Chern Characters for Twisted Matrix Factorizations and the Vanishing of the Higher Herbrand Difference

TL;DR

This paper develops a new theory of Chern characters for twisted matrix factorizations and uses it to prove the vanishing of certain higher invariants in algebraic geometry and commutative algebra, especially for complete intersections with isolated singularities.

Contribution

It introduces a novel approach to Chern characters for twisted matrix factorizations and applies this to establish vanishing results for higher Herbrand differences and theta pairings in specific algebraic settings.

Findings

Vanishing of higher Herbrand difference in certain complete intersections.

Vanishing of higher codimensional Hochster's theta pairing under specified conditions.

Development of a new theory of Chern characters for twisted matrix factorizations.

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 $c \geq 2$.