as above, then the problem is to define a homomorphism φ2 indicated by the dotted arrow.

R[X]g ll lll u lll φ2 l A2 ______o R[X]g/(f)

  φ1 A /I A o (R/I)[X]g/(f). 2 · 2 If we choose any element a A which reduce mod I to φ (X ), then we get a homomorphism j ∈ 2 1 j φ′ : R[X] A 2 g −→ 2 15 It is a theorem that for any noetherian scheme X, X affine ⇐⇒ Xred affine (EGA [1, Chapter I, (5.1.10)]).

Hence in our case, X2 affine ⇐⇒ X1 affine ⇐⇒ X0 affine. We will not The last line is illegible. 5. DEFORMATION THEORY 311

′ ′ ′ by setting φ2(Xj) = aj (since φ2(g) mod I A2 equals φ1(g) which is a unit; hence φ2(g) is a ′ · unit in A2). However φ2(fi)= fi(a) may not be zero. But we may alter aj to aj + δaj provided δa I A . Then since I2 = (0), φ′ (f ) changes to j ∈ · 2 2 i n+l ∂fi fi(a + δa)= fi(a)+ (a) δaj . ∂Xj · Xj=1 ∂fi ∂fi Note that since δaj I A2 and I M = (0), (a) δaj depends only on the image of (a) ∈ · · ∂Xj · ∂Xj ∂f in k[X]. Multiplying the adjoint matrix to i by h, we obtain an (l l)-matrix ∂X ×  n+j 1≤i,j≤l (h ) k[X] such that ij ∈ l ∂fi hjq = g δiq. ∂Xn+j · · Xj=1 Now set δa = 0, 1 j n j ≤ ≤ l δa = g(a)−1 h f (a), 1 j l. n+j − jq q ≤ ≤ q=1 X Then: l l ∂fi −1 fi(a + δa)= fi(a) (a) g(a) hjqfq(a) − ∂Xn+j · · Xn+j Xq=1 l = f (a) g(a)−1 f (a) g(a)δ i − q · iq Xq=1 = 0.

Therefore if we define φ2 by φ2(Xj)= aj + δaj , we are through. 

Step. III: Suppose X2 and Y2 are affines over R, X2 = Spec A2, Y2 = Spec B2. A0 = A /M A , B = B /M B . Let f : X Y be a morphism and let f = res f . Then 2 · 2 0 2 · 2 2 2 → 2 1 X1 2 Lift(f1/R) is a principal homogeneous space over Der (B ,I A ). k 0 · 2 Proof. We are given a homomorphism φ : B A and we wish to study 1 1 → 1 L = φ : B A φ mod I = φ { 2 2 −→ 2 | 2 1} which we assume is non-empty. If φ , φ′ L, then φ′ φ factors via D: 2 2 ∈ 2 − 2 φ′ −φ 2 2 / B2 SS A2 SSS SS ) ∪ S D B /M B _ _ _ _/ I A 2 · 2 · 2

B0 One checks immediately that D is a derivation. And conversely for any such derivation D, φ L = φ + D L.  2 ∈ ⇒ 2 ∈ 312 VIII.APPLICATIONSOFCOHOMOLOGY

Step. IV: Globalize Step III: Let X , Y be two schemes of finite type over R. Let f : X 2 2 2 2 → Y2 be a morphism, and let f1 = resX1 f2. Then Lift(f1/R) is a principal homogeneous space over ∗ 1 Γ(X2, om(f Ω ,I X )). H 0 Y0/k · O (Note that I is really an -module). · OX OX0 Proof. Take affine coverings Uα , Vα of X2 and Y2 such that f2(Uα) Vα. If Uα = (α) (α) (α) { } { } ⊂ Spec A2 , Vα = Spec B2 , f1 = resUα f1, then as in Step III,

Lift(f (α)/R) = principal homogeneous space under Der (B(α),I A(α)) 1 k 0 · 2

1 (α) Hom (α) (Ω ,I A ) B (α) 2 0 B0 /k ·

1 (α) (α) Hom (α) (Ω (α) A ,I A ) A (α) B 0 2 0 B0 /k ⊗ 0 ·

∗ 1 Γ(Uα, om(f Ω ,I X )). H 0 Y0/k · O 2 Therefore on the one hand, one can “add” a morphism f : X Y and a global section D of 2 2 → 2 om(f ∗Ω1 ,I ) by adding them locally on the U ’s and noting that the “sums” agree on 0 Y0/k X2 α H · O ′ ′ overlaps Uα Uβ. Again given two lifts f2, f2, their “difference” f2 f2 defines locally on the ∩ ∗ 1 − Uα’s a section Dα of om(f Ω ,I X ), hence a global section D. H 0 Y0/k · O 2 Note that if Y is smooth over k,Ω1 is locally free with dual Θ , hence 0 Y0/k Y0 ∗ 1 ∗ om(f Ω , ) = f ΘY O for any sheaf ; H 0 Y0/k F ∼ 0 0 ⊗ X0 F F and if X is flat over R, then I = I . Thus case (b ) of our main result is proven!  2 ·OX2 ∼ ⊗k OX0 ii Step. V: Proof of case (b ): viz. construction of the obstruction to lifting f : X Y .16 i 1 1 → 1 Proof. Choose affine open coverings U , V of X , Y such that { α} { α} 2 2 f (U ) V • 1 α ⊂ α V is a special smooth affine. • α Then by Step II, there exists a lift f (α) : U V of res f . By Step III, res f (α) : U U 2 α → α Uα 1 2 α ∩ β → V V and res f (β) : U U V V differ by an element α ∩ β 2 α ∩ β → α ∩ β D Γ(U U ,f ∗Θ I). αβ ∈ α ∩ β 0 Y0 ⊗k But on U U U we may write somewhat loosely: α ∩ β ∩ γ D + D = [res f (α) res f (β)] + [res f (β) res f (γ)] αβ βγ 2 − 2 2 − 2 = res f (α) res f (γ) 2 − 2 = Dαγ . (Check the proof in Step III to see that this does make sense.) Thus D Z1( U ,f ∗Θ I). { αβ}∈ { α} 0 Y0 ⊗k Now if the lifts f (α) are changed, this can only be done by adding to them elements E 2 α ∈ Γ(U ,f ∗Θ I) and then D is changed to D + E E . Moreover, if the covering U α 0 Y0 ⊗k αβ αβ α − β { α} 16 Note that we use, in fact, only that Y2 is smooth over R and that the same proof gives the Lifting Property for smooth morphisms. 5. DEFORMATION THEORY 313

(α) is refined and one restricts the lifts f2 , then the cocycle we get is just the refinement of Dαβ. Thus we have a well-defined element of H1(X ,f ∗Θ I). Moreover it is zero if and only if 0 0 Y0 ⊗k for some coverings U , V , D is homologous to zero, i.e., { α} { α} αβ D = E E , E Γ(U ,f ∗Θ I). αβ α − β α ∈ α 0 Y0 ⊗k Then changing f (α) by E as in Step III, we get f (α)’s, lifting f such that on U U , f (α) f (β) is 2 α 2 1 α∩ β 2 − 2 (α)  represented by the zero derivation, i.e., the f2 ’s agree on overlaps and give an f2 lifting f1. e e e The assertion (biv) is a simple calculatione that we leave to the reader.

Step. VI: Proof of (aii) and (bii) simultaneously.

Proof. Suppose we are given X1 smooth over Spec R/I and at least one deformation X2 ′ of X1 over R exists. If X2, X2 are any two deformations, we can apply the construction of Step V to the lifting of 1 : X X to an R-morphism X X′ , getting an obstruction in X1 1 → 1 2 → 2 H1(X , Θ ) I. This gives us a map: 0 X0 ⊗k Def(X /R) Def(X /R) H1(X , Θ ) I 1 × 1 −→ 0 X0 ⊗k which we write: (X, X′) X X′. 7−→ − The functorial property (biv) proves that: ( ) (X X′) + (X′ X′′) = (X X′′). ∗ − − − ′ ′ Moreover, X X =0= X = X : because if 1X1 : X1 X1 lifts to an R-morphism f : X2 ′ − ⇒ → → X2, f is automatically an isomorphism in view of the easy: Lemma 5.1. Let A and B be R-algebras, B flat over R. If φ: A B is an R-homomorphism → such that φ: A/I A ≈ B/I B · −→ · is an isomorphism, then φ is an isomorphism.

(Proof left to the reader.) If we now show that deformation X and α H1(X , Θ ) I, a deformation X′ ∀ 2 ∀ ∈ 0 X0 ⊗k ∃ 2 with X′ X = α, we will have proven that Def(X /R) is a principal homogeneous space over 2 − 2 1 H1(X , Θ ) I as required. To construct X′ , represent α by a Cechˇ cocycle D , for any 0 X0 ⊗k 2 { ij} open covering U of X , where { i} 2 D Γ(U U , Θ I). ij ∈ i ∩ j X0 ⊗k As in Step IV, we then have an automrophism of U U (as a subscheme of X ): i ∩ j 2 1 + D : U U U U . Ui∩Uj ij i ∩ j −→ i ∩ j ′ X2 is obtained by glueing together the subschemes Ui of X2 by these new automorphisms between U U regarded as part of U and U U regarded as part of U . The cocycle condition i ∩ j i i ∩ j j Dij + Dji = Dik guarantees that these glueings are consistent and one checks easily that for this X′ , X′ X is indeed α.  2 2 − 2

Step. VII: Proof of (ai): viz. construction of the obstruction to deforming X1 over R. 314 VIII.APPLICATIONSOFCOHOMOLOGY

