Syzygies of Cohen-Macaulay modules and Grothendieck groups

2016/07/21 by Toshinori Kobayashi, Kobayashi, Toshinori · 1 citation
Mathematics · #13C14 #13D15 #13H10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1607.06189

openalex publication_date 2016/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the converse of a theorem of Butler and Auslander-Reiten. We show that a Cohen-Macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of Cohen-Macaulay modules if the Auslander-Reiten sequences generate the relation of the Grothendieck group of finitely generated modules. This extends a recent result of Hiramatsu, which gives an affirmative answer in the Gorenstein case to a conjecture of Auslander.

Citations

• Relations for Grothendieck groups of Gorenstein rings

Cited by

• Structure of modules stably annihilated by a fixed ideal

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