Torsion in the Crystalline Cohomology

of Singular Varieties

Bhargav Bhatt

Received: February 18, 2013

Communicated by Takeshi Saito

Abstract. This paper discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.

2010 Mathematics Subject Classification: 14F30,14F40

Keywords and Phrases: Crystalline cohomology, derived de Rham cohomology, Cartier isomorphism, Frobenius lifts

Fix an algebraically closed field k of characteristic p > 0 with ring of Witt vectors W. Crystalline cohomology is a W -valued cohomology theory for varieties over k (see [Gro68, Ber74]). It is exceptionally well behaved on proper smooth k-varieties: the W -valued theory is finite dimensional [BO78], and the corresponding W [1/p]-valued theory is a Weil cohomology theory [KM74] robust enough to support a p-adic proof of the Weil conjectures [Ked06] (in conjunction with rigid cohomology to deal with open or singular varieties). Somewhat unfortunately, crystalline cohomology is often large and somewhat unwieldy outside the world of proper smooth varieties. For example, the crystalline cohomology of a smooth affine variety of dimension > 0 is always infinitely generated as a W -module by the Cartier isomorphism (see Remark 2.4). Even worse, it is not a topological invariant: Berthelot and Ogus showed [BO83, Appendix (A.2)] that the 0^{th} crystalline cohomology group of a fat point in A^{2} has torsion (see also Example 3.4). In this paper, we give more examples of such unexpected behaviour:

Theorem. Let X be a proper lci k-variety. Then the crystalline cohomology of X is infinitely generated if any of the following conditions is satisfied:

(1) X has at least one isolated toric singularity, such as a node on a curve.

(2) X has at least one conical singularity of low degree, such as an ordinary double point of any dimension.

The statement above is informal, and we refer the reader to the body of this paper — see examples 3.5, 3.6, 3.12, and 3.13 — for precise formulations. In contrast to the Berthelot-Ogus example, our examples are reduced and lci. We do not know if these calculations are indicative of deeper structure; see Question 0.1 below. Our approach to the above calculation relies on Illusie’s derived de Rham cohomology [Ill72]. This theory, which in hindsight belongs to derived algebraic geometry, is a refinement of classical de Rham cohomology that works better for singular varieties; the difference, roughly, is the replacement of the cotangent sheaf with the cotangent complex. Theorems from [Bha12] show: (a) derived de Rham cohomology agrees with crystalline cohomology for lci varieties, and (b) derived de Rham cohomology is computed by a “conjugate” spectral sequence whose E_{2}-terms come from coherent cohomology on the Frobenius twist. These results transfer calculations from crystalline cohomology to coherent cohomology, where it is much easier to localise calculations at the singularities (see the proof of Proposition 3.1). As a bonus, this method yields a natural (infinite) increasing bounded below exhaustive filtration with finite-dimensional graded pieces on the crystalline cohomology of any lci proper variety. We conclude by asking if finiteness properties of crystalline cohomology characterize smooth varieties (somewhat analogously to Quillen’s conjecture [Avr99]):

Question 0.1. Do there exist any singular proper k-varieties with finite dimensional crystalline cohomology over k? Do there exist any singular finite type k-algebras A whose crystalline cohomology relative to k is finitely generated over the Frobenius twist A^{(1)} ⊂ A?

We do not know what to expect, and simply note here that derived de Rham theory (see §1) shows that the sought-after examples cannot simultaneously be lci and admit lifts to W_{2} compatible with Frobenius.

Organisation of this paper. In §1, we review the relevant results from derived de Rham cohomology together with the necessary categorical background. Next, we study (wedge powers of) the cotangent complex of some complete intersections in §2. This analysis is used in §3.1 to provide examples of some singular projective varieties (such as nodal curves, or lci toric varieties) whose crystalline cohomology is always infinitely generated; all these examples admit local lifts to W_{2} where Frobenius also lifts. Examples which are not obviously liftable (such as ordinary double points in high dimensions) are discussed in §3.2.

Notation. Let k and W be as above, and set W_{2} = W/p^{2}. For a k-scheme X, let X^{(1)} denote the Frobenius-twist of X; we identify the étale topology on X and X^{(1)}. We use H_{crys}^{n}(X/k) and H_{crys}^{n}(X/W) to denote Berthelot’s crystalline cohomology groups relative to k and W respectively. All sheaves are considered with respect to the Zariski topology (unless otherwise specified), and all tensor products are derived. We say that X lifts to W_{2} compatibly with Frobenius if there exists a flat W_{2}-scheme 𝒳 lifting X, and a map 𝒳 → 𝒳 lifting the Frobenius map on X and lying over the canonical Frobenius lift on W_{2}. For fixed integers a ≤ b ∈ Z, we say that a complex K over some abelian category has amplitude in [a, b] if H^{i}(K) = 0 for i ∉ [a, b] ⊂ Z.

A complex K of abelian groups is connected (resp. simply connected) if H^{i}(K) = 0 for i > 0 (resp. for i ≥ 0). An infinitely generated module over a ring is one that is not finitely generated. All gradings are indexed by Z unless otherwise specified. If A is a graded ring, then A(−j) is the graded A-module defined by A(−j)_{i} = A_{i−j}; we set M(−j) := M ⊗_{A} A(−j) for any graded A-complex M. We use Δ for the category of simplices, and Ch(𝒜) for the category of chain complexes over an abelian category 𝒜.

Acknowledgements. I thank Johan de Jong, Davesh Maulik, and Mircea Mustaţǎ for inspiring conversations. In particular, Example 3.5 was discovered in conversation with de Jong and Maulik, and was the genesis of this paper. Both Pierre Berthelot and Arthur Ogus had also independently calculated a variant of this example (unpublished), and I thank them for their prompt response to email inquiries. I am further grateful to the anonymous referee for references and comments.

