Locally split exact triples and their extension classes

Letpbe an integer. In what follows, we are going to examine theΛ^pconstruction, cf. Construction 1.1.7, when applied to locally split exact triples of modules on X. So, let t be such a triple. Then, as it turns out,Λ^p(t) is a locally split exact triple of modules onX, too. Now given thattis in particular a short exact triple of modules onX, one may consider its extension class, which is an element of Ext¹(F,G) writing t: G → H → F. Similarly, the extension class of Λ^p(t)is an element of Ext¹(∧^pF,G⊗ ∧^p−1F). The decisive result of § 1.2 will be Proposition 1.2.12, which tells how to obtain the extension class ofΛ^p(t)from the extension class oftby means of an interior product morphism

ι^p(F,G)_{: H}om(F,G)−→Hom(∧^pF,G⊗ ∧^p−1F)_,

to be defined in the realm of Construction 1.2.11. In order to conveniently describe this relationship between the extension classes oftandΛ^p(t), we introduce the device of “locally split extension classes”, cf. Notation 1.2.3.

First of all, however, let us state local versions of Definition 1.1.9 and Proposition 1.1.11.

Definition 1.2.1. — Lettbe a triple of modules onX.

a) tis calledlocally split exactonXwhen there exists an open coverUofXtopsuch that, for allU∈U, the triplet|U(i.e., the composition oftwith the restriction functor

−|U: Mod(X)→Mod(X|U)) is a split exact triple of modules onX|U.

b) φis called alocal right splittingoftonXwhenφis a function whose domain of definition, call itU, is an open cover ofXtopsuch that, for allU∈U,φ(U)is a right splitting oft|UonX|U.

Proposition 1.2.2. — Let p be an integer and t: G → H → F a right exact triple of modules on X.

a) Letφbe a local right splitting of t on X and letφ⁰be a function onU:=dom(φ)such that, for all U∈U, we have

φ⁰(U) =κ|U◦ ∧^p(φ(U)):(∧^pF)|U−→((∧^pH)/K²)|U,

whereκdenotes the quotient morphism∧^pH→(∧^pH)/K²and K= (Kⁱ)_i∈Zdenotes the Koszul filtration in degree p induced by t(_{0, 1})_: G→H. Thenφ⁰is a local right splitting of Λ^p(t)on X.

b) When t is locally split exact on X, thenΛ^p(t)is locally split exact on X.

Proof. — a). Let U ∈ U. Thenφ(U)is a right splitting of t|U on X|U. Thus by Proposition 1.1.11 a), we know that

κ⁰◦ ∧^p(φ(U)):∧^p(F|U)−→(∧^p(H|U))/K⁰²

is a right splitting ofΛ^p_X|U(t|U) on X|U, where κ⁰: ∧^p(H|U) → (∧^p(H|U))/K⁰² denotes the quotient morphism andK⁰ = (K⁰ⁱ)_i∈Zthe Koszul filtration in degree pinduced byα|U: G|U → H|U. Since(∧^pH)|U = ∧^p(H|U)andK²|U = K⁰², we haveκ|U=κ⁰. Thereforeφ⁰(U)is a right splitting ofΛ_X|U^p (t|U)onX|U. Given that Λ_X^p(t)|U=_Λ^p_X|U(t|U), we deduce thatφ⁰is a local right splitting ofΛ^p_X(t)_onX.

b). Sincetis a locally split exact triple of modules onX, there exists an open cover UofXtop such that, for allU ∈ U, the triplet|Uis split exact on X|U. Therefore, by Proposition 1.1.11, the tripleΛ^p_X|U(t|U)is split exact onX|Ufor allU ∈ U. As (_ΛX^p(t))|U=_ΛX|U^p (t|U)_{for all}U∈U, we see thatΛ_X^p(t)is a locally split exact triple of modules onX.