Proof. Starting with X , take a special affine covering U of X . By Step I, U deforms 1 { i,1} 1 i,1 to a special affine U over R. This gives us two deformations of the affine scheme U U i,2 i,1 ∩ j,1 over R, viz. the open subschemes U U j i,2 ⊂ i,2 U U . i j,2 ⊂ j,2 By Step VI, these must be isomorphic so choose ≈ φ : U U . ij j i,2 −→ i j,2 If we try to glue the schemes Ui,2 together by these isomorphisms, consistency requires that the following commutes:

iUj,2 kUj,2 6 ∩ res φij m RR res φjk mmm RRR mmm RRR mmm RR( jUi,2 kUi,2 / jUk,2 iUk,2. ∩ res φik ∩ But, in general, (res φ ) (res φ )−1 (res φ ) will be an automorphism of U U given ij ◦ ik ◦ jk i j,2 ∩ k j,2 by a derivation D Γ(U U U , Θ I). One checks easily (1) that D is a 2- ijk ∈ i ∩ j ∩ k X0 ⊗k ijk cocycle, (2) that altering the φij’s adds to the Dijk a 2-coboundary, and conversely that any D′ cohomologous to D in H2( U , Θ I) is obtained by altering the φ ’s, and (3) that ijk ijk { i} X0 ⊗k ij refining the covering U replaces D by the refined 2-cocycle. Thus D defines an element { i} ijk { ijk} α H2(X , Θ I) depending only on X , and α = 0 if and only if X exists.  ∈ 0 X0 ⊗k 1 2 Step. VIII Proof of (aiii). Proof. Given X , Y and f, take special affine coverings U , V of X and Y such 1 1 { i,1} { i,1} 1 1 that f(U ) V . Deform U (resp. V ) to U (resp. V ) over R. By Step II, lift f to i,1 ⊂ i,1 i,1 i,1 i,2 i,2 f : U V . Consider the diagram: i i,2 → i,2

res fi jUi,2 / jVi,2

φij ψij  res fj  iUj,2 / iVj,2. It need not commute, so let (res f ) φ = ψ (res f )+ F j ◦ ij ij ◦ i ij where F Γ(U U ,f ∗Θ I). It is a simple calculation to check now that if the φ ’s ij ∈ i ∩ j 0 Y0 ⊗k ij define a 2-cocycle Dijk representing obstr(X1) and the ψij’s similarly define Eijk, then df (D ) f ∗E = F F + F . 0 ijk − 0 ijk ij − ik jk 

This completes the proof of the main results of infinitesimal deformation theory. We get some important corollaries:

Corollary 5.2. Let R be an artin local ring with maximal ideal M and residue field k and let I R be any ideal contained in M. If X1 is a scheme smooth of finite type over Spec R/I ⊂ 2 such that H (X0, ΘX0 ) = (0) — e.g., if dim X0 = 1 — then a deformation X2 of X1 over R exists. 5. DEFORMATION THEORY 315

Proof. Filter I as follows: I MI M 2I M ν I = (0). Then deform X ⊃ ⊃ ⊃ ··· ⊃ 1 successively as follows:

(1) (l) X1 ⊂ X ⊂ ⊂ X 1 · · · 1    Spec R/I ⊂ Spec R/MI ⊂ ⊂ Spec R/M lI ⊂ ⊂ Spec R · · · · · · (l) (l+1) (ν)  using case (ai) of each stage to show that X1 can be deformed to a X1 . Set X2 = X1 . Corollary 5.3. Let R be an artin local ring with residue field k and let X be a scheme smooth17 of finite type over R. Let X = X Spec k and let 0 ×Spec R f : Y X 0 0 −→ 0 be an ´etale morphism. Then there exists a unique deformation Y of Y0 over R such that f0 lifts to f : Y X. → Proof. Let M be the maximal ideal of R. Deform Y0 successively as follows:

Y ⊂ Y ⊂ ⊂ Y 0 1 · · · l−1 f0 f1 fl−1    X ⊂ X ⊂ ⊂ Xl−1 ⊂ ⊂ X 0 1 · · · · · ·     Spec k ⊂ Spec(R/M 2)⊂ ⊂Spec(R/M l)⊂ ⊂ Spec R · · · · · · where X = X Spec(R/M l). Because f is ´etale, df : Θ f ∗Θ is an isomorphism. l−1 ×Spec R 0 0 Y0 → 0 X0 Therefore at each stage, the existence of the deformation Xl of Xl−1 gurantees by (aiii) the ′ ′ existence of a deformation Yl of Yl−1. Then choose any Yl and ask whether fl−1 lifts. We get an obstruction α: 1 l l+1 ≈ / 1 ∗ l l+1 H (Y0, ΘY0 ) k (M /M ) H (Y0,f0 ΘX0 ) k (M /M ). ⊗ df0 ⊗ ′ −1 Then alter the deformation Yl by df0 (α), giving a new deformation Yl. By functoriality (biv), f lifts to f : Y X and by injectivity of df this is the only deformation for which this is l−1 l l → l 0 so.  The most exciting applications of deformation theory, however, are those cases when one can construct deformations not only over artin local rings, but over complete local rings. If the ring R is actually an integral domain, then one has constructed, by taking fibre product, a scheme over the quotient field K of R as well. A powerful tool for extending constructions to this case is Grothendieck’s GFGA Theorem (Theorem 2.17). This is applied as follows:

Definition 5.4. Let R be a complete local noetherian ring with maximal ideal M. Then a formal scheme over R is a system of schemes and morphisms: X X / X / / X / 0 1 · · · n ······    Spec(R/M) / Spec(R/M 2) / / Spec(R/M n+1) / Spec R · · · ······