1. Review of derived de Rham theory

In this section, we summarise some structure results in derived de Rham theory that will be relevant in the sequel. We begin by recalling in §1.1 some standard techniques for working with filtrations in the derived category; this provides the language necessary for the work in [Bha12] reviewed in §1.2.

1.1. Some homological algebra. In the sequel, we will discuss filtrations on objects of the derived category. To do so in a homotopy-coherent manner, we use the following model structure:

Construction 1.1. Fix a small category I, a Grothendieck abelian category ℬ, and set 𝒜 = Fun(I, ℬ). We endow Ch(ℬ) with the model structure of [Lur11, Proposition 1.3.5.3]: the cofibrations are termwise monomorphisms, while weak equivalences are quasi-isomorphisms. The category Fun(I, Ch(ℬ)) = Ch(Fun(I, ℬ)) = Ch(𝒜) inherits a projective model structure by [Lur09, Proposition A.2.8.2] where the fibrations and weak equivalences are defined termwise. By [Lur09, Proposition A.2.8.7], the pullback D(ℬ) → D(𝒜) induced by the constant map I → {1} has a left Quillen adjoint D(𝒜) → D(ℬ) that we call a “homotopy-colimit over I”. In fact, exactly the same reasoning shows: given a map ϕ : I → J of small categories, the pullback ϕ^{∗} : D(Fun(J, ℬ)) → D(Fun(I, ℬ)) induced by composition with ϕ has a left Quillen adjoint ϕ_{!} : D(Fun(I, ℬ)) → D(Fun(J, ℬ)) if Ch(Fun(I, ℬ)) and Ch(Fun(J, ℬ)) are given the projective model structures as above; we often refer to ϕ_{!} as a “homotopy-colimit along fibres of ϕ.” The most relevant examples of ϕ for us are: the projections Δ^{opp} → {1}, Δ^{opp} × N → N and N → {1}.

Using Construction 1.1, we can talk about increasing filtrations on objects of derived categories.

Construction 1.2. Let ℬ be a Grothendieck abelian category, and let 𝒜 := Fun(N, ℬ), where N is the category associated to the poset N with respect to the usual ordering. There is a homotopy-colimit functor F : D(𝒜) → D(ℬ) which is left Quillen adjoint to the pullback D(ℬ) → D(𝒜) induced by the constant map N → {1}; we informally refer to an object K ∈ D(𝒜) as an increasing (or Nindexed) exhaustive filtration on the object F (K) ∈ D(ℬ). There are also restriction functors [n]^{∗} : D(𝒜) → D(ℬ) for each n ∈ N, and maps [n]^{∗} → [m]^{∗} for n ≤ m coherently compatible with composition. For each n ∈ N, the cone construction defines a functor gr_{n} : D(𝒜) → D(ℬ) and an exact triangle [n − 1]^{∗} → [n]^{∗} → gr_{n} of functors D(𝒜) → D(ℬ); for a filtered object K ∈ D(ℬ), we often use gr_{n}(K) to denote gr_{n} applied to the specified lift of K to D(𝒜). A map K_{1} → K_{2} in D(𝒜) is an equivalence if and only if [n]^{∗}K_{1} → [n]^{∗}K_{2} is so for all n ∈ N if and only if gr_{n}(K_{1}) → gr_{n}(K_{2}) is so for all n ∈ N. Given a cochain complex K over ℬ, the association n ↦ τ_{≤n}K defines an object of D(𝒜) lifting the image of K ∈ D(ℬ) under F.

Remark 1.3. The “cone construction” used in Construction 1.2 to define gr_{n} needs clarification: there is no functor Fun([0 → 1], D(ℬ)) → D(ℬ) which incarnates the chain-level construction of the cone. However, the same construction does define a functor D(Fun([0 → 1], ℬ)) → D(ℬ), which suffices for the above application (as there are restriction functors D(𝒜) → D(Fun([0 → 1], ℬ)) for each map [0 → 1] → N in N).

1.2. The derived de Rham complex and the conjugate filtration. We first recall the definition:

Construction 1.4. For a morphism f : X → S of schemes, following [Ill72, §VIII.2], the derived de Rham complex dR_{X/S} ∈ Ch(Mod_{f^{−1}𝒪_{S}} ) is defined as the homotopy-colimit over Δ^{opp} of the simplicial cochain complex Ω_{P_{•}/f^{−1}𝒪_{S}}^{∗} ∈ Fun(Δ^{opp}, Ch(Mod_{f^{−1}𝒪_{S}} )), where P_{•} is a simplicial free f^{−1}𝒪_{S}-algebra resolution of 𝒪_{X}. When S is an F_{p}-scheme, the de Rham differential is linear over the p^{th}-powers, so dR_{X/S} can be viewed as an object of Ch(Mod_{𝒪_{X^{(1)}}} ), where X^{(1)} = X ×_{S,Frob} S is the (derived) Frobenius-twist of X (which is the usual one if f is flat).

The following theorem summarises the relevant results from [Bha12] about this construction:

Theorem 1.5. Let X be a k-scheme. Then:

(1) The complex dR_{X/k} ∈ Ch(Mod_{𝒪_{X^{(1)}}} ) comes equipped with a canonical increasing bounded below separated exhaustive filtration Fil_{•}^{conj} called the conjugate filtration. The graded pieces are computed by

Cartier_{i} : gr_{i}^{conj}(dR_{X/k}) ≃ ∧^{i}L_{X^{(1)}/k}[−i].

