Acta Mathematica Vietnamica

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On the Limit of Frobenius in the Grothendieck Group

$icon-email$ Kazuhiko Kurano , Kosuke Ohta

Abstract

Considering the Grothendieck group of finitely generated modules modulo numerical equivalence, we obtain the finitely generated lattice $\overset{―}{G_{0} (R)}$ for a Noetherian local ring $R$ . Let $C_{C M} (R)$ be the cone in ${\overset{―}{G_{0} (R)}}_{R}$ spanned by cycles of maximal Cohen-Macaulay $R$ -modules. We shall define the fundamental class $\overset{―}{μ_{R}}$ of $R$ in ${\overset{―}{G_{0} (R)}}_{R}$ , 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 $\overset{―}{μ_{R}}$ 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 $\overset{―}{μ_{R}}$ is in $C_{C M} (R)$ in the case where $R$ is FFRT or F-rational.

Acta Mathematica Vietnamica

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On the Limit of Frobenius in the Grothendieck Group

Abstract