Notation 1.2.3(Locally split extension class). — Lett: G→ H →Fbe a short ex-act triple of modules onXwith the property that the triple

Hom(F,t): Hom(F,G)−→_Hom(F,H)−→_Hom(F,F)

is again a short exact triple of modules onX. Then we writeξ_X(t)for the image of the identity sheaf morphism idF:F→Funder the composition of mappings

(Hom(F,F))(X)−→^can H⁰(X,Hom(F,F))−→^δ0 H¹(X,Hom(F,G)),

whereδ⁰stands for the 0-th connecting homomorphism for the tripleHom(F,t)with respect to the right derived functor ofΓ(X,−): Mod(X) →Mod(Z). Note that as (Hom(F,F))(X) = Hom(F,F), we have idF ∈ (Hom(F,F))(X) so that the above definition makes indeed sense. We callξ_X(t)thelocally split extension classoftonX.

As usual, we will writeξ(t)instead ofξX(t)whenever we feel that the reference to the ringed spaceXis clear from the individual context.

Remark 1.2.4. — Lett: G→H→Fbe a locally split exact triple of modules onX.

Then we claim that

Hom(F,t): Hom(F,G)−→_Hom(F,H)−→_Hom(F,F)

is a locally split exact triple of modules onX, too. In fact, letφbe a local right splitting oftonX. PutU:=dom(φ). Then, for allU∈U, the morphism

Hom(F|U,φ(U)): Hom(F|U,F|U)−→_Hom(F|U,H|U)

is a right splitting of Hom(F|U,t(1, 2)|U)by the “functoriality” ofHom(F|U,−). SinceHom(F,t(_{1, 2}))|U=_Hom(F|U,t(_{1, 2})|U)_{for all}U∈ U, the assignmentU7→

Hom(F|U,φ(U)), forU ∈ U, constitutes a local right splitting of Hom(F,t)onX.

Moreover, since the functorHom(F,−): Mod(X)→Mod(X)is left exact, the triple Hom(F,t)is left exact onX. In conclusion, we see that the tripleHom(F,t)is indeed locally split exact onX.

Specifically,Hom(F,t)is a short exact triple of modules onX, which, in view of Notation 1.2.3, tells us that (gladly) any locally split exact triple of modules on X possesses a locally split extension class onX.

The following remark will explain briefly how our newly coined notion of locally split extension classes relates to the customary extension classes for short exact triples (i.e., short exact sequences) onX. We point out that though interesting, the contents of Remark 1.2.5 are dispensable for our subsequent exposition.

Remark 1.2.5. — Lett:G→H→Fbe a short exact triple of modules onX. Recall that theextension classof ton Xis, by definition, the image of the identity sheaf morphism idFunder the composition of mappings

Hom(F,F)−→^can (R⁰Hom(F,−))(F)−→^δ00 (R¹Hom(F,−))(G) =Ext¹(F,G), whereδ⁰ = (δ⁰ⁿ)_n∈Zstands for the sequence of connecting homomorphisms for the tripletwith respect to the right derived functor of Hom(F,−): Mod(X)→Mod(Z). Observe that the commutative diagram of categories and functors

Mod(X) ^{Hom(F,−) //}

Hom(F,−)LLLLLLLLLL%% Mod(X)

Γ(X,−)

yyrrrrrrrrrr

Mod(Z)

combined with the fact that, for all injective modulesIonX, the sheafHom(F,I)is a flasque sheaf onXtop, whence an acyclic object for the functorΓ(_X,−): Mod(_X)→ Mod(Z), induces a sequenceτ= (τ^q)_q∈Zof natural transformations

τ^q: H^q(X,−)◦_Hom(F,−)−→Ext^q(F,−)

of functors from Mod(X)_{to Mod}(_Z). The sequenceτhas the property that when Hom(F,t) is a short exact triple of modules on X and δ = (δⁿ)_n∈Z denotes the sequence of connecting homomorphisms for the tripleHom(F,t)with respect to the right derived functor ofΓ(X,−): Mod(X) →Mod(Z), then, for any integerq, the following diagram commutes in Mod(_Z):

H^q(X,Hom(F,F)) ^τ

q(F) //

δ^q

Ext^q(F,F)

δ^0q

H^q+1(X,Hom(F,G))

τ^q+1(G)

//Ext^q+1(F,G)

Moreover, the following diagram commutes in Mod(Z): (_Hom(F,F))(X)

can

Hom(F,F)

can

H⁰(X,Hom(F,F))

τ⁰(F)

//Ext⁰(F,F)

Hence, we see that the locally split extension classξ(t)oftis mapped to the extension class oftby the functionτ¹(G). In addition, by means of general homological algebra (Grothendieck spectral sequence), one can show that the mappingτ¹(G)is one-to-one.

