On the limit of Frobenius in the Grothendieck group

2014/07/15 by Kazuhiko Kurano, Kurano, Kazuhiko, Kosuke Ohta +1
Computer Science · Mathematics · #13A35 #13D15 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Topological and Geometric Data Analysis #math.AC #msc:13A35 #msc:13D15

paper · pdf · doi:10.48550/arxiv.1407.4159

13 pages

arxiv created 2014/07/15 · openalex publication_date 2014/07/15 · arxiv updated 2014/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Considering the Grothendieck group modulo numerical equivalence, we obtain the finitely generated lattice G₀(R) for a Noetherian local ring R. Let C_CM(R) be the cone in G₀(R)_{\Bbb R} spanned by cycles of maximal Cohen-Macaulay R-modules. We shall define the fundamental class μ_R of R in G₀(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 μ_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 μ_R is in C_CM(R) in the case where R is FFRT or F-rational.

Related

• Grothendieck groups, convex cones and maximal Cohen-Macaulay points
• The cone spanned by maximal Cohen-Macaulay modules and an application
• Diagonalizing the Frobenius
• Boundary and shape of Cohen-Macaulay cone
• The asymptotic behavior of Frobenius direct images of rings of invariants