INTERACTION BETWEEN CECH COHOMOLOGY and GROUP COHOMOLOGY 1. Cech Cohomology Let I Be Some Index Set and Let U = {U I : I ∈ I}

Home , Group cohomology, Section (fiber bundle)

INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY

TAYLOR DUPUY

Abstract. [Summer 2012] These are old notes and they are just being put online now. They are incomplete and most likely riddled with errors that I may never ﬁx. There are important technical hypotheses missing that make which make certain proofs invalid as stateded below. For example we sometimes need the sheaf of abelian group to be quasi-coherent modules to make things work. If someone reads this and wants to be awesome and email me I would appreciate it. –Taylor

Abstract. [Original] Given a sheaf of groups acting on an abelian sheaf of groups we deﬁne a group cohomology theory. A special case of a theorem proves that twisted homomorphism in certain group cohomologies when paired with cocycles in cech cohomology give rise to 1-cocycles of vector bundles.

1. Cech Cohomology

Let I be some index set and let U = {Ui : i ∈ I} be a collection of open sets S with X = i∈I Ui. Let G be a sheaf of groups on X. Q −1 A collection (gij) ∈ i,j∈I Γ(Ui ∩ Uj, G) with gij = gji is said to be a cocycle provided

(1.1) gijgjkgki = 1 1 on Ui ∩ Uj ∩ Uk. The collection of cocycles will be denoted by Z (U; X, G). A Q collection gij ∈ i,j∈I Γ(Ui ∩ Uj, G) is called a coboundary if there exists some collection εi ∈ G(Ui) for each i ∈ I such that −1 gij = εiεj . 1 We will denote the collection of coboundaries by B (U; X, G). Note that if gij = −1 −1 −1 −1 −1 εiεj , then we have gji = εjεi = (εiεj ) = gij . Also for all i, j and k we have −1 −1 −1 gijgjkgki = εiεj εjεk εkεi = 1. Which shows that every coboundary is a cocycle. Next we define what is means for two cocycles to be cohomologous. Let (gij) and (hij) be cocycles with respect to some cover U. We say that (gij) and hij are cohomologous and write (gij) ∼ (hij) if and only if there exists some (εi) ∈ Q i∈I G(Ui) such that εigij = hijεj. Proposition 1.1 (Being Cohomologous Defines an Equivalence Relation). The above is an equivalence relation. Proof. We’ll show that the relation on Z1(U; X, G) is reflexive, transitive and symmetric. The relation is reflexive since we can take εi = 1i ∈ G(Ui). The relation is 1 2 TAYLOR DUPUY

−1 −1 symmetric since εigij = hijεj =⇒ εi hij = gijεj . Next suppose that fij ∼ gij −1 and gij ∼ hij. We can write the first relation as fij = εi gijεj and the second −1 relation as gij = δi hijδj which gives us −1 fij = εi gijεj −1 −1 = εi δi hijδjεj −1 = (δiεi) hij(δjεj). The first cohomology SET of a sheaf of groups G, on a topological space X, with respect to the open cover U is Hˇ 1(U, G) := Z1(U; X, G)/ ∼ . Here ∼ is the relation of two cocycles being cohomologous. We can make Cech Cohomology independent of the cover under consideration by taking the injective limit of the cohomology sets. Let U and V be open covers of X. We say that U is a refinement of V provided that for all U ∈ U there exists V ∈ V such that U ⊂ V . The notion of refinement defines a partial ordering on the collection of open covers. If U is a refinement of V we will write U ≤ V. Proposition 1.2. Let U and V be open covers of a scheme X. If U ≤ V then for every sheaf of groups F on X there exists a map U 1 1 ϕV : H (U; X, F) → H (V; X, F). Note that we now have a partially ordered set consisting of all open covers of X and for each open cover we have a sets H1(U, F) such that for every U ≤ V U there exists a map ϕV . One can check that this collection is an injective system. Using this injective system we can now define the First Homology Set for a sheaf of groups, Hˇ 1(X, G) := lim Hˇ 1(U,G). −→ U ` Hˇ 1(U:XG) Recall that lim Hˇ 1(U,G) = U where η ∈ H1(U, F) is said to be −→U ∼ similar to ξ ∈ H1(V, F) if and only if there exists some W such that U ≤ W and U V V ≤ W and ϕW (η) = ϕW (ξ).

Lemma 1.3. Let Φ: F → G be a morphism of sheaves of groups. If (fij) ∈ 1 1 Z (U; X, F) then gij := ΦUij (fij) ∈ Z (U; X, G).