Therefore,ξ(t)is the unique element of H¹(X,Hom(F,G))which is mapped to the extension class oftby the functionτ¹(G). We think that this observation justifies our referring toξ(t)as the “locally split extension class” oftonX.

The next couple of results are aimed at deriving, forta locally split exact triple of modules onX, from a local right splitting ofta ˇCech representation of the locally split extension classξ(t). Since the definition of ˇCech cohomology tends to vary from source to source, let us settle once and for all on the following

Conventions 1.2.6. — LetUbe an open cover ofXtopandna natural number. An n-simplexofUis an ordered(n+1)-tuple of elements ofU, i.e.,

u= (u₀, . . . ,un)∈U× · · · ×U,

such thatu₀∩ · · · ∩un 6= ∅. Note that, by definition, a 0-simplex ofUis nothing but an element ofU. Whenu = (u0, . . . ,un)is ann-simplex ofU, the intersection u0∩ · · · ∩unis called thesupportofuand denoted|u|.

LetFbe a sheaf of modules onX. Then aCech n-cochainˇ ofUwith coefficients inFis a functioncdefined on the setSofn-simplices ofUsuch that, for allu ∈S, we have c(u) ∈ F(|u|). We denote ˇCⁿ_X(U,−): Mod(X) → Mod(Z) the ˇCech n-cochain functor, so that ˇCⁿ_X(U,F)is the set of ˇCechn-cochains ofUwith coefficients inFequipped with the obvious addition andZ-scalar multiplication. Similarly, we denote ˇCq

X(U,−): Mod(X) → C(Z)the ˇCech complex functor, so that ˇCq(U,F)is the habitual ˇCech complex ofUwith coefficients inF. We write ˇZⁿ_X(U,−), ˇBⁿ_X(U,−), and ˇHⁿ_X(U,−)for the functors Mod(X)→Mod(Z)obtained by composing ˇCq(U,−) with then-cocycle, then-coboundary, and then-cohomology functors for complexes over Mod(Z), respectively. In any of the expressions ˇCq

X, ˇCⁿ_X, ˇZⁿ_X, ˇBⁿ_X, and ˇHⁿ_X, we suppress the subscript “X” whenever we feel that the correct ringed space can be guessed unambiguously from the context.

In Proposition 1.2.10 as well as in the proof of Proposition 1.2.12, we will make use of the familiar sequenceτ= (τ^q)_q∈Zof natural transformations

τ^q: ˇH^q(U,−)−→H^q(X,−)

of functors from Mod(X)to Mod(Z)which are obtained by considering the ˇCech resolution functors ˇC q(U,−): Mod(X)→C(X)together with Lemma A.6.2; in fact, the suggested construction yields a natural transformation ˇCq(U,−) = Γ(X,−)◦ Cˇq(U,−)→_RΓ(X,−)of functors from Mod(X)to K⁺(Z), from which one derives τ^q, for anyq∈Z, by applying theq-th cohomology functor H^q: K⁺(Z)→Mod(Z). Construction 1.2.7. — Lett: G→ H → Fbe a short exact triple of modules onX andφa local right splitting oftonX. For the time being, fix a 1-simplexu= (u0,u₁) ofU:=dom(φ). Setv:=|u|for better readability. Then calculating in Mod(X|v), we have:

t(1, 2)|v◦(φ(u1)|v−φ(u0)|v) = (t(1, 2)|u1)|v◦φ(u1)|v−(t(1, 2)|u0)|v◦φ(u0)|v

= (t(1, 2)|u1◦φ(u1))|v−(t(1, 2)|u0◦φ(u0))|v

=id_F|u1|v−id_F|u0|v

=idF|v−idF|v=0.

Since the tripletis short exact onX, we deduce thatt(0, 1)|v: G|v→H|vis a kernel oft(_{1, 2})|v: H|v →F|vin Mod(X|v). So, there exists one, and only one, morphism c(u):F|v→G|vin Mod(X|v)such that

t(0, 1)|v◦c(u) =φ(u₁)|v−φ(u₀)|v.

Abandoning our fixation ofu, we definecto be the function on the set of 1-simplices ofUwhich is given by the assignmentu 7→ c(u). We callctheright splitting ˇCech 1-cochainof(t,φ)onX.

By definition, for all 1-simplicesuofU, we know thatc(u)is a morphismF||u| → G||u|of modules onX||u|, i.e.,c(u)∈(_Hom(F,G))(|u|). Thus,

c∈C^ˇ1_X(U,Hom(F,G)).

In this regard, we definecto be the residue class ofcin the quotient module Cˇ¹_X(U,Hom(F,G))_{/ ˇB}¹_X(U,Hom(F,G))_.

We callctheright splitting ˇCech1-classof(t,φ)_onX.

Lemma 1.2.8. — Let t:G→H→F be a short exact triple of modules on X andUan open cover of Xtop.

a) Let n ∈ N. ThenCˇⁿ(U,t)is a short exact triple inMod(_Z)if and only if, for all n-simplices u ofU, the mapping t(1, 2)_|u|: H(|u|)→F(|u|)is surjective.

b) ˇCq(U,t)is a short exact triple inC(Z)if and only if, for all nonvoid, finiteV ⊂U such that V:=∩V6=∅, the mapping t(1, 2)_V:H(V)→F(V)is surjective.

c) Letφbe a local right splitting of t on X such thatU=dom(φ). ThenCˇ q(U,t)is a short exact triple inC(Z).

Proof. — a). We denoteSthe set ofn-simplices ofUand writeΓ= (_Γu)_u∈Sfor the family of section functorsΓu := ΓX(|u|,−): Mod(X) → Mod(Z). Then ˇCⁿ(U,−) equals, by definition, the composition of functors

∏◦_∏(_Γ): Mod(X)−→Mod(_Z)^S−→Mod(_Z),

where∏(_Γ)signifies the “external” product of the family of functorsΓ, whereas the single “∏” symbol signifies the standard product functor for the category Mod(Z). We formulate a sublemma: LetCbe any category of modules and(M_i→N_i →P_i)_i∈I a family of triples inC. Then the triple∏M_i →_∏N_i →_∏P_iis middle exact inC if and only if, for alli ∈ I, the tripleMi → Ni → Pi is middle exactC. The proof of this is clear. Emplying the sublemma in our situation, we obtain that since, for all u ∈ S, the functorΓu is left exact, the functor ˇCⁿ(U,−)is left exact, too. Thus the triple ˇCⁿ(U,t)is left exact. Hence, the triple ˇCⁿ(U,t)is short exact if and only if Cˇⁿ(U,H)→C^ˇn(U,F)→0 is exact. By the sublemma this is equivalent to saying that Γu(H)→Γu(F)→0 is exact for allu ∈ S, butΓu(H) →Γu(F) →0 is exact if and only ifH(|u|)→F(|u|)is surjective.

b). A triple of complexes of modules is short exact if and only if, for all integersn, the triple of modules in degreenis short exact. Since the triple of complexes ˇCq(U,t) is trivial in negative degrees, we see that ˇCq(U,t)is a short exact triple in C(_Z)_{if and} only if, for alln∈N, the triple ˇCⁿ(U,t)is short exact in Mod(Z), which by a) is the case if and only if, for all nonempty, finite subsetsV ⊂UwithV := ∩V 6= ∅, the mappingH(V)→F(V)is surjective.

c). LetVbe a nonvoid, finite subset ofUsuch thatV := ∩V 6= ∅. Then there exists an element Uin V. By assumption, φ(U): F|U → H|U is a morphism of modules onX|Usuch thatt(1, 2)|U◦φ(U) =idF|U. Thus, given thatV⊂U, we have t(1, 2)_V◦φ(U)_V = id_F(V), which entails thatt(1, 2)_V: H(V) → F(V)is surjective.

Therefore, ˇCq(U,t)is a short exact triple in C(_Z)by means of b).

Proposition 1.2.9. — Let t: G → H → F be a short exact triple of modules on X and φa local right splitting of t on X. PutU:= dom(φ)and denote by c (resp. c) the right splitting ˇCech1-cochain (resp. right splitting ˇCech1-class) of(t,φ)on X. Then the following assertions hold:

a) The tripleCˇ q(U,Hom(F,t)): (1.2.9.1) Cˇ q

(U,Hom(F,G))−→C^ˇ q

(U,Hom(F,H))−→C^ˇ q

(U,Hom(F,F)) is a short exact triple inC(Z).

b) We have c∈Z^ˇ1(U,Hom(F,G))and c∈H^ˇ1(U,Hom(F,G)).

c) Whenδ= (δⁿ)_n∈Zdenotes the sequence of connecting homomorphisms associated to the tripleCˇ q(U,Hom(F,t))of complexes overMod(Z)and e denotes the image of the identity sheaf mapidFunder the canonical function(_Hom(_F,F))(_X)→H^ˇ0(U,Hom(_F,F))_{, then} δ⁰(e) =c.

Proof. — a). By Remark 1.2.4, the function onUgiven by the assignment U3U7−→Hom(F|U,φ(U))

constitutes a local right splitting ofHom(F,t)onX. Moreover, the tripleHom(F,t) is a short exact triple of modules onX. Thus ˇCq(U,Hom(F,t))is a short exact triple in C(Z)by Lemma 1.2.8 c).

b). Observe that sinceφis a local right splitting oftonXandU=_dom(φ)_{, we} haveφ ∈ C^ˇ0(U,Hom(F,H)). Further on, writingd = (dⁿ)_n∈Zfor the sequence of differentials of the complex ˇCq(U,Hom(F,H)), the mapping

Cˇ¹(U,Hom(F,t(0, 1))): ˇC¹(Hom(F,G))−→C^ˇ1(U,Hom(F,H)) sendsctod⁰(φ)since, for all 1-simplicesu= (u₀,u₁)ofU,

(d⁰(φ))(u) =φ(u1)||u| −φ(u0)||u|

andc(_u)is, by definition, the unique morphismF||u| →G||u|such thatt(_{0, 1})||u| ◦ c(u) =φ(u₁)||u| −φ(u₀)||u|. Writingd⁰⁰= (d⁰⁰ⁿ)_n∈Zfor the sequence of differentials of the complex ˇCq(U,Hom(F,G)), we have

Cˇ²(U,Hom(F,t(0, 1)))◦d⁰⁰¹=d¹◦C^ˇ1(U,Hom(F,t(0, 1))).

So, since the mapping ˇC²(U,Hom(F,t(0, 1)))is one-to-one andd¹(d⁰(φ)) = 0, we see that d⁰⁰¹(c) = 0, which implies thatc ∈ Z^ˇ1(U,Hom(F,G)) and, in turn, that c∈H^ˇ1(U,Hom(F,G)).

c). Writeefor the image of idFunder the canonical function(_Hom(F,F))(X)→ Cˇ⁰(U,Hom(F,F)). Thenφis sent toeby the mapping

Cˇ⁰(U,Hom(F,t(1, 2))): ˇC⁰(U,Hom(F,H))−→C^ˇ0(U,Hom(F,F)) since, for allU∈U, (i.e., for all 0-simplices ofU) we have:

Hom(F,t(1, 2))_U(φ(U)) =t(1, 2)|U◦φ(U) =id_F|U =e(U).

Combined with the fact thatcis sent tod⁰(φ)by ˇC¹(U,Hom(F,t(0, 1))), we find that δ⁰(e) =cby the elementary definition of connecting homomorphisms for short exact triples of complexes of modules.

