Chern characters for twisted matrix factorizations and the vanishing of the higher Herbrand difference

Abstract

We develop a theory of “ad hoc” Chern characters for twisted matrix factorizations associated to a scheme X, a line bundle $\mathcal {L}$, and a regular global section $W \in \Gamma (X, \mathcal {L})$. As an application, we establish the vanishing, in certain cases, of $h_c^R(M,N)$, the higher Herbrand difference, and, $\eta _c^R(M,N)$, the higher codimensional analogue of Hochster’s theta pairing, where R is a complete intersection of codimension c with isolated singularities and M and N are finitely generated R-modules. Specifically, we prove such vanishing if $R = Q/(f_1, \dots , f_c)$ has only isolated singularities, Q is a smooth k-algebra, k is a field of characteristic 0, the $f_i$’s form a regular sequence, and $c \ge 2$. Such vanishing was previously established in the general characteristic, but graded, setting in Moore et al. (Math Z 273(3–4):907–920, 2013).

Appendix: Relative connections

We record here some well-known facts concerning connections for locally free coherent sheaves. Throughout, S is a Noetherian, separated scheme and $p: X \rightarrow S$ is a smooth morphism; i.e., p is separated, flat and of finite type and $\Omega ^1_{X/S}$ is locally free.

Definition 6.1

For a vector bundle (i.e., locally free coherent sheaf) $\mathcal {E}$ on X, a connection on $\mathcal {E}$ relative to p is a map of sheaves of abelian groups

$$\begin{aligned} \nabla : \mathcal {E}\rightarrow \Omega ^1_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}\end{aligned}$$

on X satisfying the Leibnitz rule on sections: given an open subset $U \subseteq X$ and elements $f \in \Gamma (U, \mathcal {O}_X)$ and $e \in \Gamma (U, \mathcal {E})$, we have

$$\begin{aligned} \nabla (f \cdot e) = df \otimes e + f \nabla (e) \, \text { in } \Gamma (U, \Omega ^1_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}), \end{aligned}$$

where $d: \mathcal {O}_X \rightarrow \Omega ^1_{X/S}$ denotes exterior differentiation relative to p.

Note that the hypotheses imply that $\nabla$ is $\mathcal {O}_S$-linear — more precisely, $p_*(\nabla ): p_* \mathcal {E}\rightarrow p_*\left( \Omega ^1_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}\right)$ is a morphism of quasi-coherent sheaves on S.

1.1 The classical Atiyah class

Let $\Delta : X \rightarrow X \times _S X$ be the diagonal map, which, since $X \rightarrow S$ is separated, is a closed immersion, and let $\mathcal {I}$ denote the sheaf of ideals cutting out $\Delta (X)$. Since p is smooth, $\mathcal {I}$ is locally generated by a regular sequence. Recall that $\mathcal {I}/\mathcal {I}^2 \cong \Delta _* \Omega ^1_{X/S}$. Consider the coherent sheaf $\tilde{\mathcal {P}}_{X/S} := \mathcal {O}_{X \times _S X}/\mathcal {I}^2$ on $X \times _S X$. Observe that $\tilde{\mathcal {P}}_{X/S}$ is supported on $\Delta (X)$, so that $(\pi _i)_* \tilde{\mathcal {P}}_{X/S}$ is a coherent sheaf on X, for $i = 1, 2$, where $\pi _i: X \times _S X \rightarrow X$ denotes projection onto the i-th factor.

The two push-forwards $(\pi _i)_* \tilde{\mathcal {P}}_{X/S}$, $i = 1,2$ are canonically isomorphic as sheaves of abelian groups, but have different structures as $\mathcal {O}_X$-modules. We write $\mathcal {P}_{X/S} = \mathcal {P}$ for the sheaf of abelian groups $(\pi _1)_* \tilde{\mathcal {P}}= (\pi _2)_* \tilde{\mathcal {P}}$ regarded as a $\mathcal {O}_X-\mathcal {O}_X$-bimodule where the left $\mathcal {O}_X$-module structure is given by identifying it with $(\pi _1)_* \tilde{\mathcal {P}}_{X/S}$ and the right $\mathcal {O}_X$-module structure is given by identifying it with $(\pi _2)_* \tilde{\mathcal {P}}_{X/S}$.