Proof. Let ΦUij (fij) = gij as above. We have

= ΦUijk (fij|Uijk ) · ΦUijk (fjk|Uijk ) · ΦUijk (fki|Uijk )

= ΦUijk (fij|Uijk · fjk|Uijk · fki|Uijk )

= ΦIijk (1F ) = 1G Corollary 1.4. If Φ: F → G is a morphism of sheaves of groups on a topological space X then there exists an induced map 1 1 Φ∗ : H (X, F) → H (X, G). INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 3

Every short exact sequence of groups induces a long exact sequence on cohoml- ogy1 Lemma 1.5. Let ι p 1 / G0 / G / G00 / 1 be an exact sequence of sheaves of groups on X. There is a well-deﬁned map δ : H0(X, G00) → H1(X, G0). (procedure is outlined at the end of the proof)

00 00 00 Proof. A global section of G is a collection of gi ∈ G (Ui) such that 00 00 gi |Ui∩Uj = gj |Ui∩Uj . 00 For every open subset U the map pU : G(U) → G (U) is surjective. This means 00 that there exists some gi ∈ G(Ui) such that pUi (gi) = gi . Since p is a sheaf homomorphism we have

pUi (gi)|Ui∩Uj = pUj (gj)|Ui∩Uj =⇒ pUi∩Uj (gi|Ui∩Uj ) = pUi∩Uj (gj|Ui∩Uj ) 00 and since pUi∩Uj : G(Ui ∩ Uj) → G (Ui ∩ Uj) is a sheaf homomorphism we have −1 00 1G = pUi∩Uj (gi|Ui∩Uj gj|Ui∩Uj ). 00 0 0 Since the sequence is exact there exists some gij ∈ G (Ui∩Uj) such that ιUi∩Uj (gij) = −1 0 0 0 gi|Ui∩Uj gj|Ui∩Uj . These satisfy the cocycle condition gij|Uijk gjk|Uijk gki|Uijk = 0 0 0 0 1G (Uijk . Briefly (omiting subscripts and restrictions) we have gijgjkgki = 1 if and 0 0 0 only if ι(gijgjkgki) = 1 since ι is injective. We have 0 0 0 −1 −1 −1 −1 −1 −1 ι(gijgjkgki) = ι(ι (gigj ) · ι (gjgk ) · ι (gkgi )) −1 −1 −1 = gigj · gjgk · gkgi = 1, 1 0 which shows that the collection (gij,Uij) ∈ Z (U; X, G ). It remains to show that this cocycle gives a well-defined cohomology class. Sup- 00 pose that used the surjectivity pUi : G(Ui) → G (Ui) to find some other elements 00 g˜i ∈ G(Ui) withg ˜i 6= gi for at least one i and p(˜gi) = gi . We need to show that the 0 −1 −1 cocycleg ˜ij := ι (˜gig˜j ) is cohomologous to the one we just defined from the lifts (gi). This is true since

hig˜ij = gijhj −1 −1 where hi := ι (gig˜i ). 00 Remark 1.6 (Procedure For Deﬁning Connecting Homomorphism). If (gi )i∈I ∈ Q 00 i∈I G (Ui) deﬁnes a global section in the sense that gi and gj are equal on Ui ∩Uj we do the following to get a cocycle in G0, 00 2 (1) Lift gi to gi. −1 −1 0 (2) Form ι (gigj ) := gij.

1as far as we have the first cohomology defined 2 00 This means gi satisfies p(gi) = gi 4 TAYLOR DUPUY

Using the connecting homomorphism above on relative tangent sequence:

∗ 0 / TX/S / TX / π TS / 0 where π : X → S is a morphism of schemes which makes the relative tangent sequence exact we get a map ˇ 0 ˇ 1 H (X, TS) → H (X, TX/S), which Oort and Steenbrink use as the deﬁnition of the Kodaira-Spencer Map.3

2. Cech Cohomlogy for Sheaves of Abelian Groups

p+1 X (2.1) (dξ) = (−1)jξ i0i1...ip+1 i0···iˆj ···ip+q i=0

p+1 X h (2.2) (dξ)J = (−1) ξJ\{jh} h=0

J \ jh := Jh = {jh1 < jh2 < ··· < jhp}

p+2 X h (d(dξ))J = (−1) (dξ)J\{jh} h=0 p+2 p+1 ! X h X k = (−1) (−1) ξJh\{jhk} h=0 k=0 p+2 p+1 p+1 ! X X h+k X h+k−1 = (−1) ξJ\{jh,jk} + (−1) ξJ\{jh,jk} h=0 k

3. Pairings Proposition 3.1 (Cup Product). Let F and G be sheaves of commutative groups. There exists a morphism of abelian groups Hp(X, F)×Hq(X, F) → Hp+q(X,F ⊗G) given by

(3.1) (η ∪ ξ)i0i1···ip+q = ηi0···ip ⊗ ξip···ip+q Proof. We need to show that (η ∪ ξ) is a cocycle. Let I = {i0 < i1 < . . . < ip+q} and deﬁne

I≤p := {i0 < i1 < . . . < ip} and I≥p := {ip < ip+1 < . . . < ip+q}. this gives #I≤p = and #I≥p = q and allows us to write

(η ∪ ξ)I = ηI≤p ⊗ ξI≥p .

