The Λ p construction

For the entire section, let X be a commutative ringed space.

In this section we introduce a construction which associates to a right exact triplet of modules onX, given some integerp, another right exact triple of modules onX denotedΛ^p_X(t), cf. Construction 1.1.7. This “Λ^pconstruction” will play a central role within Chapter 1 at least up to (and including) § 1.5.

TheΛ^p construction is closely related to and even essentially based upon the following notion of a “Koszul filtration” (cf. [43, (1.2.1.2)]).

Construction 1.1.1(Koszul filtration). — Letpbe an integer. Moreover, letα: G→ Hbe a morphism of modules onX. We define aZ-sequenceKby setting, for alli∈Z:

(1.1.1.1) Kⁱ:=

(im ∧^i,p−i(H)◦(∧ⁱα⊗ ∧^p−iid_H), i≥0,

∧^pH, i<0,

where

∧^i,p−i(H):∧ⁱH⊗ ∧^p−iH−→ ∧^pH

denotes the wedge product morphism. We refer toKas theKoszul filtrationin degree pinduced byαonX.

Let us shortly verify thatKis indeed a descending filtration of∧^pHby modules on X. SinceKⁱobviously is a submodule of∧^pHonXfor all integersi, it remains to show that, for all integersiandjwithi≤ j, we haveK^j⊂Kⁱ. In casei<0, this is clear as then,Kⁱ =∧^pH. Similarly, whenj> p, we know thatK^jis the zero submodule of

∧^pH, so thatK^j ⊂Kⁱis evident. We are left with the case where 0≤ i≤ j≤ p. To that end, denoteφ_ithe composition of the following obvious morphisms in Mod(X):

(1.1.1.2) ⊗( isomorphism and an epimorphism in Mod(X), respectively. The same holds fori replaced byjgiven that we defineφ_jaccordingly. Thus our claim is implied by the easy to verify identity:

a) We callt left exact(resp.right exact) onXwhen 0−→G−→H−→ F (resp. G−→H−→ F−→0) is an exact sequence of modules onX.

b) We callt short exactonXwhentis both left exact onXand right exact onX.

The upcomping series of results is preparatory for Construction 1.1.7.

Lemma 1.1.3. — Let p be an integer and t:G→H→F a right exact triple of modules on X. Then the following sequence, where the first and second arrows are given by∧^1,p−1(H)◦ (t(_{0, 1})⊗ ∧^p−1(_idH))and∧^p(t(_{1, 2})), respectively, is exact inMod(X):

(1.1.3.1) G⊗ ∧^p−1H−→ ∧^pH−→ ∧^pF−→0.

Proof. — Let x ∈ Xbe arbitrary. Then there exist isomorphisms ofOX,x-modules rendering commutative in Mod(_OX,x)the following diagram, where the top row is obtained from (1.1.3.1) by applying the stalk-at-xfunctor and the bottom row is obtained fromtx: Gx → Hx → Fxthe same way (1.1.3.1) is obtained fromt:G →

Now the bottom row of the diagram is exact in Mod(OX,x)by [16, Proposition A2.2, d]. Therefore the top row of the diagram is exact in Mod(_OX,x) too, whence the sequence (1.1.3.1) is exact in Mod(X)given thatxwas an arbitrary element of (the set underlying)X.

Proposition 1.1.4. — Let p be an integer and t: G → H → F a right exact triple of modules on X. Write K = (Kⁱ)_i∈Z for the Koszul filtration in degree p induced by t(0, 1): G→H. Then the following sequence is exact inMod(X):

(1.1.4.1) 0 //K^{1 ⊂} //∧^pH^∧

p(t(1,2))//∧^pF //0.

Proof. — By Lemma 1.1.3, the sequence (1.1.3.1) is exact in Mod(X). By the definition of the Koszul filtration, cf. Construction 1.1.1, the inclusion morphismK¹→ ∧^pHis an image in Mod(X)of the morphism

∧^1,p−1(H)◦(t(0, 1)⊗ ∧^p−1(idH)):G⊗ ∧^p−1H−→ ∧^pH;

note here that∧¹_Xequals the identity functor on Mod(X)by definition. Hence our claim follows.

