On the limit of Frobenius in the Grothendieck group

Abstract

Considering the Grothendieck group modulo numerical equivalence, we obtain the finitely generated lattice G 0 (R) for a Noetherian local ring R. Let C CM (R) be the cone in G 0 (R) R spanned by cycles of maximal Cohen-Macaulay R-modules. We shall define the fundamental class µ R of R in G 0 (R) R , which is the limit of the Frobenius direct images (divided by their rank) [ e R]/p de in the case ch(R) = p > 0. The homological conjectures are deeply related to the problems whether µ R is in the Cohen-Macaulay cone C CM (R) or the strictly nef cone SN (R) defined below. In this paper, we shall prove that µ R is in C CM (R) in the case where R is FFRT or F-rational.