Euler sequence and Koszul complex of a module

c 2016 by Institut Mittag-Leffler. All rights reserved

Euler sequence and Koszul complex of a module

Bj¨orn Andreas, Dar´ıo S´anchez G´omez and Fernando Sancho de Salas

Abstract. We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differentialp-forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.

Introduction

This paper deals with two related questions: the acyclicity of the Koszul com- plex of a module and the cohomology of the sheaves of (relative and absolute) differentialp-forms of a projective bundle over a scheme.

LetM be a module over a commutative ringA. One has the Koszul complex Kos(M)=Λ·M⊗AS·M, where Λ·M andS·M stand for the exterior and symmetric algebras ofM. It is a graded complex Kos(M)=

n≥0Kos(M)n, whosen-th graded component Kos(M)n is the complex:

0−−→ΛⁿM−−→Λⁿ⁻¹M⊗M−−→Λⁿ⁻²M⊗S²M−−→...−−→SⁿM−−→0 It has been known for many years that Kos(M)_n is acyclic for n>0, provided that M is a flat A-module or n is invertible inA (see [3] or [10]). It was conjectured in [11] that Kos(M) is always acyclic. A counterexample in characteristic 2 was given in [5], but it is also proved there thatH_μ(Kos(M)_μ)=0 for anyM, whereμis the minimal number of generators ofM. Leaving aside the case of characteristic 2 (whose pathology is clear for the exterior algebra), we prove two new evidences for the validity of the conjecture (for A Noetherian): firstly, we prove (Theorem 1.6) that, for any finitely generatedM, Kos(M)_nis acyclic forn0; secondly, we prove

This work was supported by the SFB 647 ‘Space-Time-Matter:Arithmetic and Geometric Structures’ of the DFG (German Research Foundation) and by the Spanish grants MTM2013- 45935-P (MINECO) and FS/12-2014 (Samuel Sol´orzano Barruso Foundation).

(Theorem 1.7) that if I is an ideal locally generated by a regular sequence, then Kos(I)n is acyclic for anyn>0. These two results are a consequence of relating the Koszul complex Kos(M) with the geometry of the spaceP=ProjS·M, as follows.

First of all, we shall reformulate the Koszul complex in terms of differential forms ofS·M overA: the canonical isomorphism Ω_S·M/A=M⊗AS·M allows us to interpret the Koszul complex Kos(M) as the complex of differential forms Ω·

S·M/A

whose differential,iD: Ω^p_S·M/A→Ω^p_S⁻·M/A¹ , is the inner product with theA-derivation D: S·M→S·M consisting in multiplication bynonSⁿM. By homogeneous local- ization, one obtains a complex ofOP-modulesKos(M ) onP. Our first result (The- orem1.4) is that the complexKos(M ) is acyclic with factors (cycles or boundaries) the sheaves Ω^p_P/A. Moreover, one has a natural morphism

Kos(M)_n−−→π_∗[Kos(M)⊗O_P(n)]

withπ:P→SpecAthe canonical morphism. In Theorem1.5we give (cohomologi- cal) sufficient conditions for the acyclicity of the complexesπ_∗[Kos(M )⊗OP(n)] and Kos(M)n. These conditions, under Noetherian hypothesis, are satisfied for n0, thus obtaining Theorem 1.6. The acyclicity of the Koszul complex of a locally regular ideal follows then from Theorem1.5and the theorem of formal functions.

The advantage of expressing the Koszul complex Kos(M) as (Ω·

S·M/A, iD) is two-fold. Firstly, it makes clear its relationship with the De Rham complex (Ω·

S·M/A,d): The Koszul and De Rham differentials are related by Cartan’s for- mula: iD◦d + d◦iD= multiplication by n on Kos(M)n. This yields a splitting result (Proposition 1.10 or Corollary1.11) which will be essential for some coho- mological results in Section 3 as we shall explain later on. Secondly, it allows a natural generalization (which is the subject of Section2): If A is a k-algebra, we define the complex Kos(M/k) as the complex of differential forms (overk), Ω·

S·M/k

whose differential is the inner product with the sameD as before. Again, one has that Kos(M/k)=

n≥0Kos(M/k)_n and it induces, by homogeneous localization, a complexKos(M/k) of modules on Pwhich is also acyclic and whose factors are the sheaves Ω^p_P/k (Theorem 2.1). We can reproduce the aforementioned results about the complexes Kos(M)n,Kos(M ), for the complexes Kos(M/k)n,Kos(M/k).

Section 3 deals with the second subject of the paper: let E be a locally free module of rankr+1 on ak-schemeX and letπ:P→X be the associated projective bundle, i.e.,P=ProjS·E. There are well known results about the (global and rela- tive) cohomology of the sheaves Ω^p_P/X(n) and Ω^p_P/k(n) (we are using the standard ab- breviated notationN(n)=N ⊗O_P(n)) due to Deligne, Verdier and Berthelot-Illusie ([4], [12], [1]) and about the cohomology of the sheaves Ω^p_P

r(n) of the ordinary pro- jective space due to Bott (the so called Bott’s formula, [2]). We shall not use their

results; instead, we reprove them and we obtain some new results, overall whenX is aQ-scheme. Let us be more precise:

In Theorem 3.4 we compute the relative cohomology sheaves Rⁱπ_∗Ω^p_P/X(n), obtaining Deligne’s result (see [4] and also [12]) and a new (splitting) result, in the case of a Q-scheme, concerning the sheaves π_∗Ω^p_P/X(n) and R^rπ_∗Ω^p_P/X(−n) for n>0. We obtain Bott formula for the projective space as a consequence. In Theorem3.11we compute the relative cohomology sheavesRⁱπ_∗Ω^p_P/k(n), obtaining Verdier’s results (see [12]) and improving them in two ways: first, we give a more explicit description ofπ_∗Ω^p_P/k(n) and ofR^rπ_∗Ω^p_P/k(−n) forn>0; secondly, we obtain a splitting result for these sheaves whenX is aQ-scheme (as in the relative case).

Regarding Bott’s formula, we are able to generalize it for a projective bundle, computing the dimension of the cohomology vector spaces H^q(P,Ω^p_P/X(n)) and H^q(P,Ω^p_P/k(n)) whenX is a properk-scheme of characteristic zero (Corollaries3.7 and3.14).

It should be mentioned that these results make use of the complexesKos( E) (as Deligne and Verdier) andKos( E/k). The complex Kos( E) is essentially equivalent to the exact sequence

0−−→Ω_P/X−−→(π^∗E)⊗OP(−1)−−→OP−−→0

which is usually called Euler sequence. The complexKos( E/k) is equivalent to the exact sequence

0−−→Ω_P/k−−→Ω_B/k−−→O_P−−→0

withB=S·E, which we have called global Euler sequence. These sequences still hold for any A-moduleM (which we have called relative and global Euler sequences of M). The aforementioned results about the acyclicity of the Koszul complexes of a module obtained in Sections1 and2are a consequence of this fact.

1. Relative Euler sequence of a module and Koszul complexes Let (X,O) be a scheme and let M be quasi-coherent O-module. Let B= S·Mbe the symmetric algebra of M(over O), which is a gradedO-algebra: the homogeneous component of degree n is Bn=SⁿM. The module Ω_B/O of K¨ahler differentials is a gradedB-module in a natural way: B⊗OB is a gradedO-algebra, with (B⊗OB)n= ⊕

p+q=nBp⊗OBq and the natural morphismB⊗OB→Bis a degree 0 homogeneous morphism of graded algebras. Hence, the kernel Δ is a homogeneous ideal and Δ/Δ²=Ω_B/O is a graded B-module. If bp, bq∈B are homogeneous of degreep, q, then bpdbq is an element of Ω_B/O of degreep+q. We shall denote by Ω^p_B/O the p-th exterior power of Ω_B/O, that is Λ^p_BΩ_B/O, which is also a graded

B-module in a natural way. For each O-moduleN, N ⊗OB is a graded B-module with gradation: (N ⊗OB)n=N ⊗OBn. Then one has the following basic result:

Theorem 1.1. The natural morphism of gradedB-modules M⊗_OB[−1]−−→Ω_B/O

m⊗b−→bdm

is an isomorphism. Hence Ω^p_B/OΛ^pM⊗_OB[−p], where ΛⁱM=Λⁱ_OM.

The natural morphism M⊗_OSⁱM→Sⁱ⁺¹M defines a degree zero homoge- neous morphism of B-modules Ω_B/O=M⊗OB[−1]→B which induces a (degree zero) O-derivation D: B→B, such that Ω_B/O→B is the inner product with D.

This derivation consists in multiplication bynin degreen. It induces homogeneous morphisms of degree zero:

iD: Ω^p_B/O−−→Ω^p_B⁻_/O¹ and we obtain:

Definition1.2. The Koszul complex, denoted by Kos(M), is the complex:

(1) ... Ω^p_B/O

i_D

Ω^p_B⁻_/O¹

i_D

...

i_D

Ω_B/O

i_D

B 0

Via Theorem1.1, this complex is

...−−^iD→Λ^pM⊗_OB[−p]−−^iD→...−−^iD→M⊗_OB[−1]−−^iD→B −→0

Taking the homogeneous components of degree n≥0, we obtain a complex of O-modules, which we denote by Kos(M)_n:

0−→ΛⁿM −→Λⁿ⁻¹M⊗_OM −→...−→M⊗_OSⁿ⁻¹M −→SⁿM −→0 such that Kos(M)= ⊕

n≥0

Kos(M)_n.

Now let P=ProjB and π: P→X the natural morphism. We shall use the following standard notations: for each O_P-module N, we shall denote by N(n) the twisted sheaf N ⊗OPOP(n) and for each graded B-module N we shall denote by N the sheaf of OP-modules obtained by homogeneous localization. We shall use without mention the following facts: homogeneous localization commutes with exterior powers and for any quasi-coherent module L on X one has (L⊗_OB[r])=

(π^∗L)(r).

Definition 1.3. Taking homogeneous localization on the Koszul complex (1), we obtain a complex ofOP-modules, which we denote byKos( M):

(2) ... Ω^d_B/O

i_D

Ω^d_B⁻_/O¹

i_D

...

i_D

Ω_B/O i_D

OP 0

By Theorem1.1,Ω^d_B/O=(π^∗Λ^dM)(−d), henceKos( M) can be written as ...−−^iD→(π^∗Λ^dM)(−d)−−^iD→...−−→(π^∗M)(−1)−−^iD→O_P−−→0

Theorem 1.4. The complex Kos( M) is acyclic (that is, an exact sequence).

Moreover,

Ω^p_P/X= KerΩ^p_B/O−−^iD→Ω^p_B⁻_/O¹ Hence one has exact sequences

0−−→Ω^p_P/X−−→Ω^p_B/O−−→Ω^p_P⁻_/X¹−−→0 and right and left resolutions ofΩ^p_P/X:

0−−→Ω^p_P/X−−→Ω^p_B/O−−→Ω^p_B⁻_/O¹ −−→...−−→Ω_B/O−−→O_P−−→0 ...−−→Ω^r+1_B/O−−→Ω^r_B/O−−→...−−→Ω^p+1_B/O−−→Ω^p_P/X−−→0 In particular, forp=1the exact sequence

(3) 0−−→Ω_P/X−−→Ω_B/O−−→O_P−−→0 is called the(relative)Euler sequence.

Proof. The morphismΩ_B/O→O_P is surjective, sinceM⊗_OB[−1]→Bis surjec- tive in positive degree. LetK be the kernel. We obtain an exact sequence

0−−→K−−→Ω_B/O−−→OP−−→0

SinceOP is free, this sequence splits locally; then, it induces exact sequences 0−−→Λ^pK−−→Ω^p_B/O−−→Λ^p−1K−−→0

Joining these exact sequences one obtains the Koszul complexKos( M). This proves the acyclicity ofKos( M). To conclude, it suffices to prove that K=Ω_P/X.

Let us first define a morphism Ω_P/X→Ω_B/O. Assume for simplicity thatX= SpecA. For eachb∈Bof degree 1, letU_bthe standard affine open subset ofPdefined

byUb=Spec(B(b)), with B(b) the 0-degree component of Bb. The natural inclusion B(b)→Bb induces a morphism Ω_B(b)/A→Ω_Bb/A=(Ω_B/A)b which takes values in the 0-degree component, (Ω_B/A)_(b). Thus one has a morphism Ω_B(b)/A→(Ω_B/A)_(b), i.e.

a morphism Γ(Ub,Ω_P/X)→Γ(Ub,Ω_B/A). One checks that these morphisms glue to a morphism f: Ω_P/X→Ω_B/A. This morphism is injective, because the inclusion B(b)→Bb has a retract, c_n/b^k→c_n/bⁿ, which induces a retract in the differentials.