3They actually deﬁne their map using the long exact sequence for cohomology of abelian groups in the style of Hartshorne (via the right derived functors of sheaf hom with the structure sheaf). INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 5

Now let J = {j0 < j1 < . . . < jp+q+1} p+q X h d(η ∪ ξ)J = (−1) (η ∪ ξ)J\{jh} h=0 p+q X h = (−1) (η(Jh)≤p ⊗ ξ(Jh)≥p ) h=0 X h X h = (−1) (ηJ≤p+1\{jh} ⊗ ξJ≥p+1 ) + (−1) (ηJ≤p ⊗ ξJ≥p\{jh}) 0≤h≤p p+1≤h≤q+1 X h+p+1 = (dη)J≤p+1 ⊗ ξJ≥p+1 + (−1) (ηJ≤p ⊗ ξJ≥p\{jh+p+1}) 0≤h≤q p+1 = (dη)J≤p+1 ⊗ ξJ≥p+1 + (−1) ηJ≤p ⊗ (dξ)J≥p so that we have (3.2) d(η ∪ ξ) = dη ∪ ξ + (−1)pη ∪ dξ, from this is follows that is η and ξ are cocycles then so is η ∪ ξ, ∨ The dual of an O-module is the sheaf of sets F := ModO(F, O) which takes an open set U to the natural transformation of sheaves:

U 7→ ModOX (F|U , O|U ).

In the case that F|U = F(U) (F is quasi-coherent) we have a isomorphism of abelian groups ∨ ∼ ∼ ∨ F (U) = ModO(F|U , O|U ) = ModO(U)(F(U), O(U)) = F(U) . ∨ ∼ Proposition 3.2. Let F be an invertible O-module. Then F ⊗O F = O. Proof. This is true on trivializing open sets since for any R-module or rank 1 we ∗ ∼ ∗ ∗ have M ⊗R M = R as R-modules via the map m ⊗ m 7→ m (m).

4. Group Cohomology Let G be a group and A be an abelian group and suppose that A has the structure of a left Z[G]-module. Equivalently we can say that there exists an left action of G on A which is group homomorphism ρ : G → Aut(A).4. We will denote the left action of an element g ∈ G on a ∈ A by g · a. Right actions will be denoted by ag. These notations respect the associativity required for left and right actions respectively. Now that we’ve discussed group actions we can move on to group cohomology. There are two ways (that I know of) to define group cohomology for a group G on A with respect to a particular action. The first way is Exti functors –which are the derived functors of ModZ[G](Z, −). This means we take an injective resolution of A, apply the hom functor ModZ[G](Z, −) to the sequence and take quotients of images mod kernels of succesive maps. The alternative approach is to define a cochain

4If we ﬁrst compose with the inverse then get a map to the group of automorphism this is a right action (sometimes called an antihomomorphism) 6 TAYLOR DUPUY complex directly. Our cochains are going to be maps of sets from some cartesian power of G to A: Cn(G, A) := Maps(Gn,A). Note that since A is an abelian group the set of cochains is a group. As usual we need maps d : Cn(G, A) → Cn+1(G, A). If ϕ : Gn → A, then we deﬁne (dϕ): Gn+1 → A as follows

(dϕ)(g1, g2, g3, . . . , gn+1) = g1ϕ(g2, g3, . . . , gn+1) − ϕ(g1g2, g3, . . . , gn) + ϕ(g1, g2g3, . . . , gn+1) + ··· n n+1 +(−1) ϕ(g1, g2, . . . , gngn+1) + (−1) ϕ(g1, g2, . . . , gn). One can see that dϕ is a morphism of abelian groups if one tries. The following fact is slightly more painful to verify Proposition 4.1. For all n the image of the map d : Cn−1(G, A) → Cn(G, A) is contained in the kernel of the map d : Cn(G, A) → Cn+1(G, A). In other words d2 = 0. The kernel of the map d : Cn(G, A) → Cn+1(G, A) is the group of n-cocycles, and we denote it by Zn(G, A). The image of the map d : Cn−1(G, A) → Cn(G, A) is group of n-coboundaries. Because of the proposition which tells us that d2 = 0 (proposition 4.1) the group of n-cocycles are contained in the group of n-coboundaries, this implies that the quotient of n-cocycles modulo n-coboundaries exists. This quotient is the nth cohomology Group for group cohomology: ker(d : Cn(G, A) → Cn+1(G, A)) Zn(G, A) Hn(G, A) := = . Im(d : Cn−1(G, A) → Cn(G, A)) Bn(G, A) Remark 4.2. For arbitrary groups G and A (with A not necessarily abelian) we can still deﬁne the zeroth and ﬁrst cohomology groups. Supposing our action of G on A is a right action we will have

1 g2 (4.1) Z (G, A) := {φ : G → A | φ(g1g1) = φ(g1) φ(g2)}. Two cocycles φ, ψ ∈ Z1(G, A) are said to be cohomologous if and only if there exists an a in A such that for all g ∈ G, agφ(g) = ψ(g)a. One should check that when A is abelian that this is equivalent to the previous deﬁnition given above. Now we deﬁne the First Cohomology Set by H1(G, A) = Z1(G, A)/ ∼ . We should stress that this is not a group but a pointed set5 whose trivial element corresponds to trivial cocycles. We would now like to look at some one cocycles in particular cases. For these examples note that Z1(G, A) consists of maps ϕ : G → A satisfying

ϕ(g1g2) = ϕ(g1) + g1ϕ(g2). These are called twisted homomorphism6. We will now give some examples:

5meaning a set with a distinguished element this is trivial 6Also called skew or crossed homomorphism INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 7

