On the algebraic $K$-theory of the coordinate axes over the integers

Abstract

We show that the relative algebraic K-theory groupis free abelian of rank 1 and that2 . We also find the group structure of K 2i+1 (Z[x, y]/(xy), (x, y)) in low degrees.

Topological Hochschild homology of ℓ and ko

“…The equivariant stable homotopy computations in this paper serve as input for these methods. In particular they have been used in the computations of the relative algebraic K-theory groups K * (Z[x]/(x m ), (x)) and K * (Z[x, y]/(xy), (x, y)) up to extensions (see [2] and [1] respectively).…”

“…We use the results of this paper in [2], which is joint work with Lars Hesselholt, to compute the relative K-groups K * (Z[x]/(x m ), (x)) up to extensions, and in [1] to compute the relative K-groups K * (Z[x, y]/(xy), (x, y)) up to extensions. Theorem 1.4 below is the necessary input to the trace method approach described above, allowing us to make such computations.…”

RO(S1)-graded TR-groups of Angeltveit1, Gerhardt2 2011Journal of Pure and Applied AlgebraSelf Cite

“…We highlight the following result, which is essential to the K -theory computations in [2,1]: Theorem 1.4. Let λ be a finite complex S 1 -representation.…”

“…We use the results of this paper in [2], which is a joint work with Lars Hesselholt, to compute the relative K -groups K * (Z[x]/(x m ), (x)) up to extensions, and in [1] to compute the relative K -groups K * (Z[x, y]/(xy), (x, y)) up to extensions. Theorem 1.4 below is the necessary input to the trace method approach described above, allowing us to make such computations.…”

“…The equivariant stable homotopy computations in this paper serve as input for these methