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$K$-THEORY OF MONOID ALGEBRAS AND A QUESTIONOF GUBELADZE

Part of: (Co)homology theory Arithmetic rings and other special rings Higher algebraic $K$-theory

Published online by Cambridge University Press:  10 August 2017

Amalendu Krishna  and
Husney Parvez Sarwar
Amalendu Krishna
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India (amal@math.tifr.res.in; mathparvez@gmail.com)
Husney Parvez Sarwar
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India (amal@math.tifr.res.in; mathparvez@gmail.com)
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Abstract

We show that for any commutative Noetherian regular ring $R$ containing $\mathbb{Q}$, the map $K_{1}(R)\rightarrow K_{1}\left(\frac{R[x_{1},\ldots ,x_{4}]}{(x_{1}x_{2}-x_{3}x_{4})}\right)$ is an isomorphism. This answers a question of Gubeladze. We also compute the higher $K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher $K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.

Keywords

algebraic K-theorymonoid algebrassingular schemes

MSC classification

Primary: 19D50: Computations of higher $K$-theory of rings
Secondary: 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) 14F35: Homotopy theory; fundamental groups

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 5 , September 2019 , pp. 1051 - 1085
Copyright
© Cambridge University Press 2017 

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