× Derivative: Let G = AutR(R[x]) and A = R[x] . Let us identify automorphism of R[x] by the polynomial that x is sent to under the automorphism. under this identiﬁcation composition of polynomials. If we let our action G on A be composition: (ag) = (a ◦ g)

d × then the derivative operation D = dx : AutR(R[X]) → R[X] is a cocycle. First note that the map makes sense since the derivative of any automorphism is a unit. Also since D[(f ◦ g)(x)] = D[f](g(x))D[g](x) = (D[f]g)(x)D[g](x) we have that D is a cocycle. Schwarzian Derivative: Let G = AutR(R[X]) and A = R[x] (with the addition being the group operation). The map f 000 3 f 00 2 S[f] := − f 0 2 f 0 called the Schwarzian Derivative, is actually a cocycle when viewed appro- priately. The schwarzian derivative satiﬁes the following “chain rule”:7 S[f ◦ g] = g0(x)2S[f] ◦ g + S[g] If we let G act on the right of A (F f )(x) := f 0(x)2F (f(x)), then S is a cocycle. Note that (F f )g = (f 02 · F ◦ f)g = g02 · (f 0 ◦ g)2 · F ◦ f ◦ g = (f ◦ g)02 · F ◦ (f ◦ g) = F f◦g. 2 Top Degree Coeﬃcient: Let Gd ≤ AutR/p2R(R/p R[x]) be the collection of automorphism inducing polynomials of degree less than or equal to d. In general, every automophism induction polynomial of (R/p2R[x] can be written as f(x) = a + bx + pF (x) where b is a unit in R/p2R and ordF ≥ 2. We need to check that the collection of automorphism inducing polynomials of degree less than or equal to d form a subgroup. First one can convince themselves that if f and h are automorphism inducing polynomials that deg(f ◦ g) ≤ max(deg f, deg h), after this it only remains to show that if deg f = d it’s inverse h has degree less than or equal to d. x = f(h(x))

= af + bf h(x) + pFf (h(x))

= af + bg(ah + bhx + pFh(x)) + pFf (ah + bhx)

= af + bf ah + bf bhx + p(bf Fh(x) + Ff (ah + bhx),

7proved elsewhere 8 TAYLOR DUPUY

if the degree of Fh were bigger than the degree of Ff it would be impos- sible for the term bf Fh(x) + Ff (ah + bh) to be zero since deg(bf Fh(x)) = deg(Fh(x)) and deg Ff (ah + bhx) = deg Ff . Consider now that map Gd → R/pR given by f 7→ cd(f)/m(f) where cd(f) is the top degree term divided by p, reduced mod p and m(f) is the coeﬃcient of x in f. From the computation above we can see that d d cd(f ◦ h) = bf cd(h) + bhcd(f) = m(f)cd(h) + m(h) cd(f)

when we divide this by bf bh we get d−1 cd(f ◦ h)/m(f ◦ h) = m(h) cd(f)/m(f) + cd(h)/m(h)

which implies that ϕ(f) = cd(f)/m(f) is a cocycle under the right action h d−1 of Gd on R/pR given by A = m(h) A. We would next like to describe the connection between twisted homomorphisms and semi-direct products. Let G and H be groups. Recall that given any left 8 action ρ : G × H → H we can form a new group H oρ G . As a set H oρ G is just the cartesian product H × G. The group multiplication for this group is given as follows:

(h1, g1) ∗ρ (h2, g2) = (h1(g1h2), g1g2). If we have a right action then the group is G nρ H and the group law is given by

ρ g2 (g1, h1) ∗ (g2, h2) = (g1g2, h1 h2). Twisted homomorphism ϕ : G → H give maps from G to H n G. Proposition 4.3 (Cocycles give Maps to Semi-Direct Products). Suppose that G acts on H on the left and ϕ : G → H is a left twisted homomorphism. Then the map g 7→ (ϕ(g), h) 9 deﬁnes a group homomorphism G → H oρ G. Proof.

(ϕ(g1g2), g1g2) = (ϕ(g1)g1ϕ(g2), g1g2)

= (ϕ(g1), g1) ∗ (ϕ(g2), g2) Another way of saying this is that a twisted homomorphism ϕ splits the exact sequence via the map above

1 / H / H oρ G / H / 1 , This falls in line with the general theory: it turns out N = A is abelian that H2(G, A) (which we don’t know how to deﬁne for nonabelian N) classiﬁes group extensions of A by G with the given action. We will make this statement more