Corollary 1.1.5. — Let p be an integer and t:G→H→F a right exact triple of modules on X. Denote K= (Kⁱ)_i∈Zthe Koszul filtration in degree p induced by t(0, 1):G→H.

a) There exists one, and only one,ψrendering commutative inMod(X)the following diagram:

(1.1.5.1) ∧^pH

∧^p(t(1,2))//∧^pF

(∧^pH)/K¹

ψ

99

b) Let ψ be such that the diagram in(1.1.5.1)commutes in Mod(X). Thenψis an isomorphism inMod(X).

Proof. — Both assertions are immediate consequences of Proposition 1.1.4. In order to obtain a), exploit the fact that the composition of the inclusion morphismK¹→ ∧^pH and∧^p(t(1, 2))is a zero morphism in Mod(X). In order to obtain b), make use of the fact that, by the exactness of the sequence (1.1.4.1),∧^p(t(_{1, 2}))is a cokernel in Mod(X)of the inclusion morphismK¹→ ∧^pH.

Proposition 1.1.6. — Let p be an integer and t: G → H → F a right exact triple of modules on X. Denote K= (Kⁱ)_i∈Zthe Koszul filtration in degree p induced by t(0, 1): G→ H.

a) There exists a unique ordered pair(φ₀,φ)such that the following diagram commutes inMod(X):

(1.1.6.1) H⊗ ∧^p−1H

∧^1,p−1(H)

G⊗ ∧^p−1idGH^⊗∧ //

p−1(t(1,2))

oot(0,1)⊗∧^p−1(id_H) φ0

G⊗ ∧^p−1F

φ

∧^pHoo _⊃ K¹ //K¹/K²

b) Let(φ0,φ)be an ordered pair such that the diagram in(1.1.6.1)commutes inMod(X). Thenφis an epimorphism inMod(X).

Proof. — a). By the definition of the Koszul filtration, cf. Construction 1.1.1, the inclusion morphismK¹→ ∧^pHis an image in Mod(X)of the morphism

∧^1,p−1(H)◦(t(_{0, 1})⊗ ∧^p−1(_idH))_:G⊗ ∧^p−1H−→ ∧^pH.

Therefore, there exists one, and only one, morphismφ₀making the left-hand square of the diagram in (1.1.6.1) commute in Mod(X). By Lemma 1.1.3, we know that the sequence (1.1.3.1), where we replacepbyp−1 and define the arrows as indicated in the text of the lemma, is exact in Mod(_X). Tensorizing the latter sequence withGon the left, we obtain yet another exact sequence in Mod(X):

(1.1.6.2) G⊗(G⊗ ∧^p−2H)−→G⊗ ∧^p−1H−→G⊗ ∧^p−1F−→0.

The exactness of the sequence (1.1.6.2) implies that the morphism id_G⊗∧^p−1(t(1, 2)):G⊗ ∧^p−1H−→G⊗ ∧^p−1F

is a cokernel in Mod(X)of the morphism given by the first arrow in (1.1.6.2). Besides, the definition of the Koszul filtration implies that the composition

G⊗(G⊗ ∧^p−2H)−→G⊗ ∧^p−1H−→K¹

of the first arrow in (1.1.6.2) with φ₀ maps into K² ⊂ K¹, whence composing it further with the quotient morphismK¹→K¹/K²yields a zero morphism in Mod(X). Thus by the universal property of the cokernel, there exists a uniqueφrendering commutative in Mod(X)the right-hand square in the diagram in (1.1.6.1).

b). Observe that by the commutativity of the left-hand square in (1.1.6.1),φ₀is a coimage of the morphism

∧^1,p−1(H)◦(t(0, 1)⊗ ∧^p−1(idH)):G⊗ ∧^p−1H−→ ∧^pH,

whence an epimorphism in Mod(X). Moreover, the quotient morphsimK¹→K¹/K² is an epimorphism in Mod(X). Thus the composition ofφ₀andK¹→K¹/K²is an epimorphism in Mod(X). By the commutativity of the right-hand square in (1.1.6.1), we see thatφis an epimorphism in Mod(X).