17A more careful proof shows that the smoothness of X is not really needed here and that Corollary 5.3 is true for any X of finite type over R. It is even true for comparing ´etale coverings of X and Xred, any noetherian scheme X (cf. SGA1 [4, Expos´eI, Th´eor`eme 8.3, p. 14]). 316 VIII.APPLICATIONSOFCOHOMOLOGY

n n+1 where X = X n+1 Spec(R/M ). is flat over R if each X is flat over Spec(R/M ). n−1 ∼ n×Spec(R/M ) X n If X is a scheme over Spec R, the associated formal scheme X is the system of schemes n+1 Xn = X Spec R Spec(R/M ) together with the obvious morphisms Xn Xn+1. × b → Theorem 5.5. Let R be a complete local noetherian ring and let = X be a formal X { n} scheme flat over R. If X is smooth and projective over k = R/M, and if H2(X , ) = (0), 0 0 OX0 then there exists a scheme X smooth and proper over R such that: = X. X Proof. Since X is projective over k, there exists a very ample invertible sheaf on X . 0 b L0 0 By Ex. ????? Veronese... , n is very ample for all n 1; by Theorem VII.8.1, H1( n) = (0) L0 ≥ L0 if n 0. So we may replace by n and assume that H1( ) = (0) too. The first step is to ≫ L0 L0 L0 “lift” to a sequence of invertible sheaves on X such that L0 Ln n n = n+1 O X , all n 0. L ∼ L ⊗ Xn+1 O n ≥ To do this, recall that the isomorphism classes of invertible sheaves on any scheme X are classified by H1(X, ∗ ). Therefore to construct the ’s inductively, it will suffice to show that OX Ln the natural map: H1(X , ∗ ) H1(X , ∗ ) n OXn −→ n−1 OXn−1 (given by restriction of functions from Xn to Xn−1) is surjective. But consider the map of sheaves: exp: M n ∗ · OXn −→ OXn a 1+ a. 7−→ Since M n M n 0 in , this map is a homomorphism from M n in its additive structure · ≡ OXn ·OXn to ∗ in its multiplicative structure, and the image is obviously Ker ∗ ∗ , i.e., we OXn OXn → OXn−1 get an exact sequence:   exp 0 M n ∗ ∗ 1. −→ · OXn −→ OXn −→ OXn−1 −→ But now the flatness of X over R/M n+1 implies that for any ideal a R/M n+1, n ⊂ a n+1 a ⊗R/M OXn −→ · OXn is an isomorphism. Apply this with a = M n/M n+1: n n n+1 M = (M /M ) n+1 · OXn ∼ ⊗R/M OXn n n+1 = (M /M ) (R/M) n+1 ∼ ⊗R/M ⊗R/M OXn n n+1 = (M /M ) k X0
∼ ⊗ O νn n n+1 = , if νn = dimk M /M . ∼ OX0 Therefore we get an exact sequence18: 0 νn ∗ ∗ 1 −→ OX0 −→ OXn −→ OXn−1 −→ hence an exact cohomology sequence:

δ H1( ∗ ) / H1( ∗ ) / H2( )νn . OXn OXn−1 OX0 (0)

18 Note that all Xn are topologically the same space, hence this exact sequence makes sense as a sequence of sheaves of abelian groups on X0. 5. DEFORMATION THEORY 317

This proves that the sheaves exist. Ln The second step is to lift the projective embedding of X0. Let s0,...,sN be a basis of N Γ(X0, 0), so that ( 0,s0,...,sN ) defines a projective embedding of X0 in P , as in the theory L L k of ????? . I claim that for each n there are sections s(n),...,s(n) of such that via the 0 N Ln isomorphism: = = /M n , Ln−1 ∼ Ln ⊗OXn OXn−1 ∼ Ln ·Ln (n−1) (n) si = image of si . To see this, use the exact sequence: 0 M n 0 −→ ·Ln −→ Ln −→ Ln−1 −→ and note that because is flat over R/M n+1 too, Ln n n n+1 M = (M /M ) n+1 ·Ln ∼ ⊗R/M Ln n n+1 = (M /M ) (R/M) n+1 ∼ ⊗R/M ⊗R/M Ln n n+1 = (M /M ) k 0,
∼ ⊗ L hence we get an exact cohomology sequence:

δ H0( ) / H0( ) / H1( )νn . Ln Ln−1 L0 (0)

This allows us to define s(n) inductively on n. Now for each n, ( ,s(n),...,s(n)) defines a i Ln 0 N morphism N φn : Xn P n+1 −→ (R/M ) such that the diagram: / N Xn−1 P(R/M n)

  / N Xn P(R/M n+1) commutes. I claim that φn is a closed immersion for each n. Topologically it is closed and injective because topologically φn = φ0 and φ0 is by assumption a closed immersion. As for ∗ structure sheaves, φn lies in a diagram:

