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The Cohomology of Coherent Sheaves

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The Cohomology of Coherent Sheaves 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 .
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