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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).

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