8 ρ ρ There are four reasonable notations G nρ H, G n H, H oρ G and H o G. We will reserve the superscripts for right actions and the subscripts for left actions. We could have arranged so that we would put the G on the left for left actions and the G on the right for right actions but we won’t follow this because our current choice looks best in coordinates. For right actions we ρ will use the notation G n H 9The right action case is similar with g 7→ (g, ϕ(g)) INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 9 precise below. Before proving this recall that a group extension of a group N by a group G is some group G˜ such that we have the exact sequence:

i p 1 / N / G˜ / G / 1 . Furthermore if we prescribe an action of G on N then it must be the same as the action of congugation by G (which is well deﬁned up to congugation)10. Here are some examples of group extensions Units of Z/p2Z: We will show that the units of Z/p2Z are an extension the additive group Z/pZ by the multiplicative group of units of Z/pZ. First note that we can include Z/pZ into Z/p2Z by the map a 7→ (1 + pa). It is a homomorphism since (1 + pa)(1 + pb) = (1 + p(a + b)). We will call this map the exponential map. Also notice that all of these units are precisely the kernel of the quotient map (Z/p2Z)× → (Z/pZ)×, this gives us the exact sequence

exp quot × 0 / Z/pZ / (Z/p2Z)× / (Z/pZ) / 1 . More generally if R is any ring which is is p-torsion free then we for any r > s have a similar exact sequence

exp quotient 0 / R/prR / (R/pr+sR)× / (R/psR)× / 1 where the first map is f 7→ 1 + psf and the second map is the canoni- cal quotient map. Note that the first map is injectice since two elements f and g have the same image if and only if they are congruent mod pr (which makes them the same element). We should remark that this decom- position is useful in understanding the units of (R/pn)[x], which is useful in understanding the automorphism of (R/pnR)[x],11 which is useful for understanding the automorphisms of the p-adic completion R[x]pb.12 The Affine Linear Group: Let R be a ring. The affine linear group is the group × AL1(R) := ({f(x) ∈ R[x] | f(x) = ax + b, a ∈ R , b ∈ R}, ◦) ⊂ AutR(R[x]). This group is an extension of the group R× by the additive group of R. The inclusion of R into AL1(R) if given by h 7→ τh(x) := x − h. Note that h + k 7→ τh+k(x) = (τh ◦ τk)(x) which shows that this is a group homomorphism. First it’s clear that this map is an inclusion. We will next show you that the image of R in AL1(R) is normal. First note that if f(x) = ax + b then f −1(x) = a−1 − a−1b so that on can compute: −1 (f ◦ τh ◦ f )(x) = x − ah

which shows that h is normal and that the action of AL1(R) on the image of R is given by multiplication (a, r) 7→ ar. It turns out that two elements f(x) = ax + b and g(x) = cx + d are equivalent if and only if a = c in which

10Warning: there exists sequences with the same N, G˜ and G but which have diﬀerent maps which are not the same as extensions 11Since the derivative of every automorphism is a unit 12Since it the the projective limit of automorphisms of the former type. 10 TAYLOR DUPUY

case (g ◦ f −1)(x) = x + (d − b). This shows that each element of quotient × has a representative ma(x) = ax and is isomorphic to R . This gives the sequence

× 0 / R / AL1(R) / R / 1. × Since the exact sequence splits via the homomorphism R → AL1(R) given by a 7→ ma(x) we have that ∼ × AL1(R) = R o R where the isomorphism is given by (ax+b) 7→ (b, a) since (ax+b)◦(cx+d) = (acx + ad + b) 7→ (ad + b, ac) = (b, a)(d, c). The affine linear group AL1(R) turns out be the full automorphism group AutR(R[x]) when R is a noetherian integral domain. If x 7→ f(x) induces an automorphism of R[x] then we have R[x] = R[f(x)]. This means that R[x]/hf(x)i =∼ R[x]/hxi =∼ R. This quotient map can be viewed as an evaluation map given by x 7→ c which implies that x − c ∈ hf(x)i which implies that hx − ci ⊂ hf(x)i. We actually have that x − c and f(x) ∼ ∼ R[x]/hx−ci ∼ are associates since R = R[x]/hf(x)i = hf(x)i/hx−ci = R/hf(c)i which means that f(c) = 0 (which is the Noetherian hypothesis). In particular f(x) = ax+ac for some unit a which show that f is affine linear. Conversely every affine linear automorphism is an automorphism. In particular if we look at the case R = Z/pZ, the group AL1(Z/pZ) is a group extension of Z/pZ by (Z/pZ)× distinct which is not isomorphic to 13 (Z/p2Z)×. This group extension, by what we have said previously is the trivial group extension. The following lemmas will be critical in simpifying cocycles we get get to the interaction of sheaf cohomology and group cohomology. Lemma 4.4 (Actions on Quotients). Let G be a group and A be an R-module. Suppose that G acts on A and that the action is R-linear. (1) For every r ∈ R the action of G on A/rA well defined. (2) There is a group homomorphism from G nρ A → G nρ (A/rA). Proof. The action of G on the quotient is defined by taking a representative, acting on that representative, then taking its equivalence class. In order to show this procedure is well defined we need to show that if x, y ∈ A and x − y ∈ rA then for all g we have xg − yg ∈ rA. Suppose that x − y = rm. Taking the difference of the acted on elements gives xg − yg = (x − y)g = (rz)g = rzg which is congruent to zero mod rA. This implies the action of the quotient is well-defined. The second part is clear with (g, a) 7→ (g, a¯). Here is an example of the above proposition: Let R be a p-torsion free ring. Let G = AutR(R[t]) and A = R[t] with the right action of G on A be given by composition (F g)(t) := F (g(t)). We can reduce the image mod p and still have a well defined action of the lifted automorphism.

13Because one is abelian and the other isn’t. INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 11

Lemma 4.5 (Actions of Quotients). Let G and H be groups and N a normal subgroup of G. 0 If for all x ∈ H we have xg = xg whenever g ≡ g0 mod N, then there exists a group homomorphism G n H → (G/N) n H. An application of the two reductions above could be the following. The action of AutAA[x] on A[x] induces a well defined action on A[x]/pA[x]. The action of AutA(A[x]) on A[x]/pA[x] decends to an action of AL1(A/pA) on (A[x]/pA)[x]. The first part follows from actions on quotients (lemma 4.4) and the second part of the proposition follows from proposition actions of quotients (Lemma 4.5). We will reproduce these statements in the special cases stated above: Suppose that F,G ∈ A[x] and that F ≡ G mod p. We can write F − G = pH for some H ∈ A[x]. For any g ∈ AutAA[x] we have F (x)g − G(x)g = (F (x) − G(x))g = (pH(x))g = pH(x)g, which shows that the the action on equivalence classes is well defined mod p. For the second statement, suppose that f, g ∈ AutA(A[x]) are congruent mod p and that F (x) represents some class in A[x]/pA[x]. First now that f − g ≡= 0 mod p implies that f 0 − g0 = 0 mod p. Using the above we have F (x)g = g0(x)F (g(x)) = g0(x)2F¯(g(x)) = f 0(x)2F¯(f(x)) = F (x)f , which shows that automorphisms induced by polynomials which are equivalent ∼ mod p define the same map on equivalence classes mod p. Since AL1(A/pA) = AutA(A[x]) mod p we are done.

5. Group Cohomology for Sheaves of Groups Let G be a sheaf of groups on X and A be a sheaf of abelian groups on X.A left action of G on A a collection of left actions of groups

ρU : G(U) × A(U) → A(U), for all inclusions of open sets of X, U 0 ⊂ U we have

ρU G(U) × A(U) / A(U) .

res×res res

ρ 0 G(U 0) × A(U 0) U / A(U 0)

When given a left action ρ of G on A we will denote the left action of g ∈ G(U) on A ∈ A(U) by g · a. Similarly, given a right action and some g and a we will use the notation Ag. 12 TAYLOR DUPUY

Multiplicative Group Acting on the Structure Sheaf: If we let X be a × scheme, let A = OX and let G = OX then we can deﬁne an action of G on A by (m, f) 7→ mf. Sheaf of Automorphisms acting On a Sheaf in the Obvious Way: Let 1 1 X be a scheme, let A = OX×A and G = Aut(A ) then we can deﬁne an action. Note that for every open set we have

A(U) = OX×A1 (U) ≈ OX (U)[t] where t is the parameter on A1. The sheaf Aut(A1) assigns to each open subset of X the collection of morphism ϕ U × A1 / U × A1 . ss pr1 ss sspr ss 1 ss U ys ∗ In terms of rings these are maps ϕ ∈ AutO(U)(O(U)[t]). Identifying 1 AutA (U) with a subset of polynomials in OX (U)[t] where the multiplication law is composition we have g ∈ AutA1(U) acting on F (t) ∈ 1 OX×A (U) ≈ OX (U)[t] via (F, f) 7→ F ◦ f. We can “twist” this action after multiplying by the derivative of f: (F f )(x) := f 0(x)nF (f(x)) 1 deﬁnes another action of the Aut(A ) on OX×A1 . Also we should mention that semi-direct products exist in the category of sheaves. Given a left action ρ of G on A we will form the presheaf A oρ G by letting (A oρ

G)(U) = A(U) oρU G(U). To check that this is a sheaf we just need to check the 14 sheaf axiom. ince the product of sheafs of sets is a sheaf, and A oρ G is a product as a sheaf of sets, it satisfies the sheaf axiom. Affine Linear Group: Let X be a scheme and let A = O under addition and G = O×. Let the left action of O× on O be the standard multiplication action (f, m) 7→ mf. The sheaf O o O× with the standard multiplication action is a semidirect product of sheaves. In ordinary group cohomology of a group G acting on A,15 we take the n cochains to be the maps of sets from the nth cartesian power of the group G to the group A: Cn(G, A) = Maps(Gn,A). These sets have the structure of an abelian group themselves since A is an abelian group. Next, we define a maps d : Cn(G, A) → Cn+1(G, A) which single out the cocycles Zn(G, A) = ker(d : Cn(G, A) → Cn+1(G, A) and the coboundaries Bn(G, A) = Im(d : Cn−1(G, A) → Cn(G, A)). We then get to define our cohomology groups ker(d : Cn → Cn+1) Zn(G, A) Hi(G, A) = = Im(d : Cn−1 → Cn) Bn(G, A) in virtue of the fact that d2 = 0 (which is not immediately obvious).

14S 15say on the left INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 13

To define a group cohomology of a sheaf of groups G on a sheaf of abelian group A with respect to an action ρ we proceed in a similar fashion. Our cochains are going to be morphism of sheaves of sets Cn(G, A) := Nat(Gn, A). From the viewpoint of category theory, presheaves are just contravariant functors from Top(X)16 to Sets. Morphisms of presheaves (and hence of sheaves) are natural transformations of these functors (hence to notation “Nat”). Our maps d : Cn(G, A) → Cn+1(G, A) are going to be defined similarly: If Φ : Gn−1 → A, then for all U ⊂ X and all g1, . . . , gn ∈ G(U) we define

(dΦ)U (g1, g2, g3, . . . , gn) = g1ΦU (g2, g3, . . . , gn) − ΦU (g1g2, g3, . . . , gn) + ΦU (g1, g2g3, . . . , gn) n + ··· + (−1) ΦU (g1, g2, . . . , gn−1gn). One should check that for all Φ ∈ Nat(Gn−1, A) that (dΦ) ∈ Nat(Gn+1, A). By this we mean that you should convince yourself that for all U 0 ⊂ U inclusions of open sets the following diagram commutes:

(dΦ)U G(U) × · · · × G(U) / A(U) .

resn res

(dΦ) 0 G(U 0) × · · · × G(U 0) U / A(U 0) Again we now proceed in the fashion as with ordinary group cohomology to define cocycles and coboundaries. Since for each n, Cn(G, A) is an abelian group and each d is a morphism of abelian groups the coboundaries and cocycles are already defined: The cocycles are those natural transformation17 whose image under d is indentically zero and the coboundaries are the image of d as before. This is all taking place in the category of abelian groups so we don’t need to worry about sheafifying the image or anything. Again we can define Zn(G, A) = ker(d : Cn(G, A) → Cn+1(G, A)), Bn(G, A) = Im(d : Cn−1(G, A) → Cn(G, A)). The proof that d2 = 0 follows from ordinary group cohomology fact and the prop- erty that A is a sheaf,18 and we get to define our cohomology groups Zi(G, A) Hi(G, A) := . Bi(G, A) Twisted homomorphisms will play a particularly important role in group cohomology. Derivatives: We will now copy the derivations example in the ordinary group cohomology section for sheaves of groups. Let X be a scheme and where

16The category whose objects are open subsets of X and whose morphisms are inclusions 17Morphism of sheaves of sets 18On open subsets d acts like the d from group cohomology which means that the d2 will equal zero on open subsets. Since A is a sheaf, the local triviality will patch showing that d2 is in fact the zero map. 14 TAYLOR DUPUY

1 1 × 1 π : X × A → X. We will let G = AutA and A = (π∗OX×A ) . As we 1 have seen before G(U) = AutAX (U) is the collection of maps ϕ where ϕ U × A1 / U × A1 . ss sss sss ss U ys They can be identified with subsets of O(U)[T ] consisting of polynoimals ∗ which induct automorphisms ϕ ∈ AutO(U)(O(U)[T ]). Here T is the pullback of the parameter on A1. We should also mention that A(U) = × × 1 ∼ π∗OX×A (U) = OX (U)[T ] . We want to show that D : G → A defined by D[g](T ) = g0(T ) is a cocycle in group cohomology with respect to the right group action (F (t), g(t)) 7→ (F g)(t) := g0(t)F (g(t)). First we should check that D is a well-defined morphism of sheaves of sets: Suppose that U 0 ⊂ U is an inclusion of open subset of X. We need to show that the following diagram commutes

D G(U) U / A(U)

res res

D 0 G(U 0) U / A(U 0) In our particular case, this means the following

DU × (5.1) AutA1(U) / O(U)[T ] .

U U resU0 resU0

D 0 U 0 × AutA1(U 0) / O(U )[T ] Note that we can identify AutA1(U) and O(U 0)[T ]× both as with a subsets of O(U)[t]. In both these cases we have the restiction maps given by n U U U n a0 + a1T + · + anT 7→ resU 0 (a0) + resU 0 (a1)T + ··· + resU 0 (an)T U 0 where resU 0 : O(U) → O(U ) is the restriction map for the structure sheaf. The commutivity of the diagram is just U U DU 0 ◦ resU 0 = resU 0 ◦ DU U 0 where resU 0 denotes the restriction map on polynomials O(U)[T ] → O(U )(T ) induced by the restiction map O(U) → O(U 0) on the structure sheaf. The commutivity we needed in equation 5.1 is now clear since the operation U resU 0 does nothing to the indeterminates T and the operation D does nothing to the coeﬃcients. It remains to check that check that D is twisted homomorphism in group 1 × cohomology. This means that for every U, DU : AutA (U) → O(U)[T ] is a 1 cocycle in Z (AutO(U)(O(U)[T ]), O(U)[T ]). This follows from the analysis of the cocycle in group cohomology on page-6 INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 15

Of particular importance in this theory are the quotient lemmas (Lemmas 4.4 and 4.5 of the section on group cohomology (section 4) and lemma the tells us that cocycles in group cohomology give us maps to the semidirect products (Lemma 4.3 of the section on group cohomology (section 4). Lemma 5.1 (Semi-Direct Products Induce Maps). Let X be a topological space. Let G and A be sheaves of groups and let ρ : G → A be a left action. Every 1 Φ ∈ Z (G, A)ρ induces a sheaf homomorphism G → A oρ G via g 7→ (Φ(g), g). This is what is going to happen: we will take some complicated cocycle in group cohomology then apply a cocycle in group cohomology in to get a 1-cocycle in group cohomology of the twisted cocycle. After having this we will apply a series of reductions like the quotients of action and actions of quotients. After all this is said and done we will get sections of familiar line bundles (to which we can apply trace maps to) to get sections of line bundles.

6. Interaction Between Cech and Group Cohomology The fact that cocycles in group cohomology give maps to semi-direct products (Lemma ??) and the morphisms of sheaves lemma gives maps on cohomology (Lemma 1.4) give us a pairing 1 1 1 H (G, A) × Hˇ (X, G) 7→ Hˇ (X, A o G). Proposition 6.1. The map described above is well deﬁned.

Proof. Given a cocycle in Z1(G, A) we get a map from G → G n A. The morphism of sheaves of groups obviously deﬁnes a morphism of on cohomology (Proposition 1.3).we just need to check that this is well deﬁned on the level of cocycles. Lemma 6.2 (Twisted Cocycle Condition). Let G be a sheaf of groups and A be sheaf of abelian groups that have an action. (gij,Aij) is a for G n A if and only if (gij) is a cocycle for G and

gjkgki gki (6.1) Aij + Ajk + Aki = 0

Remark 6.3. The condition for left twisted cocycles is similar: If Gij = (Aij, gij) ∈ A(Uij) o G(Uij) is a left twisted cocycle the one can derive

(6.2) Aij + gijAjk + gijgjkAki = 0.

Proof. For any sheaf of groups H the cocycle condition is hijhjkhki = 1. This is what we want to show for H = G n A and with hij = (gij,Aij) satisfying the hypotheses of the lemma:

hijhjkhki = (gij,Aij) ∗ (gjk,Ajk) ∗ (gki,Aki) gjk = (gijgjk,Aij + Ajk) ∗ (gki,Aki) gjkgki gki = (gijgjkgki,Aij + Ajk + Aki) = (1, 0).

16 TAYLOR DUPUY

Lemma 6.4 (Getting Sections). Suppose that π : E → X is a morphism of schemes with a trivializing cover {Ui : i ∈ I} with isomorphisms ϕi : Γ(Ui,E) → A(Ui) where A is a sheaf groups. Furthermore suppose that there is a group action of G on A such that

−1 gij (6.3) (ϕj ◦ ϕi )(A) = A . 1 1 ρ −1 Then Z (U; X,E) ↔ Z (U; G n A) via (Aij, gij) 7→ ϕj (Aij). Equation 6.3 really says that E is just glued together by a cocycle for G. The condition for left actions is slightly diﬀerent. We want the transition maps of E to −1 be satisfy ϕi ◦ ϕj (f) = gijf and embed the twisted cocycles via ϕi(sij) = Aij.

1 Proof. Suppose that (sij) ∈ Z (U, A) is a cech cocycle. By deﬁnition this implies

sij + sjk + ski = 0.

Now consider Aij := ϕj(sij) ∈ A(Uij). We have

ϕi(sij) + ϕi(sjk) + ϕi(ski) = 0. but notice

ϕi(ski) = Aki, −1 ϕi(sjk) = (ϕi ◦ ϕk ◦ ϕk)(sjk) −1 = (ϕi ◦ ϕk )(Ajk) gki = Ajk , −1 −1 ϕ(sij) = (ϕi ◦ ϕk ◦ ϕk ◦ ϕj ◦ ϕj)(sij) −1 −1 = ((ϕi ◦ ϕk ) ◦ (ϕk ◦ ϕj ))(Aij) gjkgki = Aij . Putting this all together we have

gjkgki gki 0 = Aij + Ajk + Aki, which is the cocycle condition for G n A. Conversely, if we are given a cocycle (gij,Aij) for G nA then we can get a cocycle −1 for E. Here we let sij := ϕj (Aij) as expected. A similar type of manipulation yields sij + sjk + ski = 0. We will now explain this more explicitly, 1 Invertible Sheaves: Let X be a schem. Suppose we are given η ∈ H (X, AL1(O)). ∼ × Since AL1(O) = O n O the cocycle can be viewed as a twisted co- 19 cycle fij = a0(fij) + a1(fij)(t) 7→ (a0(fij), a1(fij)). The multiplicative part a1(fij) = mij Deﬁnes an invertible sheaf L on X with triv- ializations ϕi : L(Ui) → O(Ui), such that for all f ∈ O(Uij) we have −1 (ϕi ◦ ϕj )(f) = mij.f By the correspondence between left twisted cocy- −1 cles and sections of group schemes we have sij := ϕi (a0(fij)) ∈ L(Uij) deﬁning a cocycle of Z1(C, L).

19Notice that we have chosen to make this isomophism with a left action so that we need follow the correspondence for left actions. INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY 17

Vector Bundles: Slightly more generally, if X is a scheme G = GLn and A = On then the condition in the sections lemma (Lemma 6.4) tells us that twisted cocycles (Mij,~vij) deﬁne cocycles of the vector bundle, −1 ˇ 1 ~vij 7→ sij = ϕj (~vij) ∈ H (X,E).