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).
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Acknowledgments
I am grateful to Jesse Burke, Olgur Celikbas, Hailong Dao, Daniel Murfet and Roger Wiegand for conversations about the topics of this paper and to the anonymous referee for some helpful suggestions.
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The author was supported in part by National Science Foundation Award DMS-0966600.
Appendix: Relative connections
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
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
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 \(X \times X\)
given locally on generators by \(dg \mapsto g \otimes 1 - 1 \otimes g\). From this we obtain the short exact sequence
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
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
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
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
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
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
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
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
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
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
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
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
\(\square \)
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
is given by
\(\square \)
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Walker, M.E. Chern characters for twisted matrix factorizations and the vanishing of the higher Herbrand difference. Sel. Math. New Ser. 22, 1749–1791 (2016). https://doi.org/10.1007/s00029-016-0231-4
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DOI: https://doi.org/10.1007/s00029-016-0231-4