SEMIREGULARITY AS A CONSEQUENCE OF GOODWILLIE’S THEOREM

J.P.PRIDHAM

Abstract. We realise Buchweitz and Flenner’s semiregularity map (and hence a fortiori Bloch’s semiregularity map) for a smooth variety X as the tangent of generalised Abel– Jacobi map on the derived moduli stack of perfect complexes on X. The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie’s theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex).

Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if X is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov–Witten and Pandharipande–Thomas invariants.

Introduction

In [Blo], Bloch defined a semiregularity map

τ: H^{1}(Z, 𝒩_{Z/X}) → H^{p+1}(X, Ω_{X}^{p−1})

for every local complete intersection Z of codimension p in a smooth proper complex variety X, and showed that curvilinear obstructions lie in the kernel of τ. He conjectured that τ should annihilate all obstructions, and that if X were also deformed, then τ would measure the obstruction to [Z] remaining a Hodge class.

In [BF1], Buchweitz and Flenner extended τ to give maps

σ_{q}: Ext_{X}^{2}(ℱ , ℱ) → H^{q+2}(X, Ω_{X}^{q}),

for any perfect complex ℱ on X, and showed that curvilinear obstructions to deforming ℱ lie in the kernel of σ_{q}. They conjectured that the whole obstruction space lies in the kernel of σ_{q} for all q.

The philosophy behind these conjectures is that for a deformation ℱ̃ of ℱ, we must have ch(ℱ) = ch(ℱ̃) because the cohomological Chern character takes rational values. The homotopy between cycles representing ch_{q+1}(ℱ) and ch_{q+1}(ℱ̃) should then be given by σ_{q}(ℱ̃).

Buchweitz and Flenner also observed that their conjecture would follow if there existed a generalised Abel–Jacobi map from the deformation groupoid to an intermediate Jacobian or to Deligne cohomology. We modify this idea, observing that the same reasoning holds with any formally étale cohomology theory in place of rational cohomology. In particular it applies to Hartshorne’s algebraic de Rham cohomology. Writing X_{A} := X ⊗_{ℂ} A, we thus set J_{X}^{p}(A, ℂ)[2p] to be the cocone of

DR^{alg}(X_{A}/ℂ) → RΓ(X_{A}, Ω_{X_{A}/A}^{<p}),

noting that in general X_{A} will not be smooth. This definition also adapts to any smooth scheme X over a base ring R containing a field k of characteristic 0, and any R-algebra A.

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/1].

To establish existence of the Abel–Jacobi map and functoriality, we reformulate in terms of cyclic homology. Algebraic de Rham cohomology is isomorphic to derived de Rham cohomology, and hence to periodic cyclic homology, giving

∏_{p} J_{X}^{p}(A, k) ≃ cocone(HP^{k}(X_{A}) ⟶^{S} HC^{A}(X_{A})[−2])

for X_{A} = X ⊗_{R} A. The generalised Abel–Jacobi map Ξ is then induced from the Goodwillie–Jones Chern character

ch^{−}: K(X_{A}) → HN^{k}(X_{A}) = cocone(HP^{k}(X_{A}) ⟶^{S} HC^{k}(X_{A})[−2]).

Goodwillie’s Theorem implies that HP^{k}(X_{A}) is formally étale as a functor in A, so J_{X}^{p}(A, k) has the required obstruction space

HC_{−2}^{A}(X_{A})^{(p−1)} = ℍ^{2p}(X_{A}, 𝒪_{X_{A}} → Ω_{X_{A}/A}^{1} → … → Ω_{X_{A}/A}^{p−1}).

Now, Ξ_{p} restricts from K-theory to a map on the nerve Perf_{X}(A) of the ∞-category of perfect complexes on X_{A}, thereby giving us a morphism of derived stacks (where we now allow A to be a simplicial R-algebra). On tangent spaces, the map Ξ_{p} becomes

ξ_{p}: Ext_{X_{A}}^{r}(ℱ , ℱ ⊗_{A} M) → H^{2p−2+r}(X, (𝒪_{X} → … → Ω_{X}^{p−1}) ⊗_{R} M),

for A-modules M, when X is smooth over R. In Proposition 2.12, we show that this map is just the Lefschetz map of [BNT], and hence (Remark 2.13) equivalent to the semiregularity map of [BF1].

The hypersheaves Perf_{X} and J_{X}^{p}(−, k) satisfy homotopy-homogeneity, a left-exactness property analogous to Schlessinger’s conditions, which in particular gives a functorial identification of tangent spaces with obstruction spaces. Given a square-zero extension A → B of simplicial algebras with kernel I, and a perfect complex ℱ on X ⊗_{R} B, the obstruction o(ℱ) to lifting ℱ to X ⊗_{R}A lies in Ext_{X}^{2}(ℱ , ℱ ⊗_{B} I). Homotopy-homogeneity ensures that the obstruction to lifting Ξ_{p}(ℱ) from H_{0}J_{X}^{p}(B, k) to H_{0}J_{X}^{p}(A, k) is just ξ_{p}(o(ℱ)).

Whenever A → B admits a splitting in the derived category of R-modules, this obstruction is zero, giving (Corollary 2.15):

o(ℱ) ∈ ⋂_{p≥1} ker(ξ_{p}).

In cases where such a splitting does not exist but A is Artinian, we can interpret ξ_{p}(o(ℱ)) as the obstruction to lifting ch_{p}(ℱ) as a horizontal section lying in F^{p}H^{2p}DR(X ⊗_{R} A/A) (Corollary 2.20 and Remark 2.24). In particular, this produces a reduced obstruction theory for the stable pairs featuring in the study of Pandharipande–Thomas invariants.

By considering perfect complexes of the form Rf_{∗}𝒪_{Z}, this also gives rise to a reduced obstruction theory for proper morphisms f: Z → X, and in particular for stable curves (Remark 2.25). It thus establishes Bloch’s semiregularity conjectures in the generality envisaged (Remark 2.26).