In particular, if X is lci, then Fil_{i}^{conj}(dR_{X/k}) is a perfect 𝒪_{X^{(1)}}-complex for all i.

(2) The formation of dR_{X/k} and the conjugate filtration commutes with étale localisation on X^{(1)}.

(3) There exists a canonical morphism

RΓ(X^{(1)}, dR_{X/k}) → RΓ_{crys}(X/k, 𝒪)

that is an isomorphism when X is an lci k-scheme.

(4) If there is a lift of X to W_{2} together with a compatible lift of Frobenius, then the conjugate filtration is split, i.e., there is an isomorphism

⊕_{i≥0} ∧^{i} L_{X^{(1)}/k}[−i] ≃ dR_{X/k}

whose restriction to the i^{th} summand on the left splits Cartier_{i}.

Remark 1.6. Theorem 1.5 can be regarded as an analogue of the results of Cartier (as explained in [DI87], say) and Berthelot [Ber74] to the singular case. In particular, when X is quasi-compact, quasi-separated and lci, parts (1) and (3) of Theorem 1.5 together with the end of Remark 1.7 yield a “conjugate” spectral sequence

E_{2}^{p,q} : H^{p}(X^{(1)}, ∧^{q}L_{X^{(1)}/k}) ⇒ H_{crys}^{p+q}(X/k).

In the sequel, instead of using this spectral sequence, we will directly use the filtration on dR_{X/k} and the associated exact triangles; this simplifies bookkeeping of indices.

Remark 1.7. We explain the interpretation of Theorem 1.5 using the language of §1.1. Let ℬ = Mod(𝒪_{X^{(1)}}), and let 𝒜 = Fun(N, ℬ). The construction of the derived de Rham complex dR_{X/k} ∈ D(ℬ) naturally lifts to an object ℰ ∈ D(𝒜) under F: if P_{•} → 𝒪_{X} is the canonical free k-algebra resolution of 𝒪_{X}, then Ω_{P_{•}/k}^{∗} ⊗_{P_{•}^{(1)}} 𝒪_{X^{(1)}} defines an object of D(Fun(Δ^{opp} × N, ℬ)) via (m, n) ↦ (τ_{≤n}Ω_{P_{m}/k}^{∗}) ⊗_{P_{m}^{(1)}} 𝒪_{X^{(1)}}, and its homotopy-colimit over Δ^{opp} (i.e., its pushforward along D(Fun(Δ^{opp} × N, ℬ)) → D(Fun(N, ℬ))) defines the desired object ℰ ∈ D(𝒜). This construction satisfies [n]^{∗}ℰ ≃ Fil_{n}^{conj}(dR_{X/k}), so gr_{n}(ℰ) ≃ gr_{n}^{conj}(dR_{X/k}) for all n ∈ N. This lift ℰ ∈ D(𝒜) of dR_{X/k} ∈ D(ℬ) is implicit in any discussion of the conjugate filtration on dR_{X/k} in this paper (as in Theorem 1.5 (1), for example). In the sequel, we abuse notation to let dR_{X/k} also denote ℰ ∈ D(𝒜). When X is quasi-compact and quasi-separated, cohomology commutes with filtered colimits, so RΓ(X^{(1)}, dR_{X/k}) ≃ colim_{n} RΓ(X^{(1)}, Fil_{n}^{conj}(dR_{X/k})). In particular, when restricted to proper varieties, derived de Rham cohomology can be written as a filtered colimit of (complexes of) finite dimensional vector spaces functorially in X.

2. Some facts about local complete intersections

In order to apply Theorem 1.5to compute crystalline cohomology, we need good control on (wedge powers of) the cotangent complex of an lci singularity. The following lemma collects most of the results we will use in §3.1.

Lemma 2.1. Let (A, 𝔪) be an essentially finitely presented local k-algebra with an isolated lci singularity at {𝔪}. Let N = dim_{k}(𝔪/𝔪^{2}) be the embedding dimension. Then:

(1) ∧^{n}L_{A/k} is a perfect complex for all n. For n ≥ N, ∧^{n}L_{A/k} can be represented by a complex of finite free A-modules lying between cohomological degrees −n and −n + N with differentials that are 0 modulo 𝔪.

(2) For any n ≥ N, the complex ∧^{n}L_{A/k} has finite length cohomology groups.

(3) For any n > N, the group H^{−n+N}(∧^{n}L_{A/k}) is non-zero.

(4) For any n > N, there exists an integer 0 < i ≤ N such that H^{−n+N−i}(∧^{n}L_{A/k}) is non-zero.

(5) If dim(A) > 0 and n > N, then H^{−n}(∧^{n}L_{A/k}) = 0, so the integer i in (4) is strictly less than N.

Proof. Choose a polynomial algebra P = k[x_{1}, …, x_{N}] and a map P → A such that Ω_{P/k}^{1} ⊗_{P} A → Ω_{A/k}^{1} is surjective. By comparing dimensions, the induced map Ω_{P/k}^{1} ⊗_{P} A ⊗_{A} A/𝔪 → Ω_{A/k}^{1} ⊗_{A} A/𝔪 is an isomorphism. Now consider the exact triangle

L_{A/P} [−1] → Ω_{P/k}^{1} ⊗_{P} A → L_{A/k}.

The lci assumption on A and the choice of P ensure that L_{A/P} [−1] is a free A-module of some rank r. Since Spec(A) is singular at 𝔪, we must have r > 0. The previous triangle then induces a (non-canonical) equivalence

(A^{⊕r} →^{T} A^{⊕N}) ≃ L_{A/k}.

The map T must be 0 modulo 𝔪 as A^{⊕N} → L_{A/k} induces an isomorphism on H^{0} after reduction modulo 𝔪, so the above presentation yields an identification

L_{A/k} ⊗_{A} k ≃ (k^{⊕r}[1]) ⊕ k^{⊕N}.

Computing wedge powers gives

(*)  ∧^{n} (L_{A/k}) ⊗_{A} k ≃ ⊕_{a=0}^{N}( ∧^{a} (k^{⊕N}) ⊗ Γ^{n−a}(k^{⊕r}))[n − a],

where Γ^{∗} is the divided-power functor; here we use that ∧^{n}(V [1]) = Γ^{n}(V)[n] for a flat k-module V over a ring k (see [Qui70, §7]). We now show the desired claims:

(1) The perfectness of ∧^{n}L_{A/k} follows from the perfectness of L_{A/k}. The desired representative complex can be constructed as a Koszul complex on the map T above (see the proof of Lemma 2.5 (4) below); all differentials will be 0 modulo 𝔪 by functoriality since T is so.

(2) We must show that ( ∧^{n} L_{A/k})_{𝔭} = 0 for any 𝔭 ∈ Spec(A) − {𝔪} and n ≥ N. The functor ∧^{n}L_{−/k} commutes with localisation, so we must show that ∧^{n}L_{A_{𝔭}/k} = 0 for 𝔭 and n as before, but this is clear: A_{𝔭} is the localisation of smooth k-algebra of dimension ≤ dim(A) < N for any such 𝔭.

(3) By (1), H^{−n+N}(∧^{n}L_{A/k}) = 0 if and only if ∧^{n}L_{A/k} has amplitude in [−n, −n+N −1]. However, in the latter situation, the complex ∧^{n}L_{A/k} ⊗_{A}k would have no cohomology in degree −n+N, contradicting formula (*); note that r ≥ 1 by the assumption that Spec(A) is singular at 𝔪.

(4) Assume the assertion of the claim is false. Then (3) shows that ∧^{n}L_{A/k} is concentrated in a single degree, so ∧^{n}L_{A/k} ≃ M[−n + N] for some finite length A-module M. By (1), M has finite projective dimension. The Auslander-Buschbaum formula and the fact that A is Cohen-Macaulay then show that the projective dimension of M is actually dim(A). Hence, M ⊗_{A} k has at most dim(A) + 1 non-zero cohomology groups. On the other hand, formula (*) shows that ∧^{n}L_{A/k} ⊗_{A} k has N + 1 distinct cohomology groups.

Hence, N ≤ dim(A), which contradicts the assumption that Spec(A) is singular at 𝔪.

(5) Set M := H^{−n}(∧^{n}L_{A/k}), and assume M ≠ 0. Then M has finite length by (2), and occurs as the kernel of a map of free A-modules by (1). Non-zero finite length R-modules cannot be found inside free R-modules for any S_{1}-ring R of positive dimension, which is a contradiction since complete intersections are S_{1}. □

Remark 2.2. The assumption dim(A) > 0 is necessary in Lemma 2.1 (5). For example, set A = k[ϵ]/(ϵ^{p}). Then N = dim_{k}(𝔪/𝔪^{2}) = 1, and L_{A/k} ≃ A[1] ⊕ A. Applying ∧^{n} for n > 0, we get

∧^{n}(L_{A/k}) ≃ Γ^{n}(A)[n] ⊕ Γ^{n−1}(A)[n − 1],

which certainly has non-zero cohomology in degree −n.

Using Lemma 2.1, we can show that the crystalline cohomology of an isolated lci singularity is infinitely generated in a very strong sense:

Corollary 2.3. Let (A, 𝔪) be as in Lemma 2.1. Assume that A admits a lift to W_{2} compatible with Frobenius. Then

(1) H_{crys}^{i}(Spec(A)/k) ≃ ⊕_{j≥0}H^{0}(Spec(A)^{(1)}, ∧^{j}L_{A^{(1)}/k}[i − j]) for all i.

(2) H_{crys}^{N}(Spec(A)/k) is infinitely generated as an A^{(1)}-module.

Proof. Note that H_{crys}^{∗}(Spec(A)/k) is an A^{(1)}-module since any divided-power thickening of A is an A^{(1)}-algebra.

(1) This follows from Theorem 1.5(4) and the vanishing of higher quasi-coherent sheaf cohomology on affines.

(2) This follows from Lemma 2.1 (3). □

Remark 2.4. Let us explain the phrase “strong sense” appearing before Corollary 2.3. If A is an essentially smooth k-algebra, then H_{crys}^{∗}(Spec(A)/k) is infinitely generated over k, but not over A^{(1)}: the Cartier isomorphism shows that H_{crys}^{i}(Spec(A)/k) ≃ Ω_{A^{(1)}/k}^{i}, which is a finite (and even locally free) A^{(1)}-module. It is this latter finiteness that also breaks down in the singular setting of Corollary 2.3.

We also record a more precise result on the wedge powers of the cotangent complex for the special case of the co-ordinate ring of a smooth hypersurface; this will be used in §3.2.

Lemma 2.5. Let A be the localisation at 0 of k[x_{0}, …, x_{N}]/(f), where f is a homogeneous degree d polynomial defining a smooth hypersurface in P^{N}. Assume p ∤ d. Then

(1) A is graded.

(2) The quotient M = A/(\frac{∂f}{∂x_{0}} , …, \frac{∂f}{∂x_{N}}) is a finite length graded A-module whose non-zero weights j are contained in the interval 0 ≤ j ≤ (d − 2)(N + 1).

(3) The A-linear Koszul complex K := K_{A}(\{\frac{∂f}{∂x_{i}} \}) of the sequence of partials is equivalent to M ⊕ M(−d)[1] as a graded A-complex.

(4) For n > N, we have an equivalence of graded A-complexes

∧^{n}L_{A/k}[−n] ≃ M((N +1)(d−1)−nd)[−N −1]⊕M((N +1)(d−1)−nd−d)[−N].

(5) Assume that N and d satisfy N(d − 2) < d + 2. Fix j and n with N < j < n. Then all graded k-linear maps

∧^{n}L_{A/k}[−n] → ∧^{j}L_{A/k}[−j][1]

are nullhomotopic as graded k-linear maps.

Proof. Let S = k[x_{0}, …, x_{N}] denote the polynomial ring. We note first the assumption p ∤ d implies (by the Euler relation) that f lies in the ideal J(f) ⊂ S generated by the sequence \{\frac{∂f}{∂x_{i}} \} of partials. Since f defines a smooth hypersurface, the preceding sequence cuts out a zero-dimensional scheme in S, and hence must be a regular sequence by Auslander-Buschbaum. In particular, each \frac{∂f}{∂x_{i}} is non-zero of degree d − 1. We now prove the claims:

(1) This is clear.

(2) Since f ∈ J(f), the quotient M is identified with S/J(f), so the claim follows from [Voi07, Corollary 6.20].

(3) Consider the S-linear Koszul complex L := K_{S}(\{\frac{∂f}{∂x_{i}} \}) of the sequence of partials. Since the partials span a regular sequence in S, we have an equivalence L ≃ S/J(f) ≃ A/J(f) ≃ M of graded S-modules. Now the complex K is simply L ⊗_{S} A ≃ M ⊗_{S} A. Since M is already an A-module, we get an identification K ≃ M ⊗_{A} (A ⊗_{S} A) as graded Amodules, where the right hand side is given the A-module structure from the last factor. The resolution (S(−d) →^{f} S) ≃ A then shows that

K ≃ M ⊗_{A} (A(−d) →^{0} A) ≃ M ⊕ M(−d)[1].

(4) Set L = (f)/(f^{2}), E = Ω_{S/k}^{1} ⊗_{S} A, and c : L → E to be the map defined by differentiation. Then the two-term complex defined by c is identified with L_{A/k}. Taking wedge powers for n > N then shows (see [KS04, Corollary 1.2.7], for example) that the complex

(**) Γ^{n}(L)⊗_{A}∧^{0}(E) → Γ^{n−1}(L)⊗_{A}∧^{1}(E) → ⋯ → Γ^{n−(N+1)}(L)⊗_{A}∧^{N+1}(E)

computes ∧^{n}L_{A/k}[−n]; here the term on the left is placed in degree 0. Explicitly, the differential

Γ^{i}(L) ⊗_{A} ∧^{k}(E) → Γ^{i−1}(L) ⊗ ∧^{k+1}(E)

is given by

γ_{i}(f) ⊗ ω ↦ γ_{i−1}(f) ⊗ (c(f) ∧ ω) = (−1)^{k} · γ_{i−1}(f) ⊗ (ω ∧ df).

In particular, if we trivialise Γ^{i}(L) using γ_{i}(f), then this differential is identified with left-multiplication by df in the exterior algebra ∧^{∗}(E). We leave it to the reader to check that the complex (**) above is isomorphic to K((N + 1)(d − 1) − nd)[−N − 1]; the rest follows from (3).

(5) Let M′ = M((N + 1)(d − 1))[−N − 1]. Then M′ is, up to a shift, a graded A-module whose weights lie in an interval size of (d − 2)(N + 1) by (2). By (4), we have

∧^{n}L_{A/k}[−n] ≃ M′(−nd) ⊕ M′(−nd − d)[1]

and

∧^{j}L_{A/k}[−j][1] ≃ M′(−jd)[1] ⊕ M′(−jd − d)[2].

Thus, we must check that all graded k-linear maps M′(−nd−d) → M′(−jd) are nullhomotopic. Twisting, it suffices to show M′ and M′((n + 1 − j)d) do not share a weight. If they did, then (n + 1 − j)d ≤ (d − 2)(N + 1). Since j < n, this implies 2d ≤ (d − 2)(N + 1), i.e., d + 2 ≤ N(d − 2), which contradicts the assumption. □

Remark 2.6. The assumption N(d − 2) < d + 2 in Lemma 2.5 (5) is satisfied in exactly the following cases: N ≥ 5 with d = 2, N = 3, 4 with d ≤ 3, N = 2 with d ≤ 5, and N = 1 with any d ≥ 1. In particular, an ordinary double point of any dimension satisfies the assumptions of Lemma 2.5 in any odd characteristic. We also remark that in this case (i.e., when d = 2), the proof of Lemma 2.5 (5) shows that the space of graded k-linear maps ∧^{n}L_{A/k}[−n] → ∧^{j}L_{A/k}[−j][1] is simply connected.

Remark 2.7. Lemma 2.5 (5) only refers to space of graded k-linear maps ∧^{n}L_{A/k}[−n] → ∧^{j}L_{A/k}[−j][1], and not the space of such graded A-linear maps. In particular, it can happen that a graded A-linear map ∧^{n}L_{A/k}[−n] → ∧^{j}L_{A/k}[−j][1] is nullhomotopic as a graded k-linear map, but not as an A-linear map.

Theorem 1.5 will be used to control on the mod p crystalline cohomology of an lci k-scheme. To lift these results to W, we will use the following base change isomorphism; see [BdJ11] for more details.

Lemma 2.8. Let X be a finite type lci k-scheme. Then the W -complex RΓ_{crys}(X/W) has finite amplitude, and there is a base change isomorphism

RΓ_{crys}(X/W) ⊗_{W} k ≃ RΓ_{crys}(X/k).

Proof. By a Mayer-Vietoris argument, we immediately reduce to the case where X = Spec(A) is affine. In this case, let D be the p-adic completion of the divided power envelope of a surjection P → A from a finite type polynomial W -algebra P. Then RΓ_{crys}(X/W) is computed by

Ω_{P/W}^{∗} ⊗_{P} D.

Since this complex has finite amplitude, the first claim is proven. Next, if P_{0} = P/p, and D_{0} is the divided power envelope of P_{0} → A, then RΓ_{crys}(X/k) is computed by

Ω_{P_{0}/k}^{∗} ⊗_{P_{0}} D_{0}.

The claim now follows from the well-known fact that D is W -flat (since A is lci), and D ⊗_{W} k ≃ D_{0} (see [BBM82, Lemma 2.3.3] for a proof). □

3. Examples

We come to the main topic of this paper: examples of singular proper lci k-varieties with large crystalline cohomology. In §3.1, using lifts of Frobenius, we show that certain singular proper varieties (such as nodal curves, or singular lci toric varieties) have infinitely generated crystalline cohomology. In §3.2, we show that a single ordinary double point (or worse) on an lci proper variety forces crystalline cohomology to be infinitely generated.

3.1. Frobenius-liftable examples. We start with a general proposition which informally says: a proper lci k-variety has large crystalline cohomology if it contains an isolated singular point whoseétale local ring lifts to W_{2} compatibly with Frobenius. Note that lci k-algebras always lift to W_{2}, so this is really a condition on Frobenius.

Proposition 3.1. Let X be proper lci k-scheme. Assume:

(1) There is a closed point x ∈ X that is an isolated singular point (but there could be other singularities on X).

(2) There is a lift to W_{2} of the Frobenius endomorphism of the henselian ring 𝒪_{X,x}^{h}.

Set N = dim_{k}(𝔪_{x}/𝔪_{x}^{2}). Then there exists an integer 0 < i ≤ N such that: (1) H_{crys}^{N}(X/k) is infinitely generated over k.

(2) H_{crys}^{N−i}(X/k) is infinitely generated over k.

(3) At least one of H_{crys}^{N+1}(X/W)[p] and H_{crys}^{N}(X/W)/p is infinitely generated over k.

(4) At least one of H_{crys}^{N+1−i}(X/W)[p] and H_{crys}^{N−i}(X/W)/p is infinitely gener-

ated over k. If dim(𝒪_{X,x}) > 0, then the integer i above can be chosen to be strictly less than N.

Proof. The desired integer i will be found in the proof of (2) below.

(1) Consider the exact triangle

Fil_{N}^{conj}(dR_{X/k}) → dR_{X/k} → 𝒬

in the category of 𝒪_{X^{(1)}}-complexes, where 𝒬 is defined as the homotopycokernel. Theorem 1.5 (1) and the lci assumption on X show that Fil_{N}^{conj}(dR_{X/k}) is a perfect complex on X^{(1)}, so H^{i}(X^{(1)}, Fil_{N}^{conj}(dR_{X/k})) is a finite dimensional vector space for all i by properness. By Theorem 1.5 (3), to show that H_{crys}^{N}(X/k) is infinitely generated, it suffices to show that H^{N}(X^{(1)}, 𝒬) is an infinite dimensional k vector space. First, we show:

Claim 3.2. The natural map RΓ(X, 𝒬) → 𝒬_{x} is a projection onto a summand as k-complexes.

Proof of Claim. Let j : U → X be an affine open neighbourhood of x such that U has an isolated singularity at x, and let j′ : V = X − {x} ⊂ X. By Theorem 1.5 (2), 𝒬|_{U∩V} ≃ 0 since U ∩ V is k-smooth. Hence, the Mayer-Vietoris sequence for the cover {U, V } of X and the complex 𝒬 degenerates to show

RΓ(X, 𝒬) ≃ RΓ(U, 𝒬) ⊕ RΓ(V, 𝒬).

It now suffices to show that RΓ(U, 𝒬) ≃ 𝒬_{x}. By Theorem 1.5 (1), 𝒬|_{U} admits an increasing bounded below separated exhaustive filtration with graded pieces ∧^{n}L_{U/k}[−k] for n > N. Since cohomology commutes with filtered colimits (as U is affine), RΓ(U, 𝒬) also inherits such a filtration with graded pieces computed by RΓ(U, ∧^{n}L_{U^{(1)}/k}[−k]) for n > N. Applying the same analysis to 𝒬_{x} reduces us to checking that RΓ(U, ∧^{n}L_{U^{(1)}/k}[−n]) ≃ ∧^{n}L_{𝒪_{X,x}^{(1)}/k} [−n] for n > N. But this is clear: for n > N, ∧^{n}L_{U^{(1)}/k}[−n]

is a perfect complex on U^{(1)} that is supported only at x and has stalk ∧^{n}L_{𝒪_{X,x}^{(1)}/k} [−n]. □

To compute the stalk 𝒬_{x}, define 𝒬′ via the exact triangle

Fil_{N}^{conj}(dR_{𝒪_{X,x}^{h}/k} ) → dR_{𝒪_{X,x}^{h}/k} → 𝒬′.

Then 𝒬_{x} = 𝒬′ by Theorem 1.5 (2), the finite length property of 𝒬_{x}, and the fact that 𝒪_{X,x}^{h} ⊗_{𝒪_{X,x}} M ≃ M for any finite length 𝒪_{X,x}-module M. Moreover, 𝒬′ can be computed using the Frobenius lifting assumption and Theorem 1.5 (4):

𝒬_{x} ≃ 𝒬′ ≃ ⊕_{n=N+1}^{∞} ∧^{n} L_{𝒪_{X,x}^{(1),h}/k} [−n].

Thus, to prove that H^{N}(X^{(1)}, 𝒬) is infinitely generated, it suffices to show that H^{N}(𝒬_{x}) is infinitely generated. This follows from the formula above and Lemma 2.1 (3).

(2) By combining the proof of (1) with Lemma 2.1 (4) and the pigeonhole principle, one immediately finds an integer 0 < i ≤ N such that H_{crys}^{N−i}(X/k) is infinitely generated over k. Lemma 2.1 (5) shows that we can choose such an i with i < N if dim(𝒪_{X,x}) > 0.

(3) The base change isomorphism from Lemma 2.8 gives a short exact sequence

0 → H_{crys}^{N}(X/W)/p → H_{crys}^{N}(X/k) → H_{crys}^{N+1}(X/W)[p] → 0,

so the claim follows from (1).

(4) The same argument as (3) works using (2) instead of (1). □

We need the following elementary result on Frobenius liftings:

Lemma 3.3. Let A be a k-algebra that admits a lift to W_{2} together with a compatible lift of Frobenius. Then the same is true for any ind-étale A-algebra B (such as the henselisation A at a point).

Proof. This follows by deformation theory since L_{B/A} = 0 for B as above. □

Specialising Proposition 3.1 leads to the promised examples.

Example 3.4. Let X = Spec(k[x]/x^{n}) for some n > 1. Then H_{crys}^{1}(X/k), H_{crys}^{0}(X/k), H_{crys}^{1}(X/W)/p, and H_{crys}^{1}(X/W)[p] are all infinitely generated. To see this, note first that Proposition 3.1 applies directly since X is a proper lci kscheme with a lift of Frobenius to W_{2}. Moreover, since X can be realised as a subscheme of A^{1}, the only non-zero cohomology groups are H_{crys}^{1} and H_{crys}^{0} (over W, as well as over k). The rest follows directly from Proposition 3.1 once we know that H_{crys}^{0}(X/W) = W. For this, note that H_{crys}^{0}(X/W) is the kernel of the differen-

tial d_{R} : R → R · dx, where R = { \atop W [x]⟨x^{n}⟩} is the pd-envelope of the evident closed immersion X ↪ Spec(W [x]). We may view R as the set of power series f = ∑_{i} a_{i}x^{i} ∈ K⟦x⟧ (where K = W [\frac{1}{p}]) such that a_{i} · [i/e]! ∈ W for all i. In particular, R is a subring of K⟦x⟧, so the kernel of d_{R} is just the constant power series (as K has characteristic 0), which shows H_{crys}^{0}(X/W) = W as desired.

Example 3.5. Let X be a proper nodal k-curve with at least one node. Then H_{crys}^{1}(X/k) and H_{crys}^{2}(X/k) are infinitely generated. Moreover, H_{crys}^{2}(X/W)/p, and at least one of H_{crys}^{1}(X/W)/p and H_{crys}^{2}(X/W)[p], are infinitely generated. Most of these claims follow directly from Proposition 3.1: a nodal curve is always lci, and the henselian local ring at a node on X is isomorphic to the henselisation of k[x, y]/(xy) at the origin, which is a one-dimensional local ring that admits a lift to W_{2} compatible with Frobenius by Lemma 3.3. It remains to show that H_{crys}^{3}(X/W)[p] is finitely generated. As pointed out by de Jong, the stronger statement H_{crys}^{3}(X/W) = 0 is true. If u : (X/W)_{crys} → X_{Zar} is the natural map (i.e., u_{∗}(ℱ)(U ⊂ X) = Γ((U/W)_{crys}, ℱ|_{U})), then R^{i}u_{∗}𝒪_{X/W,crys} is non-zero only for 0 ≤ i ≤ 2, and R^{2}u_{∗}𝒪_{X/W,crys} is supported only at the nodes^{1}. The rest follows from the Leray spectral sequence as X_{Zar} has cohomological dimension 1.

Example 3.6. Let X be a proper lci k-scheme. Assume that x ∈ X(k) is an isolated singular point (but there could be other singularities on X) such that 𝒪_{X,x}^{h} is toric of embedding dimension N. Then H_{crys}^{N}(X/k) is infinitely generated, and at least one of H_{crys}^{N}(X/W) and H_{crys}^{N+1}(X/W)[p] is infinitely generated over W. This follows from Proposition 3.1 and Lemma 3.3 since toric rings lift to W_{2} compatibly with Frobenius (use multiplication by p on the defining monoid). Some specific examples are: any proper toric variety with isolated lci singularities, or any proper singular k-scheme of dimension ≤ 3 with at worst ordinary double points.

Example 3.7. Let (E, e) be an ordinary elliptic curve over k, and let X be a proper lci k-surface with a singularity at x ∈ X(k) isomorphic to the one on the affine cone over E ⊂ P_{k}^{2} embedded via 𝒪(3[e]); for example, we could take X to be the projective cone on E ⊂ P_{k}^{2}. Then H_{crys}^{3}(X/k) and one of H_{crys}^{2}(X/k) or H_{crys}^{1}(X/k) are infinitely generated over k. This can be proven using Proposition 3.1 and the theory of Serre-Tate canonical lifts. Since we prove a more general and shaper result in Example 3.13, we leave details of this argument to the reader.

3.2. Conical examples. Our goal here is to show that the presence of an ordinary double point forces crystalline cohomology to be infinitely generated. In fact, more generally, we show the same for any proper lci variety that has a singularity isomorphic to the cone on a low degree smooth hypersurface. We start with an ad hoc definition.

1Proof sketch: Replace the Zariski topology with the Nisnevich topology in the foundations of crystalline cohomology, and then use that every nodal curve is Nisnevich locally planar. This observation yields a three-term de Rham complex computing the stalks of R^{i}u_{∗}𝒪_{X/W,crys}.

Definition 3.8. A local k-algebra A is called a low degree cone if its henselisation is isomorphic to the henselisation at the origin of the ring k[x_{0}, …, x_{N}]/(f), where f is a homogeneous degree d polynomial defining a smooth hypersurface in P^{N} such that N(d − 2) < d + 2. The integer d is called the degree of this cone; if d = 2, we also call A an ordinary double point. A closed point x ∈ X on a finite type k-scheme X is called low degree conical singularity (respectively, an ordinary double point) if 𝒪_{X,x}^{h} is a low degree cone (respectively, an ordinary double point).

We start by showing that the conjugate spectral sequence must eventually degenerate for low degree cones:

Proposition 3.9. Let A be low degree cone of degree d. Assume p ∤ d. Then for n > dim(A), the extensions

gr_{n}^{conj}(dR_{A/k}) → Fil_{n−1}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k})[1]

occurring in the conjugate filtration are nullhomotopic when viewed as k-linear extensions. In particular, there exist k-linear isomorphisms

Fil_{n}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k}) ≃ ⊕_{j=dim(A)+1}^{n}gr_{j}^{conj}(dR_{A/k}).

splitting the conjugate filtration for any n > dim(A).

Proof. By replacing A with its henselisation and then using the étale invariance of cotangent complexes and Theorem 1.5 (2), we may assume A is the localisation of k[x_{0}, …, x_{N}]/(f) at the origin for some homogeneous degree d polynomial f defining a smooth hypersurface in P^{N}. In particular, A is graded. Also, by functoriality, the conjugate filtration is compatible with the grading. Recall that the extensions in question arise from the triangles

Fil_{n−1}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k}) →

→ Fil_{n}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k}) → gr_{n}^{conj}(dR_{A/k}).

These triangles (and thus the corresponding extensions) are viewed as living in the derived category of graded k-vector spaces. By induction, we have to show the following: assuming a graded splitting

s_{n−1} : Fil_{n−1}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k}) ≃ ⊕_{j=dim(A)+1}^{n−1}gr_{j}^{conj}(dR_{A/k})

of the conjugate filtration, there exists a graded splitting

s_{n} : Fil_{n}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k}) ≃ ⊕_{j=dim(A)+1}^{n}gr_{j}^{conj}(dR_{A/k}),

of the conjugate filtration compatible with s_{n−1}. Chasing extensions, it suffices to show: for dim(A) < j < n, all graded maps

gr_{n}^{conj}(dR_{A/k}) → gr_{j}^{conj}(dR_{A/k})[1]

are nullhomotopic. This comes from Lemma 2.5 (5) and Theorem 1.5 (1). □

Remark 3.10. An inspection of the proof of Proposition 3.9 coupled with Remark 2.6 shows that if A is an ordinary double point, then the isomorphism

Fil_{n}^{conj}(dR_{A/k})/Fil_{dim(A)}^{conj}(dR_{A/k}) ≃ ⊕_{j=dim(A)+1}^{n}gr_{j}^{conj}(dR_{A/k})

is unique, up to non-unique homotopy. We do not know any applications of this uniqueness.

Using Proposition 3.9, we can prove infiniteness of crystalline cohomology for some cones:

Corollary 3.11. Let X be a proper lci k-scheme. Assume that there is low degree conical singularity at a closed point x ∈ X with degree d and embedding dimension N. If p ∤ d, then H_{crys}^{N}(X/k) and H_{crys}^{N−1}(X/k) are infinitely generated k-vector spaces.

Proof. We combine the proof strategy of Proposition 3.1 with Proposition 3.9. More precisely, following the proof of Proposition 3.1 (1), it suffices to show that 𝒬′ is infinitely generated when regarded as a complex of k-vector spaces. Now 𝒬′ admits an increasing bounded below separated exhaustive filtration with graded pieces given by gr_{n}^{conj}(dR_{𝒪_{X,x}^{h}/k} ) for n > N. By Proposition 3.9, there is a (non-canonical) isomorphism

𝒬′ ≃ ⊕_{n>N} ∧^{n} L_{𝒪_{X,x}^{(1),h}/k} [−n].

The rest follows from Lemma 2.5 (4) (note that embedding dimension in loc. cit. is N + 1, so we must shift by 1). □

We can now give the promised example:

Example 3.12. Let X be any proper lci variety that contains an ordinary double point x ∈ X(k) of embedding dimension N with p odd; for example, we could take X to be the projective cone over a smooth quadric in P^{N−1}. Then H_{crys}^{N}(X/k) and H_{crys}^{N−1}(X/k) are infinitely generated by Corollary 3.11.

All examples given so far have rational singularities, so we record an example that is not even log canonical.

Example 3.13. Let X be any proper lci surface that contains a closed point x ∈ X(k) with 𝒪_{X,x}^{h} isomorphic to the henselisation at the vertex of the cone over a smooth curve C ⊂ P^{2} of degree ≤ 5. If p ≥ 7, then H_{crys}^{3}(X/k) and H_{crys}^{2}(X/k) are infinitely generated by Corollary 3.11.

Remark 3.14. We do not know whether the ordinary double point from Example 3.12 admits a lift to W_{2} compatible with Frobenius in arbitrary dimensions; similarly for the cones in Example 3.13 (except for ordinary elliptic curves).

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Bhargav Bhatt

School of Mathematics Institute of Advanced Study Princeton, NJ 08540 USA 688

Documenta Mathematica 19 (2014)