Construction 1.1.7(Λ^pconstruction). — Letpbe an integer. Moreover, lett: G→ H → Fbe a right exact triple of modules onX. WriteK = (Kⁱ)_i∈Zfor the Koszul filtration in degreepinduced byt(0, 1):G→HonX, cf. Construction 1.1.1. Recall that K is a filtration of ∧^pH by modules on X. We define a functorΛ^p(t): 3 → Mod(X)by setting, in the first place:

(_Λ^p(t))(0):=G⊗ ∧^p−1F (_Λ^p(t))(0, 0):=id_G⊗∧p−1F

(Λ^p(t))(1):= (∧^pH)/K² (Λ^p(t))(1, 1):=id_(∧^p_H)/K2

(_Λ^p(t))(₂)_:=∧^pF (_Λ^p(t))(_{2, 2})_:=_id∧pF

Now letιandπbe the unique morphisms such that the following diagram commutes in Mod(X):

(1.1.7.1) K^{2 ⊂} //K^{1 ⊂} //

∧^pH

''NNNNNNNNNNN

K¹/K² _ι //(∧^pH)/K² _π //(∧^pH)/K¹

By Proposition 1.1.6 a), we know that there exists a unique ordered pair (φ0,φ) rendering commutative in Mod(X)the diagram in (1.1.6.1). Likewise, by Corollary 1.1.5 a), there exists a uniqueψrendering commutative in Mod(_X)the diagram in (1.1.5.1). We complete our definition ofΛ^p(t)by setting:

(_Λ^p(t))(0, 1):=ι◦φ, (_Λ^p(t))(1, 2):=ψ◦π, (_Λ^p(t))(0, 2):= (ψ◦π)◦(ι◦φ).

It is a straightforward matter to convince oneself that the so definedΛ^p(t)is a functor from3 to Mod(X), i.e., a triple of modules on X. We claim that Λ^p(t)is even a right exact triple of modules onX. To see this, observe that firstly, the bottom row of the diagram in (1.1.7.1) makes up a short exact triple of modules onX, that secondly,ψis an isomorphism in Mod(X)by Corollary 1.1.5 b), and that thirdly,φis an epimorphism in Mod(X)by Proposition 1.1.6 b).

Naturally, the construction ofΛ^p(t)depends on the ringed spaceX. So, whenever we feel the need to make the reference to the ringed space X within the above construction visible notationally, we resort to writingΛ_X^p(t)instead ofΛ^p(t).

We show that theΛ^pconstruction is nicely compatible with the restriction to open subspaces.

Proposition 1.1.8. — Let U be an open subset of X, p an integer, and t: G→H→F a right exact triple of modules on X. Then t|U:G|U→H|U→F|U is a right exact triple of modules on X|U and we have(_Λ^p_X(t))|U=_Λ^p_X|U(t|U)⁽²⁾.

Proof. — The fact that the triplet|Uis right exact onX|Uis clear since the restriction functor−|U: Mod(X)→Mod(X|U)is exact. DenoteK= (Kⁱ)_i∈ZandK⁰= (K⁰ⁱ)_i∈Z the Koszul filtrations in degreepinduced byt(0, 1): G→Handt(0, 1)|U: G|U→ H|UonXandX|U, respectively. Then by the presheaf definitions of the wedge- and tensor products, we see thatKⁱ|U=K⁰ⁱfor all integersi. Now defineιandπjust as in Construction 1.1.7. Similarly, defineι⁰andπ⁰usingK⁰instead ofKandX|Uinstead

(2)Note that in order to get a real equality here, as opposed to only a “canonical isomorphism”, one has to work with the right sheafification functor.

ofX. Then by the presheaf definition of quotient sheaves, we see that the following diagram commutes in Mod(X|U):

(K¹/K²)|U ^ι|U //(K⁰/K²)|U ^π|U //(K⁰/K¹)|U

K⁰¹/K⁰²

ι⁰

//K⁰⁰/K⁰²

π⁰

//K⁰⁰/K⁰¹

Definingφandψjust as in Construction 1.1.7 and definingφ⁰andψ⁰analogously for t|Uinstead oftandX|Uinstead ofX, we deduce thatφ|U=φ⁰andψ|U=ψ⁰. Hence (_ΛX^p(_t))|U = _Λ^p_X|U(_t|U)holds according to the definitions given in Construction 1.1.7.

The remainder of this section is devoted to investigating theΛ^pconstruction in the special case where the given tripletis a split exact triple of modules onX.

Definition 1.1.9. — Letta triple of modules onX(for the purposes of the definition Xneed not necessarily be commutative as a ringed space).

a) The tripletis calledsplit exacton Xwhentis isomorphic in the category of triples of modules onXto a triple of the form

B−→B⊕A−→A,

where the first and second arrows stand for the coprojection to the first summand and projection to the second summand, respectively.

b) φis called aright splittingoftonXwhenφis a morphism of modules on X, φ:t(2)→t(1), such that we havet(1, 2)◦φ=id_t(2)in Mod(X).