Attacking these conjectures has been the purpose of much recent research, most notably by Iacono and Manetti, who in [IM] proved Bloch’s first semiregularity conjecture in the case where Z is smooth. Their approach was to construct an explicit infinitesimal Abel– Jacobi map by L_{∞} methods, having also identified ℍ^{2p}(X, Ω_{X}^{<p}) as a more natural target for the semiregularity map than H^{p+1}(X, Ω_{X}^{p−1}). Other recent work such as [STV, KT] focuses on the case p = 1, where this discrepancy does not arise.

I am indebted to Timo Schürg for bringing [BF1] to my attention. I would also like to thank Barbara Fantechi, Richard Thomas and Daniel Huybrechts for helpful comments.

Contents

Introduction 1 Notation and conventions 3 1. The Abel–Jacobi map for rings 4 1.1. The Abel–Jacobi map 4 1.2. Homogeneity and obstructions 5 1.3. Goodwillie’s theorem 5 1.4. The semiregularity map 6 2. The Abel–Jacobi map for schemes and stacks 7 2.1. Derived de Rham cohomology 7 2.2. Perfect complexes and derived de Rham complexes 8 2.3. Generalised Abel–Jacobi maps 9 2.4. Horizontal sections 10 2.5. Reduced obstructions 11 Bibliography 13 References 13

Notation and conventions. Throughout this paper, k will denote a field of characteristic zero.

The Dold–Kan correspondence gives an equivalence of categories between simplicial abelian groups and non-negatively graded chain complexes with homotopy groups corresponding to homology groups, and we will pass between these categories without further comment. Under this equivalence, homotopy groups correspond to homology groups.

Given a chain complex V, we will write V [n] for the chain complex given by V [n]_{i} = V_{n+i}. We also write τ_{≥0}V for the good truncation of V in non-negative chain degrees (which we can then regard as a simplicial abelian group by the Dold–Kan correspondence above).

For a morphism f: V → W of chain complexes, cocone(f) will denote the homotopy kernel of f, which fits into an exact triangle

cocone(f) → V → W → cocone(f)[−1].

Definition 0.1. Given a morphism A → R of rings, with A commutative, write HC^{A}(R) (resp. HN^{A}(R), resp. HP^{A}(R), resp. HH^{A}(R)) for the chain complex associated to cyclic (resp. negative cyclic, resp. periodic cyclic, resp. Hochschild) homology of R over A.

Given a morphism A → R of simplicial rings, with A commutative, and a homology theory E as in the previous paragraph, define the complex E^{A}(R) by first forming the simplicial chain complex given by E^{A_{n}}(R_{n}) in level n, then taking the product total complex.

Remark 0.2. When working with cyclic homology, it is usual to fix a base ring and to omit it from the notation. Since varying the base will be crucial to our constructions, we have introduced the superscript A above. Also beware that the cohomology theories HN and HP are frequently denoted by HC^{−} and HC^{per} in the literature, and that our complexes are related to cyclic homology groups by

HC_{i}^{A}(R, M) := H_{i}HC^{A}(R, M)

etc. In the notation of [Wei2, Ch. 9], the complexes HH, HC, HN, HP are denoted by CC_{∗}^{h}, Tot CC_{∗∗}, Tot^{Π}CC_{∗∗}^{N}, Tot^{Π}CC_{∗∗}^{P}.

When A is a discrete ring, note that the complexes above are those studied in [Goo2].

Definition 0.3. Each of the homology theories E above admits a Hodge decomposition, which we denote by

E^{A}(R) = ∏_{p∈ℤ} E^{A}(R)^{(p)}.

Recall from [Wei2, §9.6.1] that there are exact triangles (the SBI sequences)

HN^{A}(R)^{(p)} ⟶^{I} HP^{A}(R)^{(p)} ⟶^{S} HC^{A}(R)^{(p−1)}[−2] ⟶^{B} HN^{A}(R)^{(p)}[−1]

HH^{A}(R)^{(p)} ⟶^{I} HC^{A}(R)^{(p)} ⟶^{S} HC^{A}(R)^{(p−1)}[−2] ⟶^{B} HH^{A}(R)^{(p)}[−1],

compatible with the projection map HN^{A}(R)^{(p)} → HH^{A}(R)^{(p)} and S: HP^{A}(R)^{(p)} → HC^{A}(R)^{(p)}.

Definition 0.4. Given a simplicial ring R, we follow [Wal] in writing K(R) for the Ktheory space of R (the 0th part of the K-theory spectrum). This is an infinite loop space with π_{i}K(R) = K_{i}(R).

1. The Abel–Jacobi map for rings

Fix a simplicial commutative k-algebra R. Take a simplicial R-algebra O(X), which need not be commutative, with the property that each O(X)_{n} is flat as an R_{n}-module. We will write F (X) := F (O(X)) when F is a functor such as K, HP, HC, HN, HH. Write sCAlg_{R} for the category of simplicial commutative R-algebras.

1.1. The Abel–Jacobi map.

Definition 1.1. Define a functor J_{X}^{p}(−, k) from sCAlg_{R} to chain complexes by setting

J_{X}^{p}(A, k) := cocone(HP^{k}(X_{A})^{(p)} ⟶^{S} HC^{A}(X_{A})^{(p−1)}[−2]),

where O(X_{A}) := O(X) ⊗_{R} A. Write J_{X}(A, k) := ∏_{p} J_{X}^{p}(A, k).

The Goodwillie–Jones Chern character

ch^{−}: K(X_{A}) → HN^{k}(X_{A}) = cocone(HP^{k}(X_{A}) ⟶^{S} HC^{k}(X_{A})[−2])

of [Goo2] then combines with the natural map HC^{k}(X_{A}) → HC^{A}(X_{A}) to give a map

Ξ: K(X_{A}) → ∏_{p} J_{X}^{p}(A, k),

which we call the (generalised) Abel–Jacobi map.