Proposition 1.2.10. — Let t: G → H → F be a short exact triple of modules on X,φ a local right splitting of t on X, and c the right splitting ˇCech1-class of(t,φ)on X. Put U:=dom(φ). Then the canonical mapping

(1.2.10.1) Hˇ¹(U,Hom(F,G))−→H¹(X,Hom(F,G)) sends c to the locally split extension class of t on X.

Proof. — By Proposition 1.2.9 a), the triple ˇCq(U,Hom(F,t)): (1.2.9.1) is a short ex-act triple in C(Z). So, denoteδ = (δⁿ)n∈Zthe associated sequence of connecting homomorphisms. Likewise, denoteδ⁰ = (δ⁰ⁿ)_n∈Z the sequence of connecting ho-momorphisms for the tripleHom(F,t)with respect to the right derived functor of

the functorΓ(X,−): Mod(X) → Mod(Z)(note that this makes sense given that Hom(F,t)is a short exact triple of modules onX, cf. Remark 1.2.4). Then the fol-lowing diagram commutes in Mod(Z), where the unlabeled arrows stand for the respective canonical morphisms:

(1.2.10.2) (_Hom(F,F))(X)

Γ(X,Hom(F,F))

ˇ

H⁰(U,Hom(F,F)) ^//

δ⁰

H⁰(X,Hom(F,F))

δ⁰⁰

ˇ

H¹(U,Hom(F,G)) ^//H¹(X,Hom(F,G))

By Proposition 1.2.9 c), the identity sheaf morphism idFis sent tocby the composition of the two downwards arrows on the left in (1.2.10.2). Moreover, the identity sheaf morphism idFis sent toξ(t)by the composition of the two downwards arrows on the right in (1.2.10.2), cf. Notation 1.2.3. Therefore, by the commutativity of the diagram in (1.2.10.2),cis sent toξ(t)by the canonical mapping (1.2.10.1).

Construction 1.2.11(Interior product). — Let pbe an integer. Moreover, letFand Gbe modules onX. We define a morphism of modules onX,

ι^p_X(F,G): Hom(F,G)−→_Hom(∧^pF,G⊗ ∧^p−1F),

calledinterior product morphismin degreepforFandGonX, as follows: Whenp≤0, we defineι^p_X(F,G)to be the zero morphism (note that we do not actually have a choice here). Assume p > 0 now. LetUbe an open set ofXandφan element of (Hom(F,G))(U), i.e., a morphismF|U→G|Uof modules onX|U. Then there is one, and only one, morphism

ψ:(∧^pF)|U−→(G⊗ ∧^p−1F)|U

of modules onX|Usuch that for all open setsVofX|Uand allp-tuples(x0, . . . ,xp−1) of elements ofF(V)_{, we have:}

ψV(x0∧ · · · ∧xp−1) =

∑

ν<p

(−1)^ν−1·φV(x_ν)⊗(x0∧ · · · ∧x_bν∧ · · · ∧xp−1). We let (ι^p_X(_F,G))_U be the function on (_Hom(_F,G))(_U) given by the assignment φ 7→ ψ, whereφvaries. We letι^p_X(F,G)be the function on the set of open sets of Xobtained by varyingU. Then, as one readily verifies,ι_X^p(F,G)is a morphism of modules onX fromHom(F,G)to Hom(∧^pF,G⊗ ∧^p−1F). Just as usual, we will writeι^p(F,G)instead ofι^p_X(F,G)whenever we feel the ringed spaceXis clear from the context.

Proposition 1.2.12. — Let t:G→H→F be a locally split exact triple of modules on X and p an integer. Then the function

H¹(X,ι^p(F,G)): H¹(X,Hom(F,G))−→H¹(X,Hom(∧^pF,G⊗ ∧^p−1F)) sendsξ(t)toξ(_Λ^p(t)).

