Derived exterior powers and derived symmetric powers

a chain complex of the form[L →M] whereM is put on degree 0, we give an ex-plicit description of the exterior powers and the symmetric powers in Corollary 1.2.7.

A basic reference is [19] Chapitre I 1.3 and 4.2.

In this section, (X,AX) denotes a ringed topos. In practice, we consider the fol-lowing two cases. Let (T,AT) be a ringed space. Besides (T,AT) itself, we also con-sider the topos X = Simpl(T) of simplicial sheaves of sets on T with the constant simplicial ring AX=KAT. In the second case, the category (AX-modules)is naturally identified with the category Simpl(AT-modules) of simplicial AT-modules.

We say a simplicial AX-module M is flat if each componentMn is flat. We also say a chain complex of AX-modules K is flat if each componentKn is flat. For sim-plicialAX-modules M andN, letM⊗AXN denote the simplicial module defined by (M⊗^AXN)ⁿ =Mⁿ⊗^AXNⁿ and let M⊗^bAXN denote the bisimplicial module defined

We briefly describe the idea of the definition of derived exterior powers and derived symmetric powers for chain complexes on a ringed topos(X,AX)([19] Chap.

I 4.2.2.2, Definition 1.2.1 below) before recalling it precisely. In 1.1, we have recalled an equivalence

C_•(AX-modules)←^K→

N

Simpl(AX-modules)

of the categories of chain complexes ofAX-modules and of simplicial AX-modules. For simplicialAX-modules, the exterior power and symmetric power are defined by simply taking the exterior powers and the symmetric powers componentwise. For chain com-plexes, the definitions are given by transferring the definitions for simplicial modules by using the functors N and K.

Let (X,AX) be a ringed topos and M be a simplicial AX-module. For an in-teger p ≥ 0, the p-th symmetric power S^pM is defined as the composition ∆^o →^M (AX-modules)→^Sp (AX-modules) with the functor S^p :(AX-modules)→ (AX-modules) sending an AX-module to its p-th symmetric power. Similarly, for an integer q ≥ 0, the q-th exterior power Λ^qM is defined as the composition ∆^o →^M (AX-modules) →^Λq (AX-modules) with the functor Λ^q : (AX-modules) → (AX-modules) sending an AX -module to itsq-th exterior power. The simplicial module F^∆_AXM associated to the stan-dard free resolution FA_XM ([19] Chap. I (1.5.5.2)) has a canonical quasi-isomorphism F^∆_A

XM →M of simplicial modules.

Definition 1.2.1 ([19] Chap. I 4.2.2.2). — Let (X,AX) be a ringed topos and K be a chain complex of AX-modules.

1. For an integer p ≥ 0, the p-th derived symmetric power LS^pK is defined to be NS^pF^∆_AXKK.

2. For an integer q ≥ 0, the q-th derived exterior power LΛ^qK is defined to be NΛ^qF^∆_AXKK.

For an integerq ≥0, we put L^qS^pK =HqLS^pK. For an integerr ≥0, we also putL^rΛ^qK =HrLΛ^qK. If K→K is a homotopy equivalence of chain complexes, the induced mapsLS^pK →LS^pK andLΛ^qK →LΛ^qK are also homotopy equiv-alences. If each component of K is flat, the canonical mapsLS^pK →NS^pKK and LΛ^qK → NΛ^qKK are quasi-isomorphisms. For an AX-module F, we have canon-ical isomorphisms L⁰S^pF → S^pF and L⁰Λ^qF → Λ^qF. If F is flat, the canonical maps LS^pF →S^pF and LΛ^qF →Λ^qF are quasi-isomorphisms.

For a simplicial AX-module M and an integer p ≥0, the diagonal map M → M ⊕M induces a map

S^pM →S^p(M ⊕M) −−−→

p=p+pS^pM ⊗A_XS^pM. (1.2.1.1)

For a chain complex K and integers p =p+p, it induces a canonical map LS^pK −−−→ LS^pK ⊗^LAXLS^pK.

