Cohomology of the Moduli Stack of Coherent Sheaves on a Curve

COHOMOLOGY OF THE MODULI STACK OF COHERENT SHEAVES ON A CURVE

Abstract. We compute the cohomology of the moduli stack of coherent sheaves on a curve and ﬁnd that it is a free graded algebra on inﬁnitely many generators.

1. Introduction The moduli stack of vector bundles on a curve and its cohomological invariants have been much studied. In particular Atiyah–Bott showed in [?] that its cohomology is freely generated by the Künnethcomponents of the Chern classes of the universal vector bundle. In Laumon’s work on the geometric Langlands correspondence [?] the larger stack of coherent sheaves plays an important role. In this note we compute the cohomology of the stack of coherent sheaves on a smooth projective curve C, defined over some field k, and thereby answer a question of Olivier Schiffmann, related to the aforementioned work. We find that the cohomology of the moduli stack of coherent sheaves of fixed degree and positive generic rank is again freely generated by the Künnethcomponents of the Chern classes of the universal family of coherent sheaves. However, since for a family of coherent sheaves on a curve any higher Chern class can be non-trivial, we find that – unlike in the case of vector bundles – the result does not depend on the generic rank of the coherent sheaf. To state the result, let us fix some notation. We will denote étalecohomology ∗ ∗ ∗ groups with Q` coefficients by H , i.e., H (X) := Het(Xk, Q`). If the ground field k is chosen to be C, the same result will hold for singular cohomology with rational coefficients.

d Theorem 1. Let Cohn denote the stack of coherent sheaves of degree d and generic rank n on C. (1) For n = 0 we have

∗ 1 ∗ H (Coh0) = H (C) ⊗ Q`[c1] and ∗ d d ∗ 1 H (Coh0) = Sym (H (Coh0)). (2) For n > 0 and d ∈ Z we have ∗ d ^ j H (Cohn) = Q`[a1, a2,... ] ⊗ [bi ]i∈N,j=1,...,2g ⊗ Q`[f1, f2,... ].

2i d j 2i−1 d 2i−2 d V Here ai ∈ H (Cohn), bi ∈ H (Cohn) and fi ∈ H (Cohn) and denotes the exterior algebra generated by the given elements. 1 If k = Fq and the eigenvalues of Frobenius on H (C) are αj then the j ai, bi , fi can be chosen to be eigenvectors for the action of Frobenius with i i−1 i−1 eigenvalues q , q αj, q respectively.

j Remark 1. The generators ai, bi , fi occurring as generators in the theorem, are d the Atiyah–Bott classes, deﬁned as follows: For any n ≥ 0, let us denote by Fn,univ d d the universal coherent sheaf on Cohn ×C. Since Cohn is a smooth stack, locally of 1 2 COHOMOLOGY OF Coh

finite type over k (see e.g. [?, Théorème4.6.2.1]) we can consider the Chern classes (see section ??): 2 d 2i d M 2i−k d k ci(Fn,univ) ∈ H (Cohn ×C) = H (Cohn) ⊗ H (C). k=0 0 1 2 ∗ Let 1 ∈ H (C), γ1, . . . , γ2g ∈ H (C), [pt] ∈ H (C) be a basis of H (C), then we j can define the classes ci, ai , bi by the formula: 2g d X j ci(Fn,univ) = ai ⊗ 1 + bi ⊗ γj + fi ⊗ [pt]. k=1 ∗ d As we will see below these classes generate H (Cohn) for all n ≥ 0. 2. Preliminaries Before giving the proof of our result, let us recall some well known results on the cohomology of stacks and Chern classes for bundles on stacks that we will use. d First, since the stack Cohn is only locally of finite type, it will be useful to reduce our problem to substacks of finite type: Lemma 2. Let M be a smooth algebraic stack locally of finite type over a field k and Z ⊂ M a closed substack of codimension ≥ c. Then the restriction Hi(M) → Hi(M − Z) is an isomorphism for i < 2c − 1 and it is injective for i = 2c − 1. Proof. Denote by j : M − Z → M the inclusion. It is sufficient to show that k R j∗Q` = 0 for 0 < k < 2c − 1. By the base change theorem for smooth maps, this can be checked locally on an atlas X → M, so the result follows from the corresponding result for schemes.

By definition, a vector bundle E on an algebraic stack M defines a map fE : M → ∗ BGLn. Since H (BGLn) = Q`[c1, . . . , cn] (see [?, Theorem 2.3.2]), cohomological ∗ Chern classes of E can be defined as ci(E) := fE (ci) and the Chern-series of E is P∞ i defined as ct(E) := 1+ i=1 ci(E)t . The series ct(E) is multiplicative on short exact sequences and therefore induces a notion of Chern classes for coherent sheaves on stacks that admit a resolution by vector bundles. By the preceding lemma this notion extends to coherent sheaves such that the restriction of the sheaf to any substack of finite type admits a resolution by vector bundles. For example this d d property is satisfied for the universal coherent sheaf Fn,univ on Cohn ×C. Finally we will need a stack-theoretic analog of [?, Proposition 13.4]. In order to formulate it, let us recall that if M → M is a Gm-gerbe and E is a vector bundle on M, then Gm acts naturally on all fibers of E. If this action is given as multiplication by the n−th power of scalars, the bundle is said to be of weight n. Lemma 3. Let M → M be a morphism of algebraic stacks and suppose that M is a Gm-gerbe over M. Let E a vector bundle on M that is of weight n 6= 0 with respect to the Gm-gerbe structure. Then we have: ∗ ∗ (1) H (M) = H (M)[c1(E)]. (2) The Chern classes ci(E) for i = 1,..., rank(E) are not zero divisors in H∗(M). ∗ Proof. Recall that H (BGm) = Q`[c1], where c1 is the first Chern class of the universal line bundle on BGm. Since the universal line bundle is of weight 1, the first claim is true for M = BGm 7→ M = Spec(k). The general case follows from this and the argument of Leray-Hirsch, since the first Chern class of E is an element of the cohomology of M that restricts to a generator on all fibers (see e.g. [?]). COHOMOLOGY OF Coh 3

The second part follows from this, because the restriction to a fiber of π defines ∗ ∗ the evaluation map ev : H (M) = H (M)[c1(E)] → Q`[c1]. Since the restriction of k ⊕n E to a fiber is isomorphic to (Luniv) , the image of ci(E) under this map is not a k zero divisor, so that ci(E) = a[c1] + β with β ∈ ker(ev). Hence ci(E) is not a zero divisor.

3. Proof of the theorem The first part of Theorem ?? follows from [?, Section 3]. Let us briefly recall the ∼ 1 argument. For d = 1 we have a canonical isomorphism C × BGm = Coh0, mapping ∗ ∼ a point of C to the skyscraper sheaf defined by the point. Since H (BGm) = Q`[c1] (e.g. [?]) the claim for d = 1 follows. For d > 1 we follow [?] and consider the stack

d i Cohg 0 = h0 ( T1 ( ··· ( Td|Ti ∈ Coh0i. d Qd 1 The map gr : Cohg 0 → i=1 Coh0, given by gr(T•) = (Ti/Ti−1)i=1,...d, is a smooth fibration with contractible fibers, so that it induces an isomorphism in cohomology. d d Moreover by [?, Theorem 3.3.1], the forgetful map p: Cohg 0 → Coh0 is small and generically an étale Sd-covering, so p∗(Q`) carries an action of the symmetric group Sd ∼ ∗ d ∼ ∗ 1 Sd Sd and we have p∗(Q`) = Q`. Therefore H (Coh0) = H (Coh0) , proving the first claim. j To show (2) note that the classes ai, bi , fi define a morphism of graded algebras: ^ j ∗ d AB : Q`[a1, a2,... ] ⊗ [bi ]i∈N,j=1,...,2g ⊗ Q`[f1, f2,... ] → H (Cohn). Before showing that the morphism is injective, let us first check that the graded components of the two rings have the same dimension. To this end, let us stratify d Cohn according to the length of the torsion d,≤e Cohn := hF | length(Tors(F)) ≤ ei. d,=e Cohn := hF | length(Tors(F)) = ei. We have natural maps: d,=e d−e e gr: Cohn → Bunn × Coh0 given by mapping a sheaf F to (F/ Tors(F), Tors(F)), and the direct sum of sheaves induces a map d−e e d,=e ⊕: Bunn × Coh0 → Cohn . d,=e The map ⊕ is a vector bundle whose fiber over a sheaf F ∈ Cohn is given by Hom(F/ Tors(F), Tors(F)), because any coherent sheaf on C is non-canonically isomorphic to the direct sum of its torsion subsheaf and its torsion free quotient. d−e e d,=e Since Bunn and Coh0 are smooth, this implies that Cohn is a smooth, locally d closed substack of Cohn of codimension ne and that ∗ d,=e ∗ d−e ∗ e H (Cohn ) = H (Bunn ) ⊗ H (Coh0). ∗ d To deduce the additive structure of H (Cohn) we use an analog of an argument of d,≤e d Atiyah–Bott [?]: First note that since Cohn is an open dense substack of Cohn, such that the complement is of codimension n(e + 1), we have by Lemma ?? i d ∼ i d,≤e H (Cohn) = H (Cohn ) for i < 2n(e + 1). d,≤e d,=e Consider the Gysin sequence [?, Corollary 2.1.3] for the pair (Cohn , Cohn ): ∗−2ne d,=e ∗ d,≤e ∗ d,≤e−1 · · · → H (Cohn ) → H (Cohn ) → H (Cohn ) → · · · . 4 COHOMOLOGY OF Coh

We claim that this long exact sequence splits into short exact sequences. This holds because the composition

∗−2ne d,=e ∗ d,≤e ∗ d,=e H (Cohn ) → H (Cohn ) → H (Cohn ) is given by the cup product with the top Chern class of the normal bundle of d,=e d,≤e Cohn ⊂ Cohn . The normal bundle is given by 1 Ext (Tors(Funiv), Funiv/ Tors(Funiv)).

d−e e If we restrict this bundle to Bunn × Coh0 we see that the central Gm-auto- morphisms given by multiplication by scalars on one of the factors act non-trivially on the ﬁbers of this bundle. Lemma ?? thus implies that the top Chern class of ∗ d−e ∗ e the bundle is not a zero-divisor in H (Bunn ) ⊗ H (Coh0), so that additively we have ∗ d ∼ M ∗+2ne d e H (Cohn) = H (Bunn × Coh0). e≥0 By the result of Atiyah–Bott [?, Theorem 2.15] (see e.g. [?] for an algebraic argu- d ment) we know that the Poincar´eseries of Bunn is given by ∞ Qn 2i−1 2g X i d i i=1(1 + t ) PBund (t) = dim H (Bun ) t = n n . n n Q (1 − t2i) Q (1 − t2i−2) i=0 i=1 i=2 ∗ e e ∗ And since H (Coh0) = Sym (H (C)[c1]) we have ∞ Q∞ 2i−1 2g 1 X e i=1(1 + zt ) Z(Coh , z) := PCohe (t)z = 0 0 Q∞ (1 − zt2i) Q∞ (1 − zt2i−2) e=0 i=1 i=1

d Thus the Poincaréseries of Cohn is given by: ∞ X 2ne 1 n P d (t) = P d (t) t PCohe (t) = P d (t)Z(Coh , t ) Cohn Bunn 0 Bunn 0 e=0 Q∞ 2i−1 2g i=1(1 + t ) = Q∞ 2i Q∞ 2i−2 . i=1(1 − t ) i=2(1 − t ) This coincides with the Poincaréseries of the graded ring freely generated by the Atiyah–Bott classes. To complete the proof of the theorem, we are left to prove that the morphism AB is injective. To this end, pick a line bundle L such that deg(L) = −l < 0. For any d+knl d k ⊕n k > 0 this defines a map ik : Coh0 → Cohn, given by T 7→ T ⊕ (L ) . We claim that the composition of AB with the induced map ∗ ∗ d ∗ d+knl ∼ d+knl ∗ ik : H (Cohn) → H (Coh0 ) = Sym (H (C)[c1]) is injective for ∗ ≤ d + knl. ∗ d d+knl k ⊕n We have (ik × idC ) ct(Fn,univ) = ct(F0,univ ⊕ (L ) ) and

∗ k ⊕n ∗ n (3.1) ct F0,univ ⊕ (L ) = ct(F0,univ) · (1 + kc1(L)t) .

∗ 1 1 ∼ Thus we need to compute ct(F0,univ). The universal torsion sheaf F0,univ on Coh0 ×C = ∗ BGm × C × C can be described as follows. Let L := p1Luniv denote the pull back of the universal line bundle on BGm, denote by ∆ the diagonal in C × C and L(−∆) := Luniv O(−∆). Then we have an exact sequence 1 0 → L(−∆) → L → F0,univ → 0. COHOMOLOGY OF Coh 5

1 P∞ 1 i ct(L) Thus we find ct(F ) = 1 + ci(F )t = . By definition we 0,univ i=1 0,univ ct(L(−∆)) ∗ know that c1(Luniv) = c1 ∈ H (BGm) = Q`[c1]. Therefore ∞ 1 X i−1 i ct(F0,univ) = (1 + c1t)/(1 − ([∆] − c1)t) = 1 + ([∆] − c1) [∆]t i=1 1 Let us denote by γj ∈ H (C) the basis that is Poincarédual to the basis γj. Then P2g ∗ ∗ [∆] = 1 ⊗ [pt] + j=1 γj ⊗ γj + [pt] ⊗ 1 ∈ H (C) ⊗ H (C), where γj ∪ γj = [pt] and 2 [∆] = (2 − 2g)[pt] ⊗ [pt]. Putting c := −c1 we find:

2g 1 i−1 i−2 X i−1 i−1 ci(F0,univ) = (c +(i−1)(2−2g)c [pt])⊗[pt]+ (c γj)⊗γ2g−j +(c [pt])⊗1. j=1

1 ∗ 1 Thus we see that the K¨unnethcomponents of ci(F0,univ) form a basis for H (Coh0) = H∗(C)[c]. Also, the Chern series of a direct sum is the product of the Chern series d of the summands, so that the image of ct(F0,univ) under the inclusion

∗ d ∗ 1 ⊗d ι: H (Coh0) ,→ H (Coh0) is given by the product of the Chern series of the universal bundles on the factors. 1 j ∗ Let us write ci(F0,univ) = Ai ⊗1+Bi ⊗γj +Fi ⊗[pt] and denote by Ai,k := pr k(Ai) ∈ ∗ Qd 1 j ∗ j ∗ H ( i=1 Coh0) and similarly Bi,k := pr k(Bi ),Fi,k := pr k(Fi). Then we have:

d X ι(ci(F0,univ)) = Ai1,1 ··· Aid,d ⊗ 1 i1+···+id=i d X X + A ··· Bj · A ⊗ γ i1,1 ik,k id,d j i1+···+id=i k=1 d X X + Ai1,1 ··· Fik,k ··· Aid,d ⊗ [pt]

i1+···+id=i k=1 X X + ±A ··· Bj B2g−jA ) ⊗ [pt] i1,1 ik,k il,l id,d i1+···+id=i k6=l

d We claim that this implies that the Künneth components of ci(F0,univ) gener- ∗ 1 ⊗d ate the subring of Sd-invariant elements in H (Coh0) : We argue by induc- i−1 j i−1 Pd ∗ j tion. Since Ai = c1 [pt],Bi = c1 γj we see that ι(a1) = k=1 pr k[pt], ι(b1) = Pd ∗ ∗ k=1 pr k(γj) are the first elementary symmetric functions of a basis of H (C). j j j Pd Moreover, the products AiAk,AiBk,Bi Bk vanish. Therefore ι(ai) = k=1 Ai,k +ri j where ri is a product of symmetric functions of lower degree and similarly ι(bi ) = Pd ∗ j 0 Pd ∗ i−1 00 00 k=1 pr k(Bi ) + ri. Finally ι(Fi) = k=1 pr k(c ) + ri , where ri is a symmet- P ∗ ric function that is contained in the ideal generated by prk[pt] and the classes Pd ∗ k=1 pr k(γj). d ∗ d This shows that the Künnethcomponents of ci(F0,univ) generate H (Coh0) and that relations between the classes are contained in cohomological degrees > d. By (??) this calculation also implies that for any k > 0 the composition of AB with the induced map ∗ d ∗ d+knl ∼ d+knl ∗ H (Cohn) → H (Coh0 ) = Sym H (C)[c1] is injective for ∗ ≤ d + knl. This proves the theorem. 6 COHOMOLOGY OF Coh

Acknowledgements: I thank Olivier Schiﬀmann for asking the question that led to this note. I would like to thank the Universit´eParis Sud, where a part of this work was done, for the wonderful hospitality.

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