Definition 1.2. Given a simplicial ring S, define Perf(S) to be the simplicial set given by the nerve of the core of the category of perfect S-modules. This becomes a simplicial semiring with addition given by block sum and multiplication by tensor product.

Definition 1.3. By [Wal, Theorem 2.3.2], there is a natural map Perf(S) → K(S). Composing this with the Abel–Jacobi map above gives

Ξ_{p}: Perf(X_{A}) → τ_{≥0}J_{X}^{p}(A, k).

1.2. Homogeneity and obstructions. Say that a map A → B in sCAlg_{R} is a nilpotent extension if it is levelwise surjective, with the kernel I satisfying I^{n} = 0 for some n.

Definition 1.4. We say that a homotopy-preserving functor F from sCAlg_{R} to a model category C is homotopy-homogeneous if for A → B a nilpotent extension in sCAlg_{R} and C → B any morphism, the map

F (A ×_{B} C) → F (A) ×_{F (B)}^{h} F (C)

(to the homotopy fibre product) is a weak equivalence. When C is the category of cochain complexes, this is equivalent to saying that we have an exact triangle

F (B)[−1] → F (A ×_{B} C) → F (A) ⊕ F (C) → F (B).

Definition 1.5. Define the functor Perf_{X} on sCAlg_{R} by Perf_{X}(A) := Perf(O(X) ⊗_{R} A).

Definition 1.6. Given a homotopy-homogeneous functor F: sCAlg_{R} → C, an object x ∈ F (A) and an A-module M in non-positively graded cochain complexes, define the tangent space

T_{x}(F, M)

to be the homotopy fibre of F (A ⊕ M) → F (A) over x.

Note that homotopy-homogeneity of F ensures that T_{x}(F, M) has a homotopy-abelian group structure in C. We thus define tangent cohomology by D_{x}^{n−i}(F, M) := π_{i}T_{x}(F, M[n]), which is well-defined by [Pri2, Lemma 1.12].

Lemma 1.7. Take a homotopy-preserving and homotopy-homogeneous simplicial setvalued functor F on sCAlg_{R}. For any square-zero extension e : I → A ⟶^{f} B in sCAlg_{R}, and any x ∈ F B, there is then a functorial obstruction

o_{e}(x) ∈ D_{x}^{1}(F, I),

which is zero if and only if [x] lies in the image of

f_{∗}: π_{0}(F A) → π_{0}(F B).

Proof. This is contained in [Pri2, Lemma 1.17], in which the obstruction maps given here are just one term in a long exact sequence of homotopy groups. □

The following is well-known (see for instance [Pri1, Theorem 4.12] when O(X) is commutative):

Lemma 1.8. The functor Perf_{X} is homotopy-preserving and homotopy-homogeneous. The tangent space T_{ℰ} (Perf_{X}, M) is τ_{≥0}(R\underline{Hom}_{O(X)}(ℰ , ℰ ⊗M)[−1]), so the tangent cohomology groups are

D_{ℰ}^{i} (Perf_{X}, M) ≅ 𝔼xt_{O(X)}^{i+1}(ℰ , ℰ ⊗ M).

1.3. Goodwillie’s theorem. The following is [Goo2, Lemma I.3.3], a reformulation of Goodwillie’s theorem on nilpotent ideals ([Goo1, Theorems II.5.1 and IV.2.6]):

Theorem 1.9. If S → T is a map of simplicial k-algebras such that π_{0}S → π_{0}T is a nilpotent extension, then the map

HP^{k}(S) → HP^{k}(T)

is a quasi-isomorphism of chain complexes.

Proposition 1.10. The functor J_{X}(−, k) from sCAlg_{R} to chain complexes is homotopyhomogeneous.

Proof. The chain complex HC^{R}(X) is bounded below, so HC^{A}(X_{A}) ≃ HC^{R}(X) ⊗_{R}^{L} A for all A ∈ sCAlg_{R}, which ensures that the functor A ↦ HC^{A}(X_{A}) is homotopy-homogeneous.

Take a morphism C → B and a nilpotent extension A → B in sCAlg_{R}. Now, since A ×_{B} C → C is a nilpotent extension, Proposition 1.9 gives quasi-isomorphisms

HP^{k}(X_{A×_{B}C}) → HP^{k}(X_{C}), HP^{k}(X_{A}) → HP^{k}(X_{B}).

Thus HP^{k}(X_{A×_{B}C}) is trivially quasi-isomorphic to the cocone of

HP^{k}(X_{C}) ⊕ HP^{k}(X_{A}) → HP^{k}(X_{B}),

and the result follows by taking homotopy pullbacks. □

The following is a straightforward calculation:

Lemma 1.11. For f ∈ H_{0}(J_{X}(A, k)), the tangent space T_{f}(τ_{≥0}(J_{X}^{p}(−, k), M) is canonically quasi-isomorphic to τ_{≥0}(HC^{R}(X)^{(p−1)} ⊗_{R}^{L} M[−1]).

1.4. The semiregularity map.

Definition 1.12. Given a simplicial A-algebra S with ideal J, write HC^{A}(S → S/J) := cocone(HC^{A}(S) → HC^{A}(S/J)), and define HH^{A}(S → S/J) similarly.

Lemma 1.13. Take a simplicial commutative ring A, a simplicial A-module M, and a (possibly non-commutative) simplicial A-algebra O(Y). Then for C = A ⊕ M, the map

HC^{A}(O(Y)⊗_{A}^{L}C → O(Y)) → cocone(HC^{C}(O(Y)⊗_{A}^{L}C) → HC^{A}(O(Y))) ≃ HC^{A}(O(Y))⊗_{A}^{L}M

is naturally homotopic to the composition

HC^{A}(O(Y) ⊗_{A}^{L} C → O(Y)) ⟶^{B} HH^{A}(O(Y) ⊗_{A}^{L} C → O(Y))[1]
≃ HH^{A}(O(Y)) ⊗_{A}^{L} HH^{A}(C → A)[1]
⟶^{I⊗δ} HC^{A}(O(Y)) ⊗_{A}^{L} M.

where δ: HH^{A}(C → A)[1] → M is the map induced by the canonical derivation Ω_{C/A}^{1} ≅ M.