The composition Ω_P/X→Ω_B/A→OP is null, as one checks in eachUb: (iD◦f)(d(ck

b^k)) =iD

b^kdck−ckdb^k b^2k

=b^kiDdck−ckiDdb^k b^2k = 0

because iDdcr=rcr for any element cr of degree r. Thus, we have that Ω_P/X is contained in the kernel of Ω_B/A→O_P. To conclude, it is enough to see that the image ofΩ²_B/A→^iDΩ_B/A is contained in Ω_P/X. Again, this is a computation in each Ub; one checks the equality

iD

dcp∧dcq

b^p+q

=pcp

b^pd cq

b^q

−qcq

b^q d cp

b^p

and the right member belongs to Ω_B(b)/A.

For each n∈Z, we shall denote by Kos( M)(n) the complex Kos( M) twisted by OP(n) (notice that the differential of the Koszul complex is OP-linear). The differential of the complexKos( M)(n) is still denoted by iD.

1.1. Acyclicity of the Koszul complex of a module LetKos( M)_n:=π_∗(Kos(M)(n)). The natural morphisms

[Ω^p_B/O]_n−−→π_∗[Ω^p_B/O(n)]

give a morphism of complexes

Kos(M)n−−→Kos( M)n

and one has:

Theorem 1.5. Let M be a finitely generated quasi-coherent module on a scheme(X,O),P=ProjS·Mandπ:P→Xthe natural morphism. Letdbe the min- imal number of generators of M(i.e.,it is the greatest integer such that Λ^dM=0) andn>0. Then:

1. IfR^jπ_∗[Ωⁱ_B/O(n)]=0for anyj >0and any0≤i≤d,thenKos( M)nis acyclic.

2. If (1)holds and the natural morphism[Ωⁱ_B/O]n→π_∗[Ωⁱ_B/O(n)]is an isomor- phism for any0≤i≤d,then Kos(M)n is also acyclic.

Proof. (1) By Theorem1.4, the complexKos( M)(n) is acyclic. Since the (non- zero) terms of this complex areΩⁱ_B/O(n), the hypothesis tells us thatπ_∗(Kos( M)⊗ OP(n)) is acyclic, that is, Kos( M)n is acyclic.

(2) By hypothesis, Kos(M)n→Kos( M)nis an isomorphism and then Kos(M)n

is also acyclic.

Theorem 1.6. LetXbe a Noetherian scheme andMa coherent module onX. The Koszul complexesKos(M)n andKos( M)n are acyclic for n0.

Proof. Indeed, the hypothesis (1) and (2) of Theorem 1.5 hold for n0 (see [8, Theorem 2.2.1] and [7, Section 3.3 and Section 3.4]).

Theorem 1.7. LetI be an ideal of a Noetherian ringA. IfI is locally gener- ated by a regular sequence,then Kos(I)n andKos(I) n are acyclic for anyn>0.

Proof. In this caseπ: P→X=SpecAis the blow-up with respect toI, because SⁿI=Iⁿ, since I is locally a regular ideal ([9]). Letd be the minimum number of generators ofI. By Theorem 1.5, it suffices to see that for any A-module M and any 0≤i≤done has:

H^j(P,(π^∗M)(n−i)) = 0 ifj >0 M⊗AIⁿ⁻ⁱ ifj=0

This is a consequence of the Theorem of formal functions (see [8, Corol- lary 4.1.7]). Indeed, let Y_r=SpecA/I^r, E_r=π⁻¹(Y_r) and π_r:E_r→Y_r. One has thatEr=ProjS·

A/I^r(I/I^r+1) is a projective bundle overYr, becauseI/I^r+1 is a lo- cally freeA/I^r-module of rankd, sinceI is locally regular. Hence, for any module N onY_r and anym>−done has

H^j(Er,(π^∗_rN)(m)) = 0 ifj >0 N⊗A/I^rI^m/I^m+r ifj=0 Now, by the theorem of formal functions (letm=n−i)

H^j(P,(π^∗M)(m))^∧= lim

← r

H^j(Er, π_r^∗(M/I^rM)(m)) = 0, forj >0.

