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



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.