n n+1 N N 0 / (M /M ) k PN / P / P n / 0 ⊗ O k O (R/Mn+1) O (R/M ) ∗ ∗ ∗ 1Mn ⊗φ0 φn φn−1    n n+1 0 / (M /M ) / X / Xn−1 / 0. ⊗k OX0 O n O Since φ∗ is surjective, this shows φ∗ surjective = φ∗ surjective. So by induction, all the φ 0 n−1 ⇒ n n are closed immersions. N Finally, let φn induce an isomorphism of Xn with the closed subscheme Yn P(R/M n+1). N ⊂ Then the sequence of coherent sheaves Y is a formal coherent sheaf on P in the sense of {O n } R 2 above. By the GFGA theorem (Theorem 2.17), there is a coherent sheaf on PN such that § F R = (R/M n+1) OYn ∼ F ⊗R for every n. Moreover since Yn is a quotient of the formal sheaf PN , is quotient {O } O (R/Mn+1) F N   of PN , i.e., = Y for some closed subscheme Y P . Therefore since: O F O ⊂ X = Y = Y Spec(R/M n+1), n ∼ n ∼ ×Spec R 318 VIII.APPLICATIONSOFCOHOMOLOGY it follows that Y = .  ∼ X Corollary 5.6. Let R be a complete local noetherian ring and let = X be a formal b X { n} scheme flat over R. If X is a smooth complete curve, then = X for some X smooth and 0 X proper over R. b Proof. By Exercise 2, 4D , X is projective over k and because dim X = 1, H2( )= § 0 0 OX0 (0). 

Corollary 5.7. (Severi19 -Grothendieck) Let R be a complete local noetherian ring and let X0 be a smooth complete curve over k = R/M. Then X0 has a deformation over R, i.e., there exists a scheme X, smooth and proper over R such that X = X Spec k. 0 ∼ ×Spec R Proof. Corollaries 5.2 and 5.6. 

An important supplementary remark here is that if for simplicity X0 is geometrically irre- ducible (also said to be absolutely irreducible), i.e., X Spec k is irreducible (k = algebraic 0 ×Spec k closure of k), then H1( ) is a free R-module such that OX H1( ) k = H1( ) OX ⊗R ∼ OX0 H1( ) K = H1( ) OX ⊗R ∼ OXη (K = quotient field of R, X = X Spec K). η ×Spec R Since the genus of a curve Y over k is nothing but dim H1( ), this shows that genus(X )= k OY η genus(X0). The proof in outline is this:

a) X0 Spec k Spec k irreducible and X0 smooth over k implies k algebraically closed in × 0 R(X), hence k algebraically closed in H ( X ). Thus O 0 k ≈ H0( ). −→ OX0 b) Show that there are exact sequences

n 0 / M X / X / Xn−1 / 0. · O n O n O

νn OX0 ≈ c) Show by induction on n that if g = dim H1( ), then R/M n+1 H0( ) and k OX0 −→ OXn H1( ) is a free (R/M n+1)-module of rank g such that OXn H1( ) = H1( )/M n H1( ). OXn−1 ∼ OXn · OXn d) Apply GFGA (Theorem 2.17) to prove that H1( ) is a free R-module of rank g such OX that H1( ) = H1( )/M n+1 H1( ) for all n. OXn ∼ OX · OX e) Apply (4.??) using the flatness of K over R to prove that

H1( ) = H1( ) K. OXη ∼ OX ⊗R Corollary 5.7 is especially interesting when k is a perfect field of characteristic p and R is the Witt vectors over k (see, for instance, Mumford [73]), in which case one summarizes Corollary 5.7 by saying: “non-singular curves can be lifted from characteristic p to characteristic 0”. On 20 the other hand, Serre [92] has found non-singular projective varieties X0 over algebraically

19Modulo translating Italian style geometry into the theory of schemes, a rigorous proof of this is contained in Severi [95, Anhang]. This approach was worked out by Popp [80]. 20(Added in Publication) See also Illusie’s account in FAG [3, Chapter 8]. 5. DEFORMATION THEORY 319 closed fields k of characteristic p such that for every complete local characteristic 0 domain R with R/M = k, no such X exists: such an X0 is called a non-liftable variety! One can strengthen the application of deformation theory to coverings in the same way:

Theorem 5.8. Let R be a complete local noetherian ring with residue field k and let X be a scheme smooth21 and proper over R. Let X = X Spec k and let 0 ×Spec R f : Y X 0 0 −→ 0 be a finite ´etale morphism. Then there exists a unique finite ´etale morphism f : Y X −→ such that f is obtained from f by fibre product Spec k. 0 ×Spec R Proof. By Corollary 5.3 we can lift f : Y X to a unique formal finite ´etale scheme 0 0 → 0 F : X, i.e., = Yn , F = fn where fn : Yn Xn is finite and ´etale, where Xn = Y → Y n+1{ } { } → X Spec R Spec R/M and the diagram: × b Yn / Yn+1