For j=0, the natural morphism M⊗AI^m→H⁰(P,(π^∗M)(m)) is an isomor- phism because it is an isomorphism after completion byI:

H⁰(P,(π^∗M)(m))^∧= lim

←r

H⁰(Er, π_r^∗(M/I^rM)(m))

= lim

←r

(M/I^rM)⊗A/I^rI^m/I^m+r

= lim

←r

(M⊗AS^mI)⊗AA/I^r= (M⊗AI^m)^∧.

Remark 1.8. Letdbe the minimum number of generators ofM. SinceKos( M) is acyclic and π_∗ is left exact, one has that H_d(Kos(M)_n)=0 for any n. One the other hand, it is proved in [5] that Hd(Kos(M)d)=0. One cannot expect Kos(M)n→Kos( M)n to be an isomorphism in general. For instance, consider X=SpecA with A=k[u, v, s1, s2, t1, t2]/I where k is a field and I=(−us1+vt1+ ut₂, vs₁+us₂−vt₂, vs₂, ut₁). Let M=(Ax⊕Ay)/A(¯ux+ ¯vy), where ¯u (resp. ¯v) is the class ofu(resp. v) in A. Then one can prove that the mapM→π_∗OP(1) is not injective (for details we refer to section 26.21 of The Stacks project). So that the question which arises here is whether Kos(M)n→Kos( M)nis a quasi-isomorphism.

We do not know the answer, besides the acyclicity theorems for both complexes mentioned above.

1.2. Koszul versus De Rham

The exterior differential defines morphisms d : Ω^p_B/O−−→Ω^p+1_B/O

which areO-linear, but notB-linear. One has then the De Rham complex:

DeRham(M)≡0−−→B−^d→Ω_B/O d

−→...−^d→Ω^p_B/O−^d→Ω^p+1_B/O−−→...

which can be reformulated as

0−−→B−^d→M⊗_OB[−1]−−→...Λ^pM⊗_OB[−p]−−→Λ^p+1M⊗_OB[−p−1]−−→...

Taking into account that d is homogeneous of degree 0, one has for each n≥0 a complex ofO-modules

0−−→SⁿM −−→M⊗_OSⁿ⁻¹−−→...−−→Λⁿ⁻¹M⊗_OM −−→ΛⁿM −−→0 which we denote by DeRham(M)_n.

The differentials of the Koszul and De Rham complexes are related by Cartan’s formula: iD◦d + d◦iD= multiplication bynon Λ^pM⊗OS^n−pM. This immediately implies the following result:

Proposition 1.9. IfX is a scheme overQ,thenKos(M)_n andDeRham(M)_n are homotopically trivial for any n>0. In particular,they are acyclic.

Now we pass to homogeneous localizations. The differential d : Ω^p_B/O→Ω^p+1_B/O is compatible with homogeneous localization, since for any ωk+n∈Ω^p_B/O of degree k+nand any b∈Bof degree 1, one has:

d ω_k+n

bⁿ

=bⁿdω_k+n−(dbⁿ)∧ω_k+n b²ⁿ

Thus, for anyn∈Z, one has (O-linear) morphisms of sheaves d : Ω^p_B/O(n)−−→Ω^p+1_B/O(n) and we obtain, for eachn, a complex of sheaves onP:

DeRham( M, n) = 0−−→O_P(n)−^d→Ω_B/O(n)−^d→...−^d→Ω^p_B/O(n)−−→...

which can be reformulated as

0−−→OP(n)−^d→(π^∗M)(n−1)−−→...−−→(π^∗Λ^pM)(n−p)−−→...

It should be noticed that DeRham( M, n) is not the complex obtained for n=0 twisted byO_P(n), because the differential is not O_P-linear.

Again, one has that i_D◦d + d◦i_D= multiplication by n, onΩ^p_B/O(n). Hence, one has:

Proposition 1.10. If X is a scheme overQ, then the complexes Kos( M)(n) andDeRham( M, n)are homotopically trivial for any n=0.

Corollary 1.11. LetX be a scheme overQ. For anyn=0,the exact sequences 0−−→Ω^p_P/X(n)−−→Ω^p_B/O(n)−−→Ω^p_P⁻_/X¹(n)−−→0

split as sheaves ofO-modules(but not as OP-modules).