Locally on an affine open subset $U = {\text {Spec}}(Q)$ of X lying over an affine open subset $V = {\text {Spec}}(A)$ of S, we have $\mathcal {P}_{U/V} = (Q \otimes _A Q)/I^2$, where $I = {\text {ker}}(Q \otimes _A Q \xrightarrow {- \cdot -} Q)$ and the left and right Q-module structures are given in the obvious way.

There is an isomorphism of coherent sheaves on $X \times X$

$$\begin{aligned} \Delta _* \Omega ^1_{X/S} \cong \mathcal {I}/\mathcal {I}^2 \end{aligned}$$

given locally on generators by $dg \mapsto g \otimes 1 - 1 \otimes g$. From this we obtain the short exact sequence

$$\begin{aligned} 0 \rightarrow \Omega _{X/S}^1 \rightarrow \mathcal {P}_{X/S} \rightarrow \mathcal {O}_X \rightarrow 0. \end{aligned}$$

(6.2)

This may be thought of as a sequence of $\mathcal {O}_X-\mathcal {O}_X$-bimodules, but for $\Omega ^1_{X/S}$ and $\mathcal {O}_X$ the two structures coincide.

Locally on open subsets U and V as above, we have $\Omega ^1_{Q/A} \cong I/I^2$, and (6.2) takes the form

$$\begin{aligned} 0 \rightarrow I/I^2 \rightarrow (Q \otimes _A Q)/I^2 \rightarrow Q \rightarrow 0. \end{aligned}$$

Viewing (6.2) as either a sequence of left or right modules, it is a split exact sequence of locally free coherent sheaves on X. For example, a splitting of $\mathcal {P}_{X/S} \rightarrow \mathcal {O}_X$ as right modules may be given as follows: recall that as a right module, $\mathcal {P}_{X/S} = (\pi _2)_* \mathcal {P}$ and so a map of right modules $\mathcal {O}_X \rightarrow \mathcal {P}_{X/S}$ is given by a map $\pi _2^* \mathcal {O}_X \rightarrow \mathcal {P}_{X/S}$. Now, $\pi _2^* \mathcal {O}_X = \mathcal {O}_{X \times _S X}$, and the map we use is the canonical surjection. We refer to this splitting as the canonical right splitting of (6.2).

Locally on subsets U and V as above, the canonical right splitting of is given by $q \mapsto 1 \otimes q$.

Given a locally free coherent sheaf $\mathcal {E}$ on X, we tensor (6.2) on the right by $\mathcal {E}$ to obtain the short exact sequence

$$\begin{aligned} 0 \rightarrow \Omega _{X/S}^1 \otimes _{\mathcal {O}_X} \mathcal {E}\xrightarrow {i} \mathcal {P}_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}\xrightarrow {\pi } \mathcal {E}\rightarrow 0 \end{aligned}$$

(6.3)

of $\mathcal {O}_X-\mathcal {O}_X$-bimodules. Taking sections on affine open subsets U and V as before, letting $E = \Gamma (U, \mathcal {E})$, this sequence has the form

$$\begin{aligned} 0 \rightarrow \Omega ^1_{Q/A} \otimes _Q E \rightarrow (Q \otimes _A E)/I^2 \cdot E \rightarrow E \rightarrow 0. \end{aligned}$$

Since (6.2) is split exact as a sequence of right modules and tensor product preserves split exact sequences, (6.3) is split exact as a sequence of right $\mathcal {O}_X$-modules, and the canonical right splitting of (6.2) determines a canonical right splitting of (6.3), which we write as

$$\begin{aligned} {\text {can}}: \mathcal {E}\rightarrow \mathcal {P}_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}. \end{aligned}$$

The map ${\text {can}}$ is given locally on sections by $e \mapsto 1 \otimes e$.