Proof. By taking diagonals of bisimplicial abelian groups and replacing M with an equivalent levelwise flat A-module, we reduce to the case where A, O(Y), M have constant simplicial structure, with M flat over A. Write ⊗ for ⊗_{A}^{L} and HC := HC^{A}, HH := HH^{A}.

The Künneth formula for cyclic homology in [Lod, Corollary 4.3.12] gives HC(O(Y)⊗C) as the cone of

HC(O(Y)) ⊗ HC(C)[1] ⟶^{S⊗1−1⊗S} HC(O(Y)) ⊗ HC(C)[−1].

The natural map ϕ: HC(O(Y) ⊗ C) → HC(O(Y)) ⊗ C is then given by projection on the left-hand factor HC(O(Y)) ⊗ HC(C) composed with the map HC(C) → C.

We may put a grading on C by setting A to have weight 0 and M to have weight 1. This induces gradings on all the complexes above, and since HC^{A}(O(Y)) ⊗_{A} M has weight 1, we may restrict to the weight 1 components, which we denote by W_{1}. By Goodwillie’s Theorem ([Wei2, Theorem 9.9.1]), S is homotopic to 0 on W_{1}HC^{A}(C), so W_{1}HC(O(Y)⊗C) is quasi-isomorphic to the cone of

HC(O(Y)) ⊗ W_{1}HC(C)[1] ⟶^{S⊗1} HC(O(Y)) ⊗ W_{1}HC(C)[−1],

making (B⊗1): W_{1}HC(O(Y)⊗C) → HH(O(Y))⊗W_{1}HC(C) a quasi-isomorphism. For the map ϕ above, W_{1}ϕ can thus be interpreted as the composition of (IB ⊗1): W_{1}HC(O(Y)⊗ C) → HC(O(Y)) ⊗ W_{1}HC(C) with HC(C) → C.

Now, [Wei2, Exercise 9.9.1] gives a quasi-isomorphism W_{1}HC(C) ≃ M. Since S ∼ 0, the sequence W_{1}HC(C)[−1] ⟶^{B} W_{1}HH(C) ⟶^{I} W_{1}HC(C) splits, giving W_{1}HH(C) ≃ M ⊕ M[−1]. Moreover, the map W_{1}HC(C) → W_{1}C = M is now just δ ∘ B. Therefore

W_{1}ϕ = (I ∘ B) ⊗ (δ ∘ B) = (I ⊗ δ) ∘ B,

since B: HC[−1] → HH respects products. □

Definition 1.14. Recall from [BNT, §4.3] that for a perfect B-complex ℰ, the Lefschetz map L_{B/A}: R\underline{Hom}_{B}(ℰ , ℰ) → HH^{A}(B) is given by

R\underline{Hom}_{B}(ℰ , ℰ) ≃ (ℰ ⊗_{A}^{L} R\underline{Hom}_{B}(ℰ , B)) ⊗_{B⊗_{A}^{L}B^{opp}}^{L} B ⟶^{ev⊗id} B ⊗_{B⊗_{A}^{L}B^{opp}}^{L} B.

Proposition 1.15. The tangent map T_{ℰ} (Ξ, M): τ_{≥0}(R\underline{Hom}_{O(X)}(ℰ , ℰ ⊗_{R} M)[−1]) → τ_{≥0}(HC^{R}(X) ⊗_{R}^{L} M[−1]) of Ξ is given by the composition

R\underline{Hom}_{O(X)}(ℰ , ℰ ⊗_{R} M) ⟶^{L_{O(X)/R}} HH^{R}(X) ⊗_{R}^{L} M ⟶^{I} HC^{R}(X) ⊗_{R}^{L} M.

Proof. Writing C = A ⊕ M, the tangent map is the composition

K(R ⊗_{A}^{L} C)_{ℰ} ⟶^{ch^{−}−ch^{−}(ℰ)} HN^{A}(R ⊗_{A}^{L} C → R) ⟶^{B^{−1}}HC^{A}(R ⊗_{A}^{L} C → R)[−1]
→HC^{A}(R) ⊗_{A}^{L} M[−1],

which by Lemma 1.13 is the same as the composition

K(R ⊗_{A}^{L} C)_{ℰ} ⟶^{ch−ch(ℰ)} HH^{A}(R) ⊗_{A}^{L} HH^{A}(C → A) ⟶^{I⊗δ} HC^{A}(R) ⊗_{A}^{L} M[−1],

since the composition K ⟶^{ch^{−}} HN ⟶^{B} HH is just the Dennis trace ch.

Now, the Dennis trace is given by sending a perfect complex ℱ to the image L(id_{ℱ} ) of the identity under the Lefschetz map. We wish to describe the projection of

τ_{≥0}R\underline{Hom}_{O(X)}(ℰ , ℰ ⊗_{R} M)[−1] ≃ T_{ℰ} (Perf, M) ⟶^{ch−ch(ℰ)} HH^{A}(R) ⊗_{A}^{L} HH^{A}(C → A)

to HH^{A}(R) ⊗_{A}^{L} W_{1}HH^{A}(C → A) ≃ HH^{A}(R) ⊗_{A}^{L} (M ⊕ M[−1]), for the grading W from the proof of Lemma 1.13. Calculating the Lefschetz map for extensions, and replacing M with M[n] to analyse higher degrees, we see that this map is given by (0, L_{O(X)/R}). Thus δ ∘ (ch − ch(ℰ)) = L_{O(X)/R}, which completes the proof. □

2. The Abel–Jacobi map for schemes and stacks

From now on, all rings will be commutative.