(1.2.1.2)

Similarly, canonical maps Λ^qM −−−→

q=q+qΛ^qM ⊗AXΛ^qM (1.2.1.3)

and

LΛ^qK −−−→ LΛ^qK ⊗^LAXLΛ^qK (1.2.1.4)

forq =q+q are defined. The following elementary lemma is useful in the sequel.

Lemma 1.2.2. — Let 0 → L → M → N → 0 be an exact sequence of flat AX -modules. Then, the canonical maps (1.2.1.1) and (1.2.1.3) define commutative diagrams of exact sequences

0→L ⊗S^p−1N →S^pM/(S²L ·S^p−2M)→ S^pN →0

↓ ↓

0→L ⊗S^p−1N → M ⊗S^p−1N →N ⊗S^p−1N →0, (1.2.2.1)

0→L ⊗Λ^p−1N →Λ^pM/(Λ²L ·Λ^p−2M)→ Λ^pN → 0

↓ ↓

0→L ⊗Λ^p−1N → M ⊗Λ^p−1N →N ⊗Λ^p−1N →0. (1.2.2.2)

Proof. — It suffices to show the exactness. By localization and a limit argument (cf. [19] I 4.2.1), it is reduced to the case where L,M andN are free of finite rank and the sequence 0→L →M →N →0 splits. Then the assertion is clear.

For chain complexesM andN, we naturally identify the complexesM[1]⊗N and(M ⊗N)[1].

Corollary 1.2.3. — 1. Let 0 → L → M → N → 0 be an exact sequence of flat simplicialAX-modules. Then, forp≥0, the upper exact sequence in (1.2.2.1) defines a distinguished triangle

→NL ⊗^LA_XNS^p−1N →N(S^pM/(S²L ·S^p−2M))→NS^pN →.

(1.2.3.1)

The boundary mapNS^pN →NL ⊗^LA_XNS^p−1N[1] is the composition

NS^pN −−−→^(1.2.1.1) NN ⊗^LA_XNS^p−1N −−−→ NL[1] ⊗^LA_XNS^p−1N. 2. Let L be an invertible AX-module, E be a flat AX-module and →L →E →K → be a distinguished triangle of chain complexes of AX-modules. For q ≥ 0, the upper exact sequence in (1.2.2.2) defines a distinguished triangle

−−−→ L ⊗LΛ^qK −−−→ Λ^q+1E −−−→ LΛ^q+1K −−−→ . (1.2.3.2)

The boundary mapLΛ^q+1K → L ⊗LΛ^qK[1] is the composition

LΛ^q+1K −−−→^(1.2.1.4) K ⊗LΛ^qK −−−→ L[1] ⊗LΛ^qK.

It induces an isomorphism L^p+1Λ^q+1K →L ⊗L^pΛ^qK either if p > 0 or if E is locally free of rank n ≤q.

Proof. — 1. It is sufficient to apply Lemma 1.2.2.

2. We may assume K is the mapping cone of L →E. Let C be the mapping cylinder of L → E. Then, for the distinguished triangle (1.2.3.2) and the descrip-tion of the boundary map, it is sufficient to apply Lemma 1.2.2 to the exact sequence 0→KL →KC →KK →0 of simplicial modules. The last assertion is clear from

the distinguished triangle (1.2.3.2).

To study explicitly the derived exterior power complex, we recall the divided power modulesΓ^rM, see e.g. [16] Exp. XVII 5.5.2. Let A be a commutative ring and M be an A-module. We regard M as a functor attaching to a commutative A-algebra A the set A⊗AM. For an integer r ≥ 0 and for A-modules M and N, a morphism f : M → N of functors is called r-ic if f(ax) = a^rf(x) for an A-algebra A, a ∈ A and x ∈ A ⊗A M. For an A-module M, the r-th divided power Γ^rM represents the functor attaching to an A-module Nthe set of r-ic morphisms M→N. The universal r-ic morphism is denoted by γ^r :M→ Γ^rM. We have Γ⁰M =A and the map M→ Γ¹M : x → γ¹x is an isomorphism. If r = r1+r2, the r-ic map M → Γ^r1M⊗Γ^r2M sendingx toγ^r1(x)⊗γ^r2(x)induces a mapΓ^rM→Γ^r1M⊗Γ^r2M. If M=M1⊕M2, the r-ic map M →

r₁+r₂=rΓ^r1M1⊗Γ^r2M2 sending (x1,x2) to (γ^r1(x1)⊗γ^r2(x2)) defines an isomorphism Γ^rM →

r₁+r₂=rΓ^r1M1 ⊗Γ^r2M2 ([16] Exp. XVII 5.5.2.6). If M is a free (resp. flat) A-module, its r-th power Γ^rM is also a free (resp. flat) A-module.