In general, (6.3) need not split as a sequence of left modules. Viewed as a sequence of left modules, (6.3) determines an element of

$$\begin{aligned} {\text {Ext}}^1_{\mathcal {O}_X}(\mathcal {E}, \Omega _{X/S}^1 \otimes _{\mathcal {O}_X} \mathcal {E}) \cong H^1(X, \Omega _{X/S}^1 \otimes _{\mathcal {O}_X} {\mathcal {E}}nd_{\mathcal {O}_X}(\mathcal {E})), \end{aligned}$$

sometimes called the “Atiyah class” of $\mathcal {E}$ relative to p. To distinguish this class from what we have called the Atiyah class of a matrix factorization in the body of this paper, we will call this class the classical Atiyah class of the vector bundle $\mathcal {E}$, and we write it as

$$\begin{aligned} At^{\text {classical}}_{X/S}(\mathcal {E}) \in {\text {Ext}}^1_{\mathcal {O}_X}(\mathcal {E}, \Omega _{X/S}^1 \otimes _{\mathcal {O}_X} \mathcal {E}). \end{aligned}$$

The sequence (6.3) splits as a sequence of left modules if and only if $At^{\text {classical}}_{X/S} = 0$.

Remark 6.4

The classical Atiyah class was first introduced by Atiyah in [1]. The definition can be extended to a bounded complex of vector bundles; see, for example, [24, §1]. The Atiyah class of such a complex is analogous to the Atiyah class of a matrix factorization, found in Definition 2.14 in the body of this paper.

There is also a version of the Atiyah class in the non-smooth setting, in which $\Omega ^1_X$ is replaced by the cotangent complex ${\mathbb {L}}_X$; this is due to Illusie [20]. See also the work of Buchweitz and Flenner [3], who develop a version of Illusie’s Atiyah class in the analytic setting.

Lemma 6.5

If $p: X \rightarrow S$ is affine, then for any vector bundle $\mathcal {E}$ on X $At^{\text {classical}}_{X/S}(\mathcal {E}) = 0$ and hence (6.3) splits as a sequence of left modules.

Proof

Since p is affine, $p_*$ is exact. Applying $p_*$ to (6.3) results in a sequence of $\mathcal {O}_S-\mathcal {O}_S$ bimodules (which are quasi-coherent for both actions). But since $p \circ \pi _1 = p \circ \pi _2$ these two actions coincide. Moreover, since (6.3) splits as right modules, so does its push-forward along $p_*$.

It thus suffices to prove the following general fact: if

$$\begin{aligned} F := (0 \rightarrow \mathcal {F}' \rightarrow \mathcal {F}\rightarrow \mathcal {F}'' \rightarrow 0) \end{aligned}$$

is a short exact sequence of vector bundles on X such that $p_*(F)$ splits as a sequence of quasi-coherent sheaves on S, then F splits. To prove this, observe that F determines a class in $H^1(X, {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_X}(\mathcal {F}'', \mathcal {F}'))$ and it is split if and only if this class vanishes. We may identify $H^1(X, {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_X}(\mathcal {F}'', \mathcal {F}'))$ with $H^1(S, p_* {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_X}(\mathcal {F}'', \mathcal {F}'))$ since p is affine. Moreover, the class of $F \in H^1(S, p_* {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_X}(\mathcal {F}'', \mathcal {F}'))$ is the image of the class of $p_*(F) \in H^1(S, {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_S}(p_* \mathcal {F}'', p_* \mathcal {F}'))$ under the map induced by the canonical map

$$\begin{aligned} {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_S}(p_* \mathcal {F}'', p_* \mathcal {F}') \rightarrow p_* {{\mathcal {H}{\text {om}}}}_{\mathcal {O}_X}(\mathcal {F}'', \mathcal {F}'). \end{aligned}$$

But by our assumption the class of $p_*(F)$ vanishes since $p_*F$ splits. $\square$

1.2 The vanishing of the classical Atiyah class and connections

Suppose $\sigma : \mathcal {E}\rightarrow \mathcal {P}_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}$ is a splitting of the map $\pi$ in (6.3) as left modules and recall ${\text {can}}: \mathcal {E}\rightarrow \mathcal {P}_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}$ is the splitting of $\pi$ as a morphism of right modules given locally by $e \mapsto 1 \otimes e$. Since $\sigma$ and ${\text {can}}$ are splittings of the same map regarded as a map of sheaves of abelian groups, the difference $\sigma - {\text {can}}$ factors as $i \circ \nabla _\sigma$ for a unique map of sheaves of abelian groups

$$\begin{aligned} \nabla _\sigma : \mathcal {E}\rightarrow \Omega ^1_{X/S} \otimes _{\mathcal {O}_X} \mathcal {P}. \end{aligned}$$

Lemma 6.6

The map $\nabla _\sigma$ is a connection on $\mathcal {E}$ relative to p.