2. Global Euler sequence of a module and Koszul complexes Assume that (X,O) is a k-scheme, where k is a ring (just for simplicity, one could assume thatkis another scheme). LetMbe anO-module andB=S·Mthe symmetric algebra overO. Instead of considering the module of K¨ahler differentials ofB over O, we shall now consider the module of K¨ahler differentials over k, that is, Ω_B/k. As it happened with Ω_B/O (Section 1), the module Ω_B/k is a graded B- module in a natural way. TheO-derivationD:B→Bis in particular ak-derivation, hence it defines a morphismi_D: Ω_B/k→B, which is nothing but the composition of the natural morphism Ω_B/k→Ω_B/O with the inner product i_D: Ω_B/O→B defined in Section1. Again we obtain a complex ofB-modules (Ω·

B/k, iD) which we denote by Kos(M/k):

(4) ... Ω^p_B/k

i_D

Ω^p_B⁻_/k¹

i_D

...

i_D

Ω_B/k

i_D

B 0

and for eachn≥0 a complex ofO-modules Kos(M/k)_n=... [Ω^p_B/k]_n

i_D

... [Ω_B/k]_n

i_D

SⁿM 0

By homogeneous localization one has a complex of O_P-modules, denoted by Kos( M/k):

... Ω^p_B/k

iD

Ω^p_B⁻_/k¹

iD

... Ω_B/k iD

OP 0

Theorem 2.1. The complexKos( M/k)is acyclic(that is,an exact sequence).

Moreover,

Ω^p_P/k= Ker

Ω^p_B/k−^iD→Ω^p_B⁻_/k¹

Hence one has exact sequences

0−−→Ω^p_P/k−−→Ω^p_B/k−−→Ω^p_P⁻_/k¹−−→0 and right and left resolutions ofΩ^p_P/k:

0−−→Ω^p_P/k−−→Ω^p_B/k−−→Ω^p_B⁻_/k¹−−→...−−→Ω_B/k−−→OP−−→0 ...−−→Ω^e_B/k−−→Ω^e_B⁻_/k¹−−→...−−→Ω^p+1_B/k−−→Ω^p_P/k−−→0 In particular, forp=1the exact sequence

(5) 0−−→Ω_P/k−−→Ω_B/k−−→O_P−−→0 is called the(global)Euler sequence.

Proof. It is completely analogous to the proof of Theorem1.4.

LetKos( M/k)_n:=π_∗(Kos( M/k)(n)). The natural morphisms [Ω^p_B/k]_n−−→π_∗Ω^p_B/k(n)

give a morphism of complexes

Kos(M/k)n−−→Kos( M/k)n

In complete analogy to the relative setting we have the following:

Theorem 2.2. Let M be a finitely generated quasi-coherent module on a scheme (X,O), B=S·M, P=ProjB and π: P→X the natural morphism. Let d be the greatest integer such thatΩ^d_B/k=0andn>0. Then:

1. If R^jπ_∗(Ωⁱ_B/k(n))=0 for any j >0 and any 0≤i≤d, then Kos( M/k)n is acyclic.

2. If (1)holds and the natural morphism [Ωⁱ_B/k]_n→π_∗(Ωⁱ_B/k(n))is an isomor- phism for any0≤i≤d, thenKos(M/k)n is also acyclic.

Theorem 2.3. LetXbe a Noetherian scheme andMa coherent module onX. The Koszul complexesKos(M/k)n and Kos( M/k)n are acyclic for n0.

2.1. Koszul versus De Rham (Global case)

Now we pass to the De Rham complex (over k). The k-linear differentials d : Ω^p_B/k−−→Ω^p+1_B/k

give a (global) De Rham complex

DeRham(M/k)≡0−−→B−^d→Ω_B/k−^d→...−^d→Ω^p_B⁻_/k¹−^d→Ω^p_B/k−−→...

which is bounded ifX is of finite type overk. Since d is homogeneous of degree 0, one has for eachn≥0 a complex ofO-modules (withk-linear differential)

DeRham(M/k)n≡0−−→SⁿM−^d→[Ω_B/k]n

−d→...−^d→[Ω^p_B/k]n−−→...

One has again Cartan’s formula: iD◦d + d◦iD= multiplication by n, on [Ω^p_B/k]n

and then:

Proposition 2.4. If X is a scheme over Q, then the complexes Kos(M/k)_n andDeRham(M/k)nare homotopically trivial(in particular,acyclic)for anyn>0.

As in Section1.2, we can take homogeneous localizations: for each n∈Z, the differentials Ω^p_B/k→Ω^p+1_B/k inducek-linear morphisms

d :Ω^p_B/k(n)−−→Ω^p+1_B/k(n)

and one obtains a complex ofO_P-modules (with k-linear differential) DeRham( M/k, n) = 0−−→OP(n)−^d→Ω_B/k(n)−^d→...−^d→Ω^p_B/k(n)−−→...

Again, the differentials of Koszul and De Rham complexes are related by Car- tan’s formula: iD◦d + d◦iD= multiplication by n, onΩ^p_B/k(n), so one has:

Proposition 2.5. Let X be a scheme overQ. The complexes Kos( M/k)(n) and DeRham( M/k, n) are homotopically trivial (in particular, acyclic) for any n=0.

Corollary 2.6. If X is a scheme over Q, then for any n=0, the exact se- quences

0−−→Ω^p_P/k(n)−−→Ω^p_B/k(n)−−→Ω^p_P⁻_/k¹(n)−−→0 split as sheaves of k-modules (but not as O_P-modules).

3. Cohomology of projective bundles

In this section we assume that E is a locally free sheaf of rank r+1 on a k- scheme (X,O). Let B=S·E be its symmetric algebra over O and P=ProjB−−^π→X the corresponding projective bundle. Our aim is to determine the cohomology of the sheaves Ω^p_P/X(n) and Ω^p_P/k(n).

3.1. Cohomology of Ω^p_P/X(n)

Notations: In order to simplify some statements, we shall use the following conventions:

1. S^pE=0 wheneverp<0, and analogously for exterior powers.

2. For any integerp, let ¯p=r+1−p.

3. For anyO-moduleM, we shall denote byM^∗ its dual Hom(M,O).

We shall use the following well known result on the cohomology of a projective bundle:

Proposition 3.1. Let nbe a non negative integer. Then Rⁱπ_∗OP(n) = 0 fori=0

SⁿE fori=0 If nis a positive integer,then

Rⁱπ_∗OP(−n) = 0 fori=r S^n−r−1E^∗⊗Λ^r+1E^∗ fori=r

We shall also use without further explanation a particular case of projection formula: for any quasi-coherent moduleN onX and any locally free moduleL on Psuch thatR^jπ_∗L is locally free (for anyj), one has

Rⁱπ_∗(π^∗N ⊗L) =N ⊗Rⁱπ_∗L Proposition 3.2. Let nbe a non negative integer. Then

Rⁱπ_∗Ω^p_B/O(n) = 0 fori=0 Λ^pE ⊗S^n−pE fori=0 For any positive integern,one has

Rⁱπ_∗Ω^p_B/O(−n) = 0 fori=r

Λ^p¯E^∗⊗S^n−p¯E^∗ fori=r withp=r+1¯ −p

Proof. Since Ω^p_B/O=(π^∗Λ^pE)(−p), the results follows from Proposition 3.1.

For the second formula we have also used the natural isomorphism Λ^p¯E=Λ^pE^∗⊗ Λ^r+1E.

Remark 3.3. Notice that Λ^pE ⊗S^n−pE=[Ω^p_B/O]n. Thus, Proposition 3.2 and Theorem 1.5 tell us that Kos(E)n→Kos( E)n is an isomorphism for any n≥0 and the Koszul complexesKos( E)_nand Kos(E)_nare acyclic for anyn>0 (thus we obtain the well known fact of the acyclicity of the Koszul complex of a locally free module).

Let us denote byKp,n the kernels of the morphismsi_D in Kos(E)_n, that is, Kp,n:= Ker

Λ^pE ⊗S^n−pE −−→Λ^p−1E ⊗S^n−p+1E

One has the following result (see [12] or [4, Expos´e XI] for different approaches).

Theorem 3.4. LetE be a locally free sheaf of rankr+1on ak-scheme(X,O) andP=ProjS·E−−^π→X the corresponding projective bundle.

Let n be a positive integer number.

1.

Rⁱπ_∗Ω^p_P/X= O if 0≤i=p≤r 0 otherwise 2.

Rⁱπ_∗Ω^p_P/X(n) = 0 if i=0 Kp,n if i=0 and,if X is aQ-scheme,then

Kp,n⊕Kp−1,n= Λ^pE ⊗S^n−pE. 3.

Rⁱπ_∗Ω^p_P/X(−n) = 0 ifi=r K^∗r−p,n ifi=r and,if X is aQ-scheme,then

K^∗r−p,n⊕K^∗r−p+1,n= Λ^p¯E^∗⊗S^n−p¯E^∗ Proof. Letn≥0. By Theorem1.4

0−−→Ω^p_P/X(n)−−→Ω^p_B/O(n)−−→...−−→Ω_B/O(n)−−→O_P(n)−−→0

is a resolution of Ω^p_P/X(n) byπ_∗-acyclic sheaves (by Proposition3.2). One concludes then by Proposition3.2and Remark 3.3.

(3) follows from (2) and (relative) Grothendieck duality: one has an isomor- phism Ω^p_P/X=Hom(Ω^r_P⁻_/X^p,Ω^r_P/X) and then

Rπ_∗Ω^p_P/X(−n)Rπ_∗Hom(Ω^r_P⁻_/X^p(n),Ω^r_P/X)RHom(Rπ_∗Ω^r_P⁻_/X^p(n)[r],O) and one concludes by (2).

Finally, the statements of (2) and (3) regarding the case thatX is aQ-scheme follow from Corollary1.11.

Corollary 3.5. (Bott’s formula) Let Pr be the projective space of dimension r over a fieldk. Let n be a positive integer number.

1.

dim_kH^q(Pr,Ω^p_P

r) = 1 if0≤q=p≤r 0 otherwise 2.

dimkH^q(Pr,Ω^p_P

r(n)) =

0 if q=0

_n+r−p

n

_n−1

p

if q=0

3.

dimkH^q(Pr,Ω^p_P

r(−n)) =

0 if q=r

_n+p

n

_n−1

r−p

if q=r

Proof. By Theorem 3.4, it is enough to prove that dimkKp,n=_n+r−p

n

_n−1

p

. From the exact sequence

0−−→Kp,n−−→Λ^pE ⊗S^n−pE −−→Kp−1,n−−→0 it follows that dimkKp,n+dimkKp−1,n=_r+1

p

_n−p+r

r

; hence it suffices to prove

that

n+r−p n

n−1 p

+

n+r−p+1 n

n−1 p−1

= r+1

p

n−p+r r

which is an easy computation if one writes_a

b

=_b!(a^a!_−b)!.

Remark 3.6. (1) We can give an interpretation of H⁰(Pr,Ω^p_P

r(n)) in terms of differentials forms of the polynomial ringk[x0, ..., xr]; one has the exact sequence

0−−→H⁰(Pr,Ω^p_P

r(n))−−→[Ω^p_k[x

0,...,xr]/k]_n−^iD→[Ω^p_k[x⁻¹

0,...,xr]/k]_n that is,H⁰(Pr,Ω^p_P

r(n)) are those p-formsω_p∈Ω^p_k[x

0,...,x_r]/k which are homogeneous of degreenand such that i_Dω_p=0, whereD=r

i=0x_i∂/∂x_i. (2) From the exact sequence

0−−→H⁰(Pr,Ω^p_P

r(n))−−→Λ^pE ⊗S^n−pE −−→...−−→E ⊗Sⁿ⁻¹E −−→SⁿE −−→0 we can give a different combinatorial expression of dimkH⁰(Pr,Ω^p_P

r(n)) (as Verdier does):

dimkH⁰(Pr,Ω^p_P

r(n)) = p i=0

(−1)ⁱ r+1

p−i

n+r−p+i r

.

It follows from Theorem3.4 that H^q(P,Ω^p_P/X)=H^q−p(X,O). For the twisted case we have the following:

Corollary 3.7. LetX be a proper scheme over a fieldkof characteristic zero.

Let E be a locally free module on X of rank r+1 and P=ProjS·E the associated projective bundle. Then,for any positive integer n, one has:

1. dimkH^q(P,Ω^p_P/X(n))=p

i=0(−1)ⁱdimH^q(X,Λ^p−iE ⊗S^n−p+iE).

2. dim_kH^q(P,Ω^p_P/X(−n))=p

i=0(−1)ⁱdimH^q−r(X,Λ^p+i¯ E^∗⊗S^n−p¯−iE^∗) with p=r+1¯ −p.