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

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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 = fUi : i 2 Ig be a collection of open sets S with X = i2I Ui. Let G be a sheaf of groups on X. Q −1 A collection (gij) 2 i;j2I 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 2 i;j2I Γ(Ui \ Uj; G) is called a coboundary if there exists some collection "i 2 G(Ui) for each i 2 I such that −1 gij = "i"j : 1 We will denote the collection of coboundaries by B (U; X; G). Note that if gij = −1 −1 −1 −1 −1 "i"j , then we have gji = "j"i = ("i"j ) = gij . Also for all i; j and k we have −1 −1 −1 gijgjkgki = "i"j "j"k "k"i = 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 ("i) 2 Q i2I G(Ui) such that "igij = hij"j: 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 sym- metric. The relation is reflexive since we can take "i = 1i 2 G(Ui). The relation is 1 2 TAYLOR DUPUY −1 −1 symmetric since "igij = hij"j =) "i hij = gij"j : Next suppose that fij ∼ gij −1 and gij ∼ hij. We can write the first relation as fij = "i gij"j and the second −1 relation as gij = δi hijδj which gives us −1 fij = "i gij"j −1 −1 = "i δi hijδj"j −1 = (δi"i) hij(δj"j): 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 2 U there exists V 2 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 'V : 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 'V . 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 2 H1(U; F) is said to be −!U ∼ similar to ξ 2 H1(V; F) if and only if there exists some W such that U ≤ W and U V V ≤ W and 'W (η) = 'W (ξ). Lemma 1.3. Let Φ: F!G be a morphism of sheaves of groups. If (fij) 2 1 1 Z (U; X; F) then gij := ΦUij (fij) 2 Z (U; X; G). Proof. Let ΦUij (fij) = gij as above. We have gijjUijk · gjkjUijk · gkijUijk = ΦUij (fij)jUijk · ΦUjk (fjk)jUijk · ΦUki (fki)jUijk = ΦUijk (fijjUijk ) · ΦUijk (fjkjUijk ) · ΦUijk (fkijUijk ) = ΦUijk (fijjUijk · fjkjUijk · fkijUijk ) = 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 2 G (Ui) such that 00 00 gi jUi\Uj = gj jUi\Uj : 00 For every open subset U the map pU : G(U) !G (U) is surjective. This means 00 that there exists some gi 2 G(Ui) such that pUi (gi) = gi . Since p is a sheaf homomorphism we have pUi (gi)jUi\Uj = pUj (gj)jUi\Uj =) pUi\Uj (gijUi\Uj ) = pUi\Uj (gjjUi\Uj ) 00 and since pUi\Uj : G(Ui \ Uj) !G (Ui \ Uj) is a sheaf homomorphism we have −1 00 1G = pUi\Uj (gijUi\Uj gjjUi\Uj ): 00 0 0 Since the sequence is exact there exists some gij 2 G (Ui\Uj) such that ιUi\Uj (gij) = −1 0 0 0 gijUi\Uj gjjUi\Uj . These satisfy the cocycle condition gijjUijk gjkjUijk gkijUijk = 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) 2 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 2 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 )i2I 2 Q 00 i2I 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···i^j ···ip+q i=0 p+1 X h (2.2) (dξ)J = (−1) ξJnfjhg h=0 J n jh := Jh = fjh1 < jh2 < ··· < jhpg p+2 X h (d(dξ))J = (−1) (dξ)Jnfjhg h=0 p+2 p+1 ! X h X k = (−1) (−1) ξJhnfjhkg h=0 k=0 p+2 p+1 p+1 ! X X h+k X h+k−1 = (−1) ξJnfjh;jkg + (−1) ξJnfjh;jkg h=0 k<h h<k = 0 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 = fi0 < i1 < : : : < ip+qg and define I≤p := fi0 < i1 < : : : < ipg and I≥p := fip < ip+1 < : : : < ip+qg: 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).
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