Proof. — First of all, we note that sincetis a locally split exact triple of modules onX, the triplet⁰:=_Λ^p(t)is a locally split exact triple of modules onXby Proposition 1.2.2 b), whence it makes sense to speak ofξ(t⁰). Whenp≤0, we know thatHom(∧^pF,G⊗

∧^p−1F)∼=0 in Mod(X)and thus H¹(X,Hom(∧^pF,G⊗ ∧^p−1F))∼=0 in Mod(Z), so that our assertion is true in this case. So, from now on, we assume thatpis a natural number different from 0. Astis a locally split exact triple of modules onX, there exists a local right splittingφoftonX. PutU:=dom(φ). Letc∈C^ˇ1(U,Hom(F,G))be the right splitting ˇCech 1-cochain associated to(t,φ), cf. Construction 1.2.7, and denote byK = (Kⁱ)_i∈Zthe Koszul filtration in degree pinduced byα := t(_{0, 1})_: G → H.

Defineφ⁰to be the unique function onUsuch that, for allU∈U, we have φ⁰(U) =κ|U◦ ∧^p(φ(U)):(∧^pF)|U−→((∧^pH)/K²)|U,

whereκdenotes the quotient sheaf morphism∧^pH→(∧^pH)/K². Thenφ⁰is a local right splitting oft⁰ by Proposition 1.2.2 a). Writec⁰ for the right splitting ˇCech 1-cochain associated to(t⁰,φ⁰)and abbreviateι_X^p(F,G)toι. We claim thatcis sent toc⁰ by the mapping

Cˇ¹(U,ι): ˇC¹(U,Hom(F,G))−→C^ˇ1(U,Hom(∧^pF,G⊗ ∧^p−1F)).

In order to check this, letu = (U₀,U₁)be a 1-simplex ofUandVan open set ofX which is contained in|u|=U0∩U1. Observe that whenh0, . . . ,hp−1are elements of H(V)andg0,g1are elements ofG(V)such thath0 =α_V(g0)andh1 =α_V(g1)(i.e., p>1), we have

(1.2.12.1) κ_V(h0∧ · · · ∧hp−1) =0

in((∧^pH)/K²)(V)by the definition of the Koszul filtration. Further, observe that writingα⁰fort⁰(_{0, 1})_andβfort(_{1, 2}), the following diagram commutes in Mod(X) by the definition ofα⁰in theΛ^pconstruction:

(1.2.12.2) G⊗ ∧^p−1idH^G⊗∧ //

p−1(β)

∧^1,p−1(H)◦(α⊗∧^p−1(id_H))

G⊗ ∧^p−1F

α⁰

∧^pH _κ //(∧^pH)/K²

Let f0, . . . ,fp−1be elements ofF(V). Then, on the one hand, we have:

whence as elements of(_Hom(∧^pF,G⊗ ∧^p−1F))(|u|). In turn, asuwas an arbitrary 1-simplex ofU,

(1.2.12.3) (C^ˇ1(U,ι))(c) =c⁰

as claimed. Writet⁰ast⁰: G⁰ → H⁰ →F⁰. Then the following diagram, where the horizontal arrows altogether stand for the respective canonical morphisms, commutes in Mod(Z):

(1.2.12.4)

Zˇ¹(U,Hom(F,G)) ^//

Zˇ¹(U,ι)

Hˇ¹(U,Hom(F,G)) ^//

Hˇ¹(U,ι)

H¹(X,Hom(F,G))

H¹(X,ι)

ˇ

Z¹(U,Hom(F⁰,G⁰)) ^//H^ˇ1(U,Hom(F⁰,G⁰)) ^//H¹(X,Hom(F⁰,G⁰)) By Proposition 1.2.10, we know that c(resp.c⁰) is sent toξ(t) (resp.ξ(t⁰)) by the composition of arrows in the upper (resp. lower) row of (1.2.12.4). By (1.2.12.3), we have (_Z^ˇ1(U,ι))(_c) = _c⁰_{. Hence,} (_H¹(_X,ι))(ξ(_t)) = ξ(_t⁰) by the commutativity of (1.2.12.1).

Im Dokument Irreducible symplectic complex spaces (Seite 29-39)