Vol. 6, No. 4, 2021

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$K_0$-stability over monoid algebras

Husney Parvez Sarwar

Vol. 6 (2021), No. 4, 629–649
DOI: 10.2140/akt.2021.6.629
Abstract

(1)  Let R be a commutative Noetherian ring of dimension d and M a commutative partially cancellative torsion-free seminormal monoid. Then Vecr(R[M]) is injective stable at d + 1. This settles a conjecture of Gubeladze for the mentioned class of monoids.

(2)  Take the same R as in (1) with an additional assumption that R has a positive characteristic which is prime to (r 1)!. Let M be a commutative cancellative torsion-free seminormal positive monoid with a radical ideal I M. Then the map Umr(R[M]) Umr(R[M]IR[M]) is surjective for r d2 + 2. This answers a question of Wiemers in some special cases.

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Keywords
algebraic $K$-theory, $K_0$-stability, projective module, projective cancellation, partially cancellative monoids, monoid algebra
Mathematical Subject Classification 2010
Primary: 19A13
Secondary: 13C10, 13D15
Milestones
Received: 1 January 2020
Revised: 4 April 2021
Accepted: 22 April 2021
Published: 12 February 2022
Authors
Husney Parvez Sarwar
Department of Mathematics
Indian Institute of Technology Kharagpur
Kharagpur
India