More precisely, if M is a free A-module and e1, ...,en is a basis of M, Γ^rM is also a free A-module and γ^r1e1⊗ · · · ⊗γ^rnen, (r1+ · · · +rn = r,r1, ...,rn ≥ 0) is a basis of Γ^rM. Similarly as (1.2.1.1) and (1.2.1.3), we have a canonical map

Γ^rM −−−→

r=r+rΓ^rM⊗AΓ^rM. (1.2.4.1)

The definition ofΓ^r and the properties as above are generalized to modules on a ringed topos.

Definition 1.2.4. — Let (X,AX) be a ringed topos and v:L → M be a morphism of AX-modules.

1. For an integer p ≥0, we define a chain complex S^p(L →^v M)=

S^p−qM ⊗Λ^qL,dq

by putting dq to be the composition

S^p−q−1M ⊗Λ^q+1L −−−−−→^{1⊗(1.2.1.3)} S^p−q−1M ⊗L ⊗Λ^qL

_1⊗v⊗1 S^p−qM ⊗Λ^qL ←−−−^·⊗1 S^p−q−1M ⊗M ⊗Λ^qL. (1.2.4.2)

2. For an integer q ≥0, we define a chain complex Λ^q(L →^v M)=

Λ^q−rM ⊗Γ^rL,dr

by putting dr to be the composition

Λ^q−r−1M ⊗Γ^r+1L 1⊗(1.2.4.1)

−−−−−→ Λ^q−r−1M ⊗L ⊗Γ^rL

_1⊗v⊗1 Λ^q−rM ⊗Γ^rL ←−−−^∧⊗1 Λ^q−r−1M ⊗M ⊗Γ^rL. (1.2.4.3)

The complex S^p(L →^v M) is the same as the total degreep-part of the Koszul complex Kos_•(v) and the complex Λ^q(L →^v M) is the total degree q-part of the Koszul complex Kos^•(v) defined in [19] I 4.3.1.3.

Lemma 1.2.5. — Let L and E be locally free AX-modules of rank 1 and n. Let u : L → E be an AX-linear map and u^∗ :E^∗ →L^∗ be its dual. Let

Λ^n−pE ⊗L^⊗p −−−→ HomAX(Λ^pE ⊗L^⊗n−p,ΛⁿE ⊗L^⊗n)



L^∗⊗n−p⊗Λ^pE^∗⊗ΛⁿE ⊗L^⊗n

be the isomorphism sending x⊗yto the map x⊗y →x∧x⊗y⊗y and the canonical isomorphism.

Then they induce an isomorphism

Λⁿ(L →E) −−−→ Sⁿ(E^∗ →L^∗)⊗ΛⁿE ⊗L^⊗n (1.2.5.1)

of chain complexes.

Proof. — The squares

Λ^n−p−1E⊗L^⊗p+1 −→ Λ^n−pE ⊗L^⊗p

 

Hom(Λ^p+1E⊗L^⊗n−p−1,ΛⁿE⊗L^⊗n) −→ Hom(Λ^pE ⊗L^⊗n−p,ΛⁿE ⊗L^⊗n)

 

L^{∗⊗n−p−1}⊗Λ^p+1E^∗⊗ΛⁿE⊗L^⊗n −→ L^∗⊗n−p⊗Λ^pE^∗⊗ΛⁿE ⊗L^⊗n are commutative up to (−1)^p and the assertion follows.

Lemma 1.2.6 (cf. [34] 7.34, [19] I 4.3.2). — Let 0 → L → M → N → 0 be an exact sequence of flat AX-modules. Then, the natural maps

S^p(L →^v M) −−−→ S^pN, (1.2.6.1)

Λ^q(L →^v M) −−−→ Λ^qN (1.2.6.2)

are quasi-isomorphisms.

