Grothendieck groups and completions of Gorenstein local rings

2025/10/15 by Tony J. Puthenpurakal, Puthenpurakal, Tony J.
Mathematics · #13C60 #13D15 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 13D09 #Rings, Modules, and Algebras #Secondary 13C14

paper · pdf · doi:10.48550/arxiv.2510.13225

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

Let (A,\mathfrakm) be an excellent Gorenstein local ring of dimension d ≥ 2 which is an isolated singularity. Let \widehatA denote the completion of A. If G(A) is the Grothendieck group of A then by G(A)_ℚ we denote G(A)⊗_ℤ ℚ. We prove that the natural map G(A)_ℚ → G(\widehatA)_ℚ is an isomorphism if and only if for any maximal Cohen-Macaulay (= MCM) \widehatA-module M there exists an MCM A-module N and integers r ≥ 1 and s ≥ 0 (depending on M) such that M^r⊕ \widehatA^s ≅ \widehatN. An essential ingredient is the classification of ℚ-subspaces of G(C)_ℚ (here C is a skelletaly small triangulated category) in terms of certain dense subcategories of C. We also give criterion for a Henselian Gorenstein ring B (not an isolated singularity) such that the natural map G(B)_ℚ → G(\widehatB)_ℚ is an isomorphism ( when dim B = 2, 3). We give many examples where our result holds.

Citations

• Classifying thick subcategories of the stable category of Cohen-Macaulay modules
• On injectivity of maps between Grothendieck groups induced by completion
• A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective

Related

• On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings
• A function on the the set of isomorphism classes in the stable category\n of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications\n to Liason theory
• Local homology and Gorenstein flat modules
• On G(A)_ℚ of rings of finite representation type
• A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective