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On the \(K\)-theory of coordinate axes in affine space. (English) Zbl 1486.19002

Let \(k\) be a perfect ground field of positive characteristic, and let us consider the monomial algebra \(A_d := k[x_1,\ldots,x_d]/\langle x_ix_j\mid i\neq j\rangle\), which is referred as the algebra of functions of the coordinate axes in the affine space \(\mathbb{A}^d\). The paper under review calculates the relative algebraic \(K\)-theory \(K_q(A_d,I_d)\) of the canonical algebra morphism \(A_d\to k\) whose kernel is denoted by \(I_d\) in the paper. The analogous result for characteristic 0 is proven by [S. Geller et al., J. Reine Angew. Math. 393, 39–90 (1989; Zbl 0649.14006)], but the author provides an extension for ind-smooth rational algebras in the paper. The main result proven in the paper for the positive characteristic is a continuation of the author’s earlier work [Adv. Math. 366, Article ID 107083, 18 p. (2020; Zbl 07183739)] and an extension of two previous calculations: one by R. K. Dennis and M. I. Krusemeyer for \(q=2\) [J. Pure Appl. Algebra 15, 125–148 (1979; Zbl 0405.18009)] and one by L. Hesselholt for \(d=2\) [Nagoya Math. J. 185, 93–109 (2007; Zbl 1136.19002)]. In order to prove the result, the author uses the cyclotomic trace map from \(K\)-theory to topological cyclic homology similar to [Contemp. Math. 749, 139–148 (2020; Zbl 1436.19006)] where L. Hesselholt and T. Nikolaus calculated the \(K\)-theory of cuspidal curves using the same strategy.

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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