Proof. — It is proved for the symmetric power in [19] I 4.3.2. The proof for the exterior power is similar. We briefly sketch it. For the direct sum, we have a canonical isomorphism

Λ^q(L ⊕L −→^(v,v) M ⊕M)

−−−→

q=q+q

Λ^q(L →^v M)⊗Λ^q(L →^v M).

Similarly as loc. cit., it is reduced to the case where L, M and N are free of fi-nite rank and the sequence 0 → L → M → N → 0 splits. Hence, we may identify L → M with L ⊕0 −→^(1,0) L ⊕N. By induction on rank of L, we see that Λ^q(L →^id L) is acyclic except for q =0. Thus we obtain a quasi-isomorphism

Λ^q(L →^v M)→Λ^qN and the assertion follows.

Corollary 1.2.7. — Let u : L → M be a map of flat AX-modules and let K = [L →^u M] be the mapping cone. Then, the maps (1.2.6.1) and (1.2.6.2) induce isomorphisms

S^p(L →^v M) −−−→ LS^pK, (1.2.7.1)

Λ^q(L →^v M) −−−→ LΛ^qK (1.2.7.2)

in the derived category.

Proof. — Let C = [L ^(u→^,−1) M ⊕ L] be the mapping cylinder. The exact sequence of chain complexes 0 → L → C → K → 0 induces an exact se-quence of simplicial modules 0 → KL → KC → KK → 0. By Lemma 1.2.6, we obtain a quasi-isomorphism S^p(KL → KC) → S^pKK of complexes of sim-plicial modules. It induces a quasi-isomorphism

NS^p(KL → KC)→ NS^pKK of chain complexes. Since the canonical mapM →C is a quasi-isomorphism, it induces a quasi-isomorphism KM → KC of simplicial modules. It further induces a quasi-isomorphism S^p(L → M)=

NS^p(KL → KM) →

NS^p(KL → KC). Thus we obtain an isomorphism (1.2.7.1). It is similar for the exterior power.

Proposition 1.2.8 ([34] 7.21, [19] Chap. I Proposition 4.3.2.1). — LetK be a chain complex of AX-modules and p≥0 be an integer. Then, the map (1.2.6.1) induces an isomorphism

(LΛ^pK)[p] −−−→ LS^p(K[1]) (1.2.8.1)

in the derived category.

Proof. — We briefly recall the proof of loc. cit. ReplacingK by a flat resolution, we may assume K is flat. Let C be the mapping cone of the identity K →K. Then, we have an exact sequence 0 → K → C → K[1] → 0. Applying Lemma 1.2.6 to the exact sequence 0 → KK → KC → KK[1] → 0 of sim-plicial modules, we obtain a quasi-isomorphism of complexes of simsim-plicial modules S^p(KK → KC) → S^p(KK[1]). Since C is acyclic, the map of associated simple complexes

NS^p(KK → KC)→ NΛ^p(KK)[p] is a quasi-isomorphism. Thus the

assertion follows.

Lemma 1.2.9. — The isomorphism (1.2.8.1) and the maps (1.2.1.2) and (1.2.1.4) form a commutative diagram

LΛ^pK[p] −−−→ LΛ^pK[p] ⊗^LAXLΛ^pK[p]

 

LS^p(K[1]) −−−→ LS^p(K[1])⊗^LAXLS^p(K[1]).

(1.2.9.1)

Proof. — We use the notation in the proof of Proposition 1.2.8. As in the proof of Lemma 1.2.6, we obtain maps

S^p(KK →KC)→S^p(KK^⊕2 →KC^⊕2)

→S^p(KK →KC)⊗S^p(KK →KC)

of complexes of simplicial modules. The composition is compatible with the map Λ^pKK[p] →Λ^pKK[p] ⊗Λ^pKK[p]. Hence the assertion follows.

1.3. Koszul algebras. — We introduce Koszul simplicial algebras. We will use

Dans le document On the conductor formula of Bloch (Page 9-15)