\(K\)-theory of monoid algebras and a question of Gubeladze. (English) Zbl 1425.19001
The algebraic \(K\) theory of smooth algebraic varieties satisfies homotopy invariance. A particular consequenece of this is that the \(K\)-theory of a commutative Noetherian regular ring does not change under polynomial extensions. One can ask a more general question: if \(R\) is such a ring, how is the \(K\)-theory of \(R\) related to the \(K\)-theory of \(R[M]\) where \(M\) is a monoid algebra? Free monoids are covered under homotopy invariance.
The research in the article under review explores this question for the specific monoid algebra \(M\), which is the free monoid algebra in four generators \(x_1,x_2,x_3,x_4\) modulo the relation \(x_1x_2 = x_3x_4\). They prove that \(K_1(R)\cong K_1(R[M])\), but the same is not true for higher \(K\)-groups, showing some failure of invariance here for monoid algebras.
They also use their theorem to show that the Serre dimension of \(R[x_1x_3,x_1x_4,x_2x_3,x_2x_4]\) is less than or equal to 1 whenever \(R\) is a one-dimensional commutative Noetherian ring, answering a special case of a question of H. Lindel [J. Algebra 172, No. 2, 301–319 (1995; Zbl 0839.13006)].
The paper is a good contribution to the theory of how \(K\)-theory behaves under monoid-algebra extensions of rings.
The research in the article under review explores this question for the specific monoid algebra \(M\), which is the free monoid algebra in four generators \(x_1,x_2,x_3,x_4\) modulo the relation \(x_1x_2 = x_3x_4\). They prove that \(K_1(R)\cong K_1(R[M])\), but the same is not true for higher \(K\)-groups, showing some failure of invariance here for monoid algebras.
They also use their theorem to show that the Serre dimension of \(R[x_1x_3,x_1x_4,x_2x_3,x_2x_4]\) is less than or equal to 1 whenever \(R\) is a one-dimensional commutative Noetherian ring, answering a special case of a question of H. Lindel [J. Algebra 172, No. 2, 301–319 (1995; Zbl 0839.13006)].
The paper is a good contribution to the theory of how \(K\)-theory behaves under monoid-algebra extensions of rings.
Reviewer: Jason Polak (Montreal)
MSC:
| 19D50 | Computations of higher \(K\)-theory of rings |
| 13F15 | Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) |
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
Citations:
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