Lemma 1.1.10. — Let α: G → H and φ: F → H be morphisms of modules on X and p∈Z. Assume thatα⊕φ:G⊕F→H is an isomorphism inMod(X).

a) The morphism M

ν∈Z

∧^ν,p−ν(H)◦(∧^να⊗ ∧^p−νφ): ^M

ν∈Z

(∧^νG⊗ ∧^p−νF)−→ ∧^pH is an isomorphism inMod(X).

b) Let K= (Kⁱ)_i∈Zbe the Koszul filtration in degree p induced byα. Then, for all integers i, Kⁱcorresponds to^L_ν≥i(∧^νG⊗ ∧^p−νF)under the above isomorphism.

Proof. — a). By considering stalks (just like in the proof of Lemma 1.1.3), we find that it suffices to prove the statement in case whereXis an ordinary ring. In that case, however, the statement follows from [16, Proposition A2.2, c].

b). Letibe an integer. Then for all integersν≥i, the sheaf morphism

∧^ν,p−ν(H)◦(∧^να⊗ ∧^p−νφ):∧^νG⊗ ∧^p−νF−→ ∧^pH

clearly maps intoKⁱ. Therefore, the direct sum^L_ν≥i(∧^νG⊗ ∧^p−νF)maps intoKⁱ under the given isomorphism. Conversely, any section in∧^pHcoming from

∧^i,p−i(H)◦(∧ⁱ(α)⊗ ∧^p−i(idH)):∧ⁱG⊗ ∧^p−iH−→ ∧^pH

comes from^L_ν≥i(∧^νG⊗ ∧^p−νF)under the given isomorphism as one sees decom-posing∧^p−iHin the form

M

µ≥0

(∧^µG⊗ ∧^p−i−µF)∼=∧^p−iH according to part a) (withpreplaced byp−i).

Proposition 1.1.11. — Let p be an integer and t: G → H → F a right exact triple of modules on X.

a) Let φbe a right splitting of t on X. Denote by K = (Kⁱ)_i∈Zthe Koszul filtration in degree p induced by t(_{0, 1})_: G→ H and writeκ:∧^pH →(∧^pH)/K²for the evident quotient morphism. Then the composition

κ◦ ∧^p(φ):∧^pF−→(∧^pH)/K² is a right splitting ofΛ^p(t)on X.

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

Proof. — a). By Construction 1.1.7, we see that

(Λ^p(t))(1, 2):(∧^pH)/K²−→ ∧^pF

is the unique morphism of modules on Xwhich precomposed with the quotient morphism∧^pH→(∧^pH)/K²yields∧^p(t(1, 2)): ∧^pH→ ∧^pF. Therefore, we have:

(_Λ^p(t))(1, 2)◦(κ◦ ∧^p(φ)) =∧^p(t(1, 2))◦ ∧^p(φ) =∧^p(t(1, 2)◦φ)

=∧^p(_idF) =_id∧pF.

b). WriteK= (Kⁱ)_i∈Zfor the Koszul filtration in degreepinduced by α:=t(0, 1):G−→H.

Then by the definition of(_Λ^p(t))(0, 1)in Construction 1.1.7, the following diagram commutes in Mod(X):

G⊗ ∧^p−1idGH^⊗∧ //

p−1(t(1,2))

∧^1,p(H)◦(α⊗∧^p−1(id_H))

G⊗ ∧^p−1F

(Λ^p(t))(0,1)

∧^pH _κ //(∧^pH)/K²

Sincetis a split exact triple of modules onX, there exists a right splittingφoftonX.

Using the commutativity of the diagram, we deduce that (_Λ^p(t))(0, 1) =κ◦ ∧^1,p(H)◦(α⊗ ∧^p−1(φ)).

Hence, by Lemma 1.1.10, the sheaf map(_Λ^p(t))(0, 1)is injective. Knowing already that the tripleΛ^p(t)is right exact onX(Construction 1.1.7), we deduce thatΛ^p(t) is short exact onX. ThereforeΛ^p(t)is split exact onXas by a) there exists a right splitting ofΛ^p(t)onX.

1.2. Locally split exact triples and their extension classes

Im Dokument Irreducible symplectic complex spaces (Seite 22-29)