fn fn+1   Xn / Xn+1 commutes (the inclusion Y Y being part of the definition of a formal scheme ). If n → n+1 Y = f ( ), then An n,∗ OYn pec ( ) = Y S Xn An ∼ n = Y X ∼ n+1 ×Xn+1 n = pec n+1 O X ∼ S Xn A ⊗ Xn+1 O n   hence n = n+1 O X . Therefore n is a coherent formal sheaf on X, hence by A ∼ A ⊗ Xn+1 O n {A } Theorem 2.17 there is a unique coherent sheaf on X such that = for all n. A An ∼ A⊗OX OXn Using the fact that Hom ( , ) = Formal Hom ( , ) OX A⊗OX A A ∼ OX Afor ⊗OX Afor Afor and similar facts with , we see immediately that is a sheaf of commutative algebras. A⊗A⊗A A Let Y = pec ( ), and let f : Y X be the canonical morphism. f is obviously proper and S X A → finite to one. Moreover since for all x X , is a free -module, it follows immediately ∈ 0 An,x Ox,Xn that x is a free x,X -module, i.e., f is flat at x. And f0 ´etale implies Ω O k(x) = (0), A O Y/X ⊗ X hence (ΩY/X )x = (0) by Nakayama’s lemma. Therefore by Criterion V.4.1, f is ´etale at x. Since this holds for all x, f is ´etale in an open set U X, with U X . But X proper over Spec R ⊂ ⊃ 0 implies that every such open set U equals X. Thus f is ´etale. Finally if f ′ : pec ′ X is S X A → another such lifting with ′ = , then by Theorem 2.17 there is a unique isomorphism A0 A0 φ : ≈ ′ n An −→ An ≈ ′ of Xn -algebras inducing the identity 0 0. Then these φn patch together into a formal O ≈ A −→ A isomorphism ′ , which comes by Theorem 2.17 from a unique algebraic isomorphism Afor −→ Afor φ: ≈ ′.  A −→ A 21As mentioned above, Corollary 5.3 is actually true without assuming smoothness and hence since smooth- ness is not used in proving this result from Corollary 5.3, it is unnecessary here too. 320 VIII.APPLICATIONSOFCOHOMOLOGY

Corollary 5.9. In the situation of Theorem 5.8, there is an isomorphism of pro-finite groups: ≈ πalg(X ) πalg(X) 1 0 −→ 1 canonical up to inner automorphism.

Proof. If f : Y X is any connected covering, i.e., f finite and ´etale, then by fibre → product X , we get a covering f : Y X . Note that Y is connected (if not, we could ×X 0 0 0 → 0 0 lift its connected components separately by Theorem 5.8, hence find a disconnected covering f ′ : Y ′ X lifting f , thus contradicting the uniqueness in the theorem). By Theorem 5.8 every → 0 connected covering f0 : Y0 X0, up to isomorphism, arises in this way. Moreover, if R(Y ) is → Galois over R(X), then we get a homomorphism: ≈ ≈ Gal(R(Y )/R(X)) Aut(Y/X) Aut(Y0/X0) Gal(R(Y0)/R(X0)) −→ −→ −→ which is easily seen to be an isomorphism, e.g., by Ex. 1, 6A???? . Now fix separable algebraic § ∗ ∗ ∗ ∗ closures R(X) of R(X) and R(X0) of R(X0), and let R^(X) R(X) , R^(X0) R(X0) be the ⊂ ⊂ maximal subfields such that normalization in finitely generated subfields of these is ´etale over

X or over X0. Now write R^(X) as an increasing union of finite Galois extensions Ln of R(X); we get a tower of coverings Y = normalization of X in L ; let Z = Y X (this is a Ln n n Ln ×X 0 tower of connected coverings of X0); choose inductively in n isomorphisms: ≈ R(Zn) Kn R^(X0). −→ ⊂

It follows readily that Kn = R^(X0), and that alg S π1 (X) ∼= lim Gal(Ln/R(X)) ←− ∼= lim Gal(Kn/R(X0)) ←−alg ∼= π1 (X0). 

This result can be used to partially compute π1 of liftable characteristic p varieties in terms of π1 of varieties over C, hence in terms of classical topology. This method is due to Grothendieck and illustrates very beautifully the Kroneckerian idea of IV.1: Let § k = algebraically closed field of characteristic p, R = complete local domain of characteristic 0 with R/M = k, K = quotient field of R, K = algebraic closure of K. Choose an isomorphism (embedding?): K ∼= C. Let X = scheme proper and smooth over R, X = X Spec k : we assume this is irreducible, 0 ×Spec R X = X Spec K, η ×Spec R X = X Spec K : we assume this is irreducible, η ×Spec R Xη = X Spec R Spec C. × e 5. DEFORMATION THEORY 321

Theorem 5.10. There is a surjective homomorphism alg ։ alg π1 (Xη) π1 (X0) canonical up to inner automorphism, and hence, fixing an isomorphism K C, a surjective ≈ homomorphism: top ։ alg π1 (Xη) π1 (X0).

alg alg Proof. By Theorem 2.16 and Corollaryb e 5.9, it suffices to compare π1 (Xη) and π1 (X). Let Ω R(Xη) be an algebraic closure. Note that ⊃ i) R(X)= R(Xη) ii) R(Xη)= R(Xη) K K is algebraic over R(Xη), hence Ω is an algebraic closure of R(Xη) ⊗ too. Thus we may consider the maximal subfields of Ω such that the normalization of any of the schemes X, Xη and Xη is ´etale. Note that: iii) L Ω finite over R(X), normalization of X in L ´etale over X = normalization of ⊂ ⇒ Xη in L ´etale over Xη, iv) K0 K finite over K, normalization of Xη in R(Xη) K K0 is Xη Spec K Spec K0 ⊂ ⊗ × which is ´etale over Xη. Thus we get a diagram Ω

Ω1  ??  ??  ?? K R(Xη)= R(Xη) Ω · 2  ??   ??   ?  K R(Xη)= R(X) ? ??  ??  ?  K where

