On G(A)_ℚ of rings of finite representation type

2022/09/23 by Tony J. Puthenpurakal, Puthenpurakal, Tony J.
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2209.11566

Abstract

Let (A,\mathfrakm) be an excellent Henselian Cohen-Macaulay local ring of finite representation type. If the AR-quiver of A is known then by a result of Auslander and Reiten one can explicity compute G(A) the Grothendieck group of finitely generated A-modules. If the AR-quiver is not known then in this paper we give estimates of G(A)_ℚ = G(A)⊗_ℤ ℚ when k = A/\mathfrakm is perfect. As an application we prove that if A is an excellent equi-characteristic Henselian Gornstein local ring of positive even dimension with char A/\mathfrakm ≠ 2,3,5 (and A/\mathfrakm perfect) then G(A)_ℚ ≅ ℚ.

Related

• Grothendieck groups and completions of Gorenstein local rings
• K-groups for rings of finite Cohen-Macaulay type
• Cohen-Macaulay representations of invariant subrings
• A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective
• K'-Theory of a Local Ring of Finite Cohen-Macaulay Type