On the limit of Frobenius in the Grothendieck group

TL;DR

This paper studies the fundamental class in the Grothendieck group modulo numerical equivalence, showing it lies in the Cohen-Macaulay cone for certain classes of rings, linking to homological conjectures.

Contribution

It introduces the fundamental class in the Grothendieck group and proves its inclusion in the Cohen-Macaulay cone for FFRT and F-rational rings, advancing understanding of homological conjectures.

Findings

The fundamental class is in the Cohen-Macaulay cone for FFRT rings.

The fundamental class is in the Cohen-Macaulay cone for F-rational rings.

Links between the fundamental class and homological conjectures are established.

Abstract

Considering the Grothendieck group modulo numerical equivalence, we obtain the finitely generated lattice $\overline{G_0(R)}$ for a Noetherian local ring $R$. Let $C_{CM}(R)$ be the cone in $\overline{G_0(R)}_{\Bbb R}$ spanned by cycles of maximal Cohen-Macaulay $R$-modules. We shall define the fundamental class $\overline{\mu_R}$ of $R$ in $\overline{G_0(R)}_{\Bbb R}$, 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 $\overline{\mu_R}$ 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 $\overline{\mu_R}$ is in $C_{CM}(R)$ in the case where $R$ is FFRT or F-rational.