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On the Limit of Frobenius in the Grothendieck Group
Kazuhiko Kurano, Kosuke Ohta

 

Abstract

Considering the Grothendieck group of finitely generated modules modulo numerical equivalence, we obtain the finitely generated lattice G0(R)¯¯¯¯¯¯¯¯ for a Noetherian local ring R. Let C C M (R) be the cone in G0(R)¯¯¯¯¯¯¯¯R spanned by cycles of maximal Cohen-Macaulay R-modules. We shall define the fundamental class μR¯¯¯¯ of R in G0(R)¯¯¯¯¯¯¯¯R , which is the limit of the Frobenius direct images (divided by their rank) [ e R]/p d e in the case c h(R)=p>0. The homological conjectures are deeply related to the problems whether μR¯¯¯¯ is in the Cohen-Macaulay cone C C M (R) or the strictly nef cone S N(R) defined below. In this paper, we shall prove that μR¯¯¯¯ is in C C M (R) in the case where R is FFRT or F-rational.