Proof

The property of being a connection may be verified locally, in which case the result is well known.

In more detail, restricting to an affine open $U = {\text {Spec}}(Q)$ of X lying over an affine open $V = {\text {Spec}}(A)$ of S, we assume E is a projective Q-module and that we are given a splitting $\sigma$ of the map of left Q-modules . The map $\nabla _\sigma = (\sigma - {\text {can}})$ lands in $I/I^2 \otimes _Q Q = \Omega ^1_{Q/A} \otimes _Q E$, and for $a \in A, e \in E$ we have

$$\begin{aligned} \begin{aligned} \nabla _\sigma (a e)&= \sigma (ae) - 1 \otimes ae \\&= a \sigma (e) -1 \otimes ae \\&= a\sigma (e) - a \otimes e + a \otimes e - 1 \otimes ae \\&= a (\sigma (e) - 1 \otimes e) + (a \otimes 1 - 1 \otimes a) \otimes e = a \nabla _\sigma (e) + da \otimes e, \\ \end{aligned} \end{aligned}$$

since da is identified with $a \otimes 1 - 1 \otimes a$ under $\Omega ^1_{Q/A} \cong I/I^2$. $\square$

Lemma 6.7

Suppose $\mathcal {E}, \mathcal {E}'$ are locally free coherent sheaves on X and $\nabla , \nabla '$ are connections for each relative to p. If $g: \mathcal {E}\rightarrow \mathcal {E}'$ is a morphisms of coherent sheaves, then the map

$$\begin{aligned} \nabla ' \circ g - ({\text {id}}\otimes g) \circ \nabla : \mathcal {E}\rightarrow \Omega ^1_{X/S} \otimes _{\mathcal {O}_X} \mathcal {E}' \end{aligned}$$

is a morphism of coherent sheaves.

Proof

Given an open set U and elements $f \in \Gamma (U, \mathcal {O}_X), e \in \Gamma (U, \mathcal {E})$, the displayed map sends $f \cdot e \in \Gamma (U, \mathcal {E})$ to

$$\begin{aligned} \begin{aligned}&\nabla '(g(fe)) - ({\text {id}}\otimes g)(df \otimes e - f \nabla (e) \\&\quad = \nabla '(fg(e)) - df \otimes g(e) - f ({\text {id}}\otimes g)(\nabla (e)) \\&\quad =df \otimes g(e) + f\nabla '(g(e)) - df \otimes g(e) - f ({\text {id}}\otimes g)(\nabla (e)) \\&\quad = f\nabla '(g(e)) - f ({\text {id}}\otimes g)(\nabla (e)) \\&\quad = f\left( \left( \nabla ' \circ g - ({\text {id}}\otimes g) \circ \nabla \right) (e) \right) . \end{aligned} \end{aligned}$$

$\square$

The original, non-relative version of the following result is due to Atiyah; see [1, Theorem 2].

Proposition 6.8

For a vector bundle $\mathcal {E}$ on X, the function $\sigma \mapsto \nabla _\sigma$ determines a bijection between the set of splittings of the map $\pi$ in (6.3) as a map of left modules and the set of connections on $\mathcal {E}$ relative to p. In particular, $\mathcal {E}$ admits a connection relative to p if and only if $At^{\text {classical}}_{X/S}(\mathcal {E})= 0$.

Proof

From Lemma 6.7 with g being the identity map, the difference of two connections on $\mathcal {E}$ is $\mathcal {O}_X$-linear. By choosing any one splitting $\sigma _0$ of (6.3) and its associated connection $\nabla _0 = \nabla _{\sigma _0}$, the inverse of

$$\begin{aligned} \sigma \mapsto \nabla _{\sigma } \end{aligned}$$

is given by

$$\begin{aligned} \nabla \mapsto \nabla - \nabla _{\sigma _0} + \sigma _0. \end{aligned}$$

$\square$