2.1. Derived de Rham cohomology.

Definition 2.1. Given A ∈ sCAlg_{ℚ} and B ∈ sCAlg_{A}, define the de Rham complex to be the chain complex

DR(B/A) := ∏_{n} Ω^{n}(B/A)[n] = ∏_{n}(Λ_{B}^{n}Ω^{1}(B/A))[n],

with differential given by combining the differentials on the chain complexes Ω^{n}(B/A) with the derivation induced by ∂: B → Ω^{1}(B/A). This has a Hodge filtration given by F^{p}DR(B/A) := ∏_{n≥p} Ω^{n}(B/A)[n].

Beware that the de Rham complex is usually regarded as a cochain complex, so negative homology groups will correspond to positive cohomology groups.

Definition 2.2. Given A ∈ sCAlg_{ℚ} and B ∈ sCAlg_{A}, define the left-derived de Rham complex LDR(B/A) by first taking a cofibrant replacement B̃ → B over A, then setting

LΩ^{p}(B/A) := Ω^{p}(B̃/A), LDR(B/A) := DR(B̃/A).

Note that this is well-defined up to quasi-isomorphism, that such replacements can be chosen functorially, and that LΩ^{1}(B/A) is a model for the cotangent complex 𝕃(B/A).

The complex LDR(B/A) has a Hodge filtration F^{p}LDR(B/A) = F^{p}DR(B̃/A), and we write LDR(B/A)/F^{p} := LDR(B/A)/F^{p}LDR(B/A).

Remark 2.3. Following the ideas of [Gro] as developed in [GR], there is a more conceptual interpretation of the derived de Rham complex. For A → B as above, [TV] uses derived Hom in the model category sCAlg_{A} to give a sSet-valued hypersheaf RSpec B = \underline{Hom}_{A}(B̃, −) on sCAlg_{A}. For any such functor F, we may define F_{inf}(C) := F ((H^{0}C)^{red}) and F_{strat}(C) = Im (π_{0}F (C) → F_{inf}(C)). Note that these set-valued functors are equal if and only if B is smooth over A in the sense of [TV].

Now, F_{strat}(C) is equivalent to the Čech nerve of F (C) over F_{inf}(C), which is represented in level n by formal completions of the diagonal map B̃^{⊗_{A}(n+1)} → B̃. Cohomology of symmetric powers then shows that LDR(B/A) ≃ RΓ((RSpec B)_{strat}, 𝒪), where 𝒪 is the hypersheaf given by 𝒪(C) = C.

Proposition 2.4. For a morphism A → B in sCAlg_{ℚ} and all p ∈ ℤ, there are canonical quasi-isomorphisms

HP^{A}(B)^{(p)} ≃ LDR(B/A)[−2p],  HC^{A}(B)^{(p)} ≃ (LDR(B/A)/F^{p+1})[−2p],
HN^{A}(B)^{(p)} ≃ F^{p}LDR(B/A)[−2p],  HH^{A}(B)^{(p)} ≃ LΩ^{p}(B/A)[−p],

with the SBI sequence corresponding to the short exact sequences 0 → F^{p}LDR → LDR → LDR/F^{p} → 0 and 0 → LΩ^{p}[−p] → LDR/F^{p+1} → LDR/F^{p} → 0.

Proof. When A and B are concentrated in degree 0, with B smooth over A, this is a well-known consequence of the Hochschild–Kostant–Rosenberg theorem. As observed in [Maj, §5], the general case follows by taking a cofibrant replacement for B and passing to the diagonal of the resulting bisimplicial diagram. □

Remark 2.5. Note that [Emm, Theorem 2.2] (following [FT, Theorem 5]) shows that for finitely generated k-algebras B the complex HP^{A}(B)^{(p)} is quasi-isomorphic to the infinitesimal cohomology complex, or equivalently to Hartshorne’s algebraic de Rham cohomology ([Har]) over A.

2.2. Perfect complexes and derived de Rham complexes. As in [Wei1], we now use naturality of the affine constructions to pass from local to global.

By [TV, §2.2] or the proof of [Pri1, Theorem 4.12], it follows that the functor Perf satisfies smooth hyperdescent on sCAlg_{ℚ}. Thus it extends to a hypersheaf from the smooth site of derived geometric stacks to the model category sSet of simplicial sets.

Definition 2.6. Denote the hypersheaf above by Perf, so given a strongly quasi-compact derived geometric Artin n-stack 𝔛, we have a simplicial set Perf(𝔛).

Remarks 2.7. Using the explicit hyperdescent formulae of [Pri3, Examples 1.15], the simplicial semiring Perf(𝔛) can be constructed as follows. First, [Pri3] provides the existence of a suitable resolution of 𝔛 by a derived Artin hypergroupoid X_{•}, which is a simplicial derived affine scheme satisfying certain properties. We then define a cosimplicial simplicial semiring given by C^{n}(X_{•}, Perf(𝒪_{𝔛})) := Perf(X_{n}), and set

Perf(𝔛) = RTot_{sSet}C^{•}(X_{•}, Perf(𝒪_{𝔛})),

where RTot_{sSet} is the derived total functor from cosimplicial simplicial sets to simplicial sets, as in [GJ, §VIII.1].

Because Perf forms an étale hypersheaf, the definition of Perf(𝔛) above agrees with the standard definition for underived schemes, and indeed for stacks. In the case when 𝔛 is a quasi-compact semi-separated scheme, X_{•} can just be constructed by taking the Čech nerve of an affine cover, in which case C^{•} is just a Čech complex.

In all of our applications in §2.5, 𝔛 will be of the form Y ⊗_{R} A, for Y a smooth quasicompact semi-separated scheme over a Noetherian ring R, and A ∈ sCAlg_{R}. Thus we can regard 𝔛 as being the derived scheme associated to a dg scheme (or even a dg manifold) in the sense of [CFK]. However, we need the greater flexibility provided by the theory of [TV, Pri3] in order to obtain a satisfactory construction of Perf invariant under quasiisomorphism.

The functors LΩ^{p}(−/A) satisfy LΩ^{p}(B ⊗_{A}^{L} A′/A) ≃ LΩ^{p}(B/A) ⊗_{A} A′ for étale morphisms (in the sense of [TV, Theorem 2.2.2.6]) A → A′ of simplicial ℚ-algebras, so satisfy étale hyperdescent. The functors F^{p}LDR(−/A) thus satisfy étale hyperdescent over A, so determine hypersheaves from the étale site of derived geometric Deligne–Mumford stacks over A to the model category of A-modules in unbounded complexes.

Definition 2.8. Denote the hypersheaves above by F^{p}LDR(−/A), giving complexes F^{p}LDR(𝔛/A) := RΓ(𝔛, LDR(𝒪_{𝔛}/A)) for any strongly quasi-compact derived geometric Deligne–Mumford n-stack 𝔛 over A ∈ sCAlg_{ℚ}.

Explicitly, for a resolution X_{•} as in Remarks 2.7, this is computed by the product total complex

F^{p}LDR(𝔛/A) = Tot^{Π}C^{•}(X_{•}, F^{p}LDR(𝒪_{𝔛}/A));

by hyperdescent, this agrees with the existing definition of LDR(𝔛/A) whenever 𝔛 is affine.

2.3. Generalised Abel–Jacobi maps. Fix a simplicial commutative k-algebra R, and a strongly quasi-compact derived geometric Deligne–Mumford n-stack 𝔛 over R. Given A ∈ sCAlg_{R}, from now on we will write 𝔛_{A} := 𝔛 ⊗_{R}^{L} A.

Definition 2.9. For A ∈ sCAlg_{R}, define the étale presheaf 𝒥_{𝔛}^{p}(A, k) to be

J_{𝒪_{𝔛}}^{p} (A, k) = cocone(LDR(𝒪_{𝔛} ⊗_{R}^{L} A/k)[−2p] → (LDR(𝒪_{𝔛} ⊗_{R}^{L} A/A)/F^{p})[−2p]).

Since the F^{m}LDR are étale hypersheaves, so is 𝒥_{𝔛}^{p}(A, k), and we may set

J_{𝔛}^{p}(A, k) := RΓ(𝔛, 𝒥_{𝔛}^{p}(A, k)).

Definition 2.10. Define the Abel–Jacobi map

Ξ: Perf(𝔛_{A}) → τ_{≥0} ∏_{p≥0} J_{𝔛}^{p}(A, k)

by applying RΓ(𝔛, −) to the Abel–Jacobi maps

Ξ_{p}: Perf(𝒪_{𝔛} ⊗_{R}^{L} A) → τ_{≥0}𝒥_{𝔛}^{p}(A, k)

of Definition 1.3, then composing with the natural map

RΓ(𝔛, τ_{≥0}𝒥_{𝔛}^{p}(A, k)) → τ_{≥0}RΓ(𝔛, 𝒥_{𝔛}^{p}(A, k)).

Definition 2.11. For ℰ ∈ Perf_{𝔛}(A) and a simplicial A-module M, write

ξ^{i}: 𝔼xt_{𝒪_{𝔛_{A}}}^{i+1} (ℰ , ℰ ⊗_{A}^{L} M) → ∏_{p≥0} H^{2p+i−1}((LDR(𝔛/R)/F^{p}) ⊗_{R}^{L} M)

for the tangent map

D_{ℰ}^{i} (Ξ, M): D_{ℰ}^{i} (Perf_{𝔛}, M) → ∏_{p≥0} D_{Ξ_{p}(ℰ)}^{i}(τ_{≥0}J_{𝔛}^{p}(−, k), M).

Hypersheafifying Proposition 1.15 yields:

Proposition 2.12. The map ξ_{p}^{i} is given by composing the Lefschetz map L_{𝔛/R}: 𝔼xt_{𝒪_{𝔛}⊗_{A}^{L}}^{i+1} (ℰ , ℰ ⊗_{A}^{L} M) → ℍ^{p+i}(𝔛, LΩ_{𝔛/R}^{p−1} ⊗_{R}^{L} M) with the canonical map

I: ℍ^{p+i}(𝔛, LΩ_{𝔛/R}^{p−1} ⊗_{R}^{L} M) → ℍ^{2p+i−1}(𝔛, (LDR(𝒪_{𝔛}/R)/F^{p}) ⊗_{R}^{L} M).

Remark 2.13. Note that the construction of the Atiyah–Hochschild character AH(ℰ) of [BF2, §5] just makes it the dual of the Lefschetz map, in the sense that L(α) = tr(AH(ℰ)∘ α). Thus [BF2, Theorem 5.1.3 and Proposition 6.2.1] ensure that L is the same as the semiregularity map σ of [BF1], given by applying the exponential of the Atiyah class then taking the trace.

Lemma 2.14. Take a square-zero extension e : I → A → B in sCAlg_{R}, admitting a section in the derived category of R-modules (when R = k, this just says that π_{∗}A ↠ π_{∗}B). Then the obstruction map

o_{e}: H_{0}(J_{𝔛}^{p}(B, k)) → H_{0}((LDR(𝔛/R)/F^{p}) ⊗_{R} I)

for the functor τ_{≥0}J_{𝔛}^{p}(−, k) is identically 0.

Proof. The R-module splitting of A → B splits the long exact sequence of homology, in particular ensuring that

H_{0}(J_{𝔛}^{p}(A, k)) → H_{0}(J_{𝔛}^{p}(B, k))

is surjective, and hence that o_{e} = 0. □

Since Lemma 1.7 is functorial, we may apply it to the Abel–Jacobi map, giving:

Corollary 2.15. Under the hypotheses of Lemma 2.14, for any ℰ ∈ Perf_{𝔛}(B) the obstruction

o_{e}(ℰ) ∈ D_{ℰ}^{1} (Perf_{𝔛}, I) = 𝔼xt_{𝔛_{B}}^{2}(ℰ , ℰ ⊗_{R} I)

lies in the kernel of

ξ_{ℰ ,I}^{1} : 𝔼xt_{𝔛}^{2}(ℰ , ℰ ⊗ I) → ∏_{p≥1} H^{2p}((LDR(𝔛/R)/F^{p}) ⊗_{R} I).

2.4. Horizontal sections. Retain R and 𝔛 as in the previous section. Take A ∈ sCAlg_{R} local Artinian with residue field k; write 𝔛_{0} := 𝔛 ⊗_{A}^{L} k.

Proposition 2.16. In the derived category of simplicial A-modules, there is a canonical (unfiltered) quasi-isomorphism

LDR(𝔛_{A}/A) ≃ LDR(𝔛_{0}/k) ⊗_{k} A.

Proof. The diagonal map A ⊗_{k} A → A is a nilpotent extension, so pulling back 𝔛_{A} along it gives us a nilpotent embedding ι: 𝔛_{A} ↪ 𝔛_{A} ⊗_{k} A. Proposition 1.9 then implies that ι^{∗}: LDR(𝔛_{A} ⊗_{k} A/A) → LDR(𝔛_{A}/A) is a quasi-isomorphism, where the map 𝔛_{A} ⊗_{k} A → Spec A is taken to be projection on the second factor.

Since A is finite-dimensional over k, base change then gives an isomorphism LDR(𝔛_{A}⊗_{k} A/A) ≅ LDR(𝔛_{A}/k)⊗_{k}A. Applying Proposition 1.9 to the nilpotent embedding 𝔛_{0} ↪ 𝔛_{A} gives a quasi-isomorphism LDR(𝔛_{A}/k) → LDR(𝔛_{0}/k), which completes the proof. □

Definition 2.17. Define the horizontal sections H_{∗}LDR(𝔛_{A}/A)^{hor} ⊂ H_{∗}LDR(𝔛_{A}/A) to be the image of H_{∗}LDR(𝔛_{A}/k) ↪ H_{∗}LDR(𝔛_{A}/A), so H_{∗}LDR(𝔛_{A}/A) ≅ H_{∗}LDR(𝔛_{A}/A)^{hor} ⊗_{k} π_{∗}(A).

Remark 2.18. When k = ℂ and 𝔛 is a smooth proper scheme over R, the horizontal sections above agree with those constructed in [Blo, Proposition 3.8]. To see this, note that for any finitely generated simplicial ℂ-algebra B, we can form an analytic simplicial algebra B^{an} by completing B over π_{0}B then taking the associated analytic algebra levelwise. This gives us an analytic sheaf LDR(𝒪_{𝔛_{A}}^{an}/A) on the analytic site of h^{0}𝔛_{A} = Spec_{𝔛_{A}}ℋ_{0}𝒪_{𝔛_{A}}, and Proposition 1.9 shows that A → LDR(𝒪_{𝔛_{A},an}/A) is a quasi-isomorphism of analytic sheaves. GAGA then reduces the proof of Proposition 2.16 to the canonical equivalence RΓ((h^{0}𝔛_{A})_{an}, A) ≃ RΓ((h^{0}𝔛_{A})_{an}, ℂ) ⊗_{ℂ} A.

We now describe the obstruction map in cases not covered by Lemma 2.14, in particular when R is not a field.

Proposition 2.19. Take a square-zero extension e : I → A → B (for A as above). Then the composition

H_{n}J_{𝔛}^{p}(B, k) ⟶^{o_{e}} H_{n−2p}((LDR(𝔛_{A}/A)/F^{p}) ⊗_{A} I) → H_{n−2p}(LDR(𝔛_{A}/A)/F^{p})

is equal to the composition

H_{n}J_{𝔛}^{p}(B, k) →H_{n−2p}(LDR(𝔛_{B}/B))^{hor}
≅H_{n−2p}(LDR(𝔛_{A}/A))^{hor} → H_{n−2p}(LDR(𝔛_{A}/A)/F^{p}),

for the horizontal sections of Definition 2.17.

Proof. By Proposition 2.16, we have a canonical isomorphism H_{∗}LDR(𝔛_{A}/A) ≃ H_{∗}(LDR(𝔛_{A}/A))^{hor} ⊗_{k} π_{∗}A, with H_{∗}(LDR(𝔛_{A}/A))^{hor} the isomorphic image of H_{∗}LDR(𝔛_{A}/k), and similarly for 𝔛_{B} over B. The second composition above then becomes

H_{n}J_{𝔛}^{p}(B, k) → H_{n−2p}LDR(𝔛_{B}/k) ≅ H_{n−2p}LDR(𝔛_{A}/k) → H_{2n−p}(LDR(𝔛_{A}/A)/F^{p})

which is precisely the first composition. □

The following now follows immediately from functoriality of obstructions in §1.2.

Corollary 2.20. Under the hypotheses of Proposition 2.19, for any ℰ ∈ Perf(𝔛_{B}) the image of the obstruction o_{e}(ℰ) under the maps

𝔼xt_{𝔛_{B}}^{2}(ℰ , ℰ ⊗_{B} I) ⟶^{ξ_{ℰ ,I,p}^{1}} H_{−2p}((LDR(𝔛_{A}/A)/F^{p}) ⊗_{A} I) → H_{−2p}(LDR(𝔛_{A}/A)/F^{p})

is the obstruction to lifting ch_{p}(ℰ) from H_{−2p}(LDR(𝔛_{B}/B))^{hor} ∩F^{p}H_{−2p}(LDR(𝔛_{B}/B)) to H_{−2p}(LDR(𝔛_{A}/A))^{hor} ∩ F^{p}H_{−2p}(LDR(𝔛_{A}/A)).

2.5. Reduced obstructions.

Corollary 2.21. Take a square-zero extension e : I → A ⟶^{f} B in sCAlg_{R}, admitting a section in the derived category of R-modules (when R = k, this says π_{∗}A ↠ π_{∗}B). Then for any ℰ ∈ Perf_{𝔛}(B) the obstruction

o_{e}(ℰ) ∈ D_{ℰ}^{1} (Perf_{𝔛}, I) = 𝔼xt_{𝒪_{𝔛}}^{2} (ℰ , ℰ ⊗ I)

lies in the kernel of the composition

𝔼xt_{𝔛}^{2}(ℰ , ℰ ⊗_{B} I) ⟶^{L} ∏_{p≥1} ℍ^{p+1}(𝔛, LΩ_{𝔛/R}^{p−1} ⊗_{R} I)
→ ∏_{p≥1} ℍ^{2p}(𝔛, (LDR(𝒪_{𝔛}/R)/F^{p}) ⊗_{R} I),

where L is the Lefschetz map of Definition 1.14.

