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 $c\ge 2$.