Ω1/R(Xη) = maximal extension ´etale over Xη

Ω1/R(Xη) = maximal extension ´etale over Xη

Ω2/R(Xη) = maximal extension ´etale over X i.e.,

alg π1 (Xη) ∼= Gal(Ω1/R(Xη)) alg π1 (Xη) ∼= Gal(Ω1/R(Xη)) alg π1 (X) ∼= Gal(Ω2/R(Xη)). Since

v) Gal(R(Xη)/R(Xη)) ∼= Gal(K/K), 322 VIII.APPLICATIONSOFCOHOMOLOGY we get homomorphisms: / alg / alg / / 1 π1 (Xη) π1 (Xη) Gal(K/K) 1 KK KKK KKK φ KK%  alg π1 (X) To finish the proof of Theorem 5.10, we must show that φ is surjective. A small consideration of this diagram of fields shows that this amounts to saying: vi) Ω K Ω is injective; or equivalently (cf. IV.2) K is algebraically closed in Ω . 2 ⊗K → § 2 If this is not true, then suppose L K is finite over K and L Ω . Let S be the inte- ⊂ ⊂ 2 gral closure of R in L. Then X Spec S is smooth over Spec S, hence is normal; since ×Spec R R(X Spec R Spec S)= R(X) K L, X Spec R Spec S is the normalization of X in R(X) K L. × ⊗ × ⊗ Now f : Spec S Spec R is certainly not ´etale unless R = S: because if [M] Spec R is the → ∈ closed point, then (a) by Hensel’s lemma (Lemma IV.6.1), f −1([M]) = one point, so (b) if f is ´etale, the closed subscheme f −1([M]) is isomorphic to Spec k, hence (c) by Ex. 4D????? , f is § a closed immersion, i.e., R S is surjective. But then neither can g : X Spec S X be → ×Spec R → ´etale because I claim there is a section s:

XU s  Spec R hence g ´etale implies by base change via s that f is ´etale. To construct s, just take a closed point x X , let a ,..., a be generators of m in the regular local ring , lift these to ∈ 0 1 n x,X0 Ox,X0 a ,...,a m and set Z = Spec ( /(a ,...,a )). By Hensel’s lemma (Lemma IV.6.1), Z 1 n ∈ x,X Ox,X 1 n is finite over R, hence is isomorphic to a closed subscheme of X. Since the projection Z Spec R → is immediately seen to be ´etale, Z Spec R is an isomorphism, hence there is a unique s with → Z = Image(s).  At this point we can put together Parts I and II to deduce the following famous result of Grothendieck. Corollary 5.11. Let k be an algebraically closed field of characteristic p and let X be a non- 1 singular complete curve over k. Let g = dimk H ( X ), the genus of X. Then a1,...,ag, b1,...,bg alg O ∃ ∈ π1 (X) satisfying: ( ) a b a−1b−1 a b a−1b−1 = e ∗ 1 1 1 1 · · · g g g g alg and generating a dense subgroup: equivalently π1 (X) is a quotient of the pro-finite completion of the free group on the a ’s and b ’s modulo a normal subgroup containing at least ( ). i i ∗ Proof. Lift X to a scheme Y over the ring of Witt vectors W (k), and via an isomorphism of C with (embedding into C of?) the algebraic closure of the quotient field of W (k) (see, for 1 instance, Mumford [73]), let Y induce a curve Z over C. Note that g = dimC H ( Z ) by the O remark following Corollary 5.7 and by (4.?????) . By Part I [76, 7B], we know that topolog- § ically Z is a compact orientable surface with g handles. It is a standard result in elementary top topology (cf. ????? ) that π1 of such a surface is free on ai’s and bi’s modulo the one relation ( ). Thus everything follows from Theorem 5.10.  ∗ What is the kernel of πalg(X ) πalg(X )? A complete structure theorem is not known, 1 η → 1 0 even for curves, but the following two things have been discovered: 5. DEFORMATION THEORY 323

a) Grothendieck has shown that the kernel is contained in the closed normal subgroup alg generated by the p-Sylow subgroups: i.e., if H is finite such that p ∤ #H and π1 (Xη) alg → H is a continuous map, then this map factors through π1 (X0). b) If you abelianize the situation, and look at the p-part of these groups, the kernel tends to be quite large. In fact alg alg alg 2g π (Xη)/ π ,π = Z T0 1 1 1 ∼ l × h i primesY l while alg alg alg 2g r π (X0)/ π ,π = Z Z Tp 1 1 1 ∼ l × p × h i Yl6=p where 0 r g and T , T are finite groups, (0) in the case of curves (compare ≤ ≤ 0 p (6.?????) (a) ). In fact, Shafarevitch [97] has shown for curves that the maximal alg pro-p-nilpotent quotient of π1 (X0) is a free pro-p-group on r generators. Going back now to general deformation theory, it is clear that the really powerful applications are in situations where one can apply the basic set-up: R R/I (I M = (0)) inductively and → · get statements over general artin rings and via GFGA (Theorem 2.17) to complete local ring. In the two cases examined above, we could do this by proving that there were no obstructions. However even if obstructions may be present, one can seek to build up inductively a maximal deformation of the original variety X0/k. This is the point of view of moduli, which we want to sketch briefly. Start with an arbitrary scheme X0 over k. Then for all artin local rings R with residue field k, define the set of triples (X,φ,π), where π : X Spec R is a flat morphism  → ≈   and φ: X Spec R Spec k X0 is a   × −→   k-isomorphism, modulo (X,φ,π) (X′, φ′,π′)   ∼   ≈ ′  Def(X /R) =  if an R-isomorphism ψ : X X such that  0  ∃ −→   ψ×1k ′  or  X Spec R Spec k / X Spec k  × ×Spec R the deformations RRR ll RRR lll  φ RRR lll φ′  of X0 over R  R( lv l   X0     commutes.      Note that   R Def(X /R) 7−→ 0 is a covariant functor for all homomorphisms f : R R′ inducing the identities on the residue → fields. In fact, if (X,φ,π) Def(X /R), let ∈ 0 X′ = X Spec R′ ×Spec R π′ = projection of X′ onto Spec R′ φ′ = the composition:

′ ′ φ X ′ Spec k = (X Spec R ) ′ Spec k = X Spec k X . ×Spec R ×Spec R ×Spec R ∼ ×Spec R −→ 0 Then (X′, φ′,π′) Def(X /R′) depends only on the equivalence class of (X,φ,π) and on f. One ∈ 0 says that X′/R′ is the deformation obtained from X/R by the base change f. What one wants to do next is to build up inductively the biggest possible deformation of X0 so that any other is obtained from it by base change! More precisely, suppose that R is a complete noetherian local 324 VIII.APPLICATIONSOFCOHOMOLOGY ring with maximal ideal M and residue field R/M = k. Then we get a sequence of artin local rings R = R/Mn+1. Then, by definition, a formal deformation of X over R is a sequence n X 0 of deformations Xn and closed immersions φn:

o Xn o Xn−1 o o X0 · · · φn · · · φ1

   Spec R o o Spec R o Spec R o o Spec k · · · n n−1 · · · where φn induces an isomorphism: ≈ X X Spec R . n−1 −→ n ×Spec Rn n−1 Note that if S is artin local with residue field k, R S is a homomorphism inducing identity on → residue fields and /R is a formal deformation, we can define a real deformation RSpec S X X×Spec by base change, since R S factors through R if n is large enough. Then a formal deformation → n /R is said to be versal or semi-universal if: X (1) every deformation Y of X0 over S is isomorphic to the one obtained by base change R Spec S for a suitable α: R S, and X ×Spec → (2) if the maximal ideal N S satisfies N 2 = (0), then one asks that there be only one α ⊂ for which (1) holds. (2′) /R is universal if α is always unique. It is clear that a universal deformation is unique X if it exists, and it is not hard to prove that a versal one is also unique, but only up to a non-canonical isomorphism. A theorem of Grothendieck and Schlessinger [85] asserts the following: a) If X is smooth and proper over k, then a versal deformation /R exists and there is 0 X a canonical isomorphism: char k = 0 : Hom (M/M2, k) k = H1(X , Θ ). char k = p : Hom (M/(M2 + (p)), k) ∼ 0 X0 k  0 b) If H (X0, ΘX0 ) = (0), then /R is universal. 2 X c) If H (X0, ΘX0 ) = (0), then char k = 0 : R k[[t ,...,t ]], n = dim H1(X , Θ ) ∼= 1 n 0 X0 char k = p, k perfect : R = W (k)[[t ,...,t ]], n = dim H1(X , Θ ).  ∼ 1 n 0 X0 A further development of these ideas leads us to the global problem of moduli. Starting with any X0 smooth and proper over k, suppose you drop the restriction that R be an artin local ring and for any pair (R, m), R a ring, m R a maximal ideal such that R/m = k you define ⊂ Def(X0/R) to be the pairs (X, φ) as before, but now X is assumed smooth and proper over Spec R. If moreover you isolate the main qualitative properties that X0 and its deformations have, it is natural to cut loose from the base point [m] Spec R and consider instead functors ∈ like: set of smooth proper morphisms f : X S such that −1 → MP(S)=  all the fibres f (s) of f have property P,  , ≈  modulo f f ′ if an S-isomorphism g : X X′  ∼ ∃ −→ where S is any scheme and P is some property of schemes X over fields k. Provided that P satisfies: [if X/k has P and k′ k, then X Spec k′ has P], then MP is a functor in S, ⊃ ×Spec k i.e., given g : S′ S and X/S MP(s), then X S′ MP(S′). → ∈ ×S ∈ 5. DEFORMATION THEORY 325

For instance, take P(X/k) to mean dim X = 1 H0(X, ) ≈ k OX −→ dim H1(X, )= g; k OX then MP is the usual moduli functor for curves of genus g. The “problem of moduli” is just the question of describing MP as explicitly as possible and in particular asking how far it deviates from a representable functor. The best case, in other words, would be that as functors in S, MP(S) ∼= Hom(S, MP) for some scheme MP which would then be called the moduli space. For an introduction to these questions, see Mumford et al. [72].

CHAPTER IX

Elementary treatment of Hilb, ?? Macaulay No manuscript

327

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