Proof. Combine Corollary 2.15 with Proposition 2.12, the composite map being ξ_{ℰ ,I}^{1}. □

Remark 2.22. Whenever the Hodge–de Rham spectral sequence for 𝔛 over R degenerates, the map

ℍ^{p+1}(𝔛, LΩ_{𝔛/R}^{p−1} ⊗_{R} I) → ℍ^{2p}(𝔛, (LDR(𝒪_{𝔛}/R)/F^{p}) ⊗_{R} I)

is injective, so obstructions then lie in the kernel of L. This applies whenever 𝔛 is a smooth proper scheme over a Noetherian ℚ-algebra R ([Del]), or a smooth proper Deligne– Mumford stack over a field.

Corollary 2.23. Take an Artinian local simplicial R-algebra A with residue field k, and a square-zero extension e : I → A → B. Then for any ℰ ∈ Perf_{𝔛}(B) the image of the obstruction o_{e}(ℰ) under the maps

𝔼xt_{𝔛}^{2}(ℰ , ℰ ⊗_{B} I) ⟶^{L_{p−1}} ℍ^{p+1}(𝔛, LΩ_{𝔛/R}^{p−1} ⊗_{R} I)
→ H^{2p}((LDR(𝔛_{A}/A)/F^{p}) ⊗_{A} I)
→ H^{2p}(LDR(𝔛_{A}/A)/F^{p})

is the obstruction to lifting ch_{p}(ℰ) from H^{2p}(LDR(𝔛_{B}/B))^{hor} ∩ F^{p}H^{2p}(LDR(𝔛_{B}/B)) to H^{2p}(LDR(𝔛_{A}/A))^{hor} ∩ F^{p}H^{2p}(LDR(𝔛_{A}/A)).

Proof. This just combines Corollary 2.20 with Proposition 2.12. □

Remark 2.24. Assume that R is a Noetherian ℚ-algebra, with X a smooth proper scheme over Spec R. Then [Del] gives an isomorphism

ℍ^{∗}(X, (DR(𝒪_{X}/R)/F^{p}) ⊗_{R} M) ≅ ℍ^{∗}(X, (DR(𝒪_{X}/R)/F^{p})) ⊗_{R} π_{∗}(M)

for all simplicial R-modules M, with ℍ^{∗}(X, (DR(𝒪_{X}/R)/F^{p})) a projective R-module. Thus whenever π_{∗}A ↠ π_{∗}B, the map

H^{p+1}(X, Ω_{X/R}^{p−1} ⊗_{R} I) → ℍ^{2p}(X, (DR(𝒪_{X}/R)/F^{p}) ⊗_{R} A)

is injective, so Corollary 2.23 shows that L_{p−1}(o_{e}(ℰ)) ∈ H^{p+1}(X, Ω_{X/R}^{p−1} ⊗_{R} I) will vanish provided ch_{p}(ℰ) stays in F^{p} when lifted as a horizontal section.

Taking an open substack 𝔐 ⊂ Perf_{X} for which this holds (for instance by restricting to the Hodge locus of [Voi]), we thus obtain an obstruction theory for 𝔐 by

ℰ ↦ ker(L_{p−1} : 𝔼xt_{𝒪_{X}}^{2} (ℰ , ℰ) → H^{p+1}(X, Ω_{X/R}^{p−1})).

This gives a reduced obstruction theory for stable pairs, as required in the study of Pandharipande–Thomas invariants.

Remark 2.25. There is a morphism from the derived moduli stack of proper schemes over X ([Pri1, Theorem 3.32]) to the derived stack Perf_{X}, given by sending f: Z → X_{B} to Rf_{∗}𝒪_{Z}. Since the obstruction maps of Lemma 1.7 are functorial, this gives rise to morphisms

ψ: Ext_{Z}^{2}(𝕃^{Z/X_{B}}, 𝒪_{Z}) → Ext_{X_{B}}^{2}(Rf_{∗}𝒪_{Z}, Rf_{∗}𝒪_{Z})

of obstruction theories, so Corollaries 2.21 and 2.23 give conditions for ξ_{ℰ ,I}^{1} ∘ψ to annihilate obstructions to deforming Z over X_{B}.

For X smooth and proper over R, Remark 2.24 then implies that L_{p−1} ∘ ψ annihilates such obstructions provided ch_{p}(Rf_{∗}Z) stays of Hodge type on deforming Z. For suitable moduli of proper schemes Z over X, this gives rise to a reduced global obstruction theory

Z ↦ ker(L_{p−1} ∘ ψ: Ext_{Z}^{2}(𝕃^{Z/X}, 𝒪_{Z}) → H^{p+1}(X, Ω_{X/R}^{p−1}));

in particular this applies to stable curves Z over X, as required in the study of Gromov– Witten invariants (see for instance [KT, §2.2]).

Remark 2.26. The proof of Proposition 1.15 characterises L as a deformation of the Dennis trace. This means that for any proper LCI morphism f: Z → X_{B}, the Riemann–Roch theorem allows us to interpret the semiregularity map L ∘ ψ of Remark 2.25 as the deformation of f_{∗}(Td(T_{f})) as f varies.

When Z ⊂ X is a codimension p LCI subscheme of a smooth proper scheme over k, [BF1, Proposition 8.2] combines with Remark 2.13 to show that

L_{p−1} ∘ ψ: H^{1}(Z, 𝒩_{Z/X}) → H^{p+1}(X, Ω_{X/k}^{p−1})

is just Bloch’s semiregularity map from [Blo], which admits a relatively simple description in terms of Verdier duality.

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