On the Limit of Frobenius in the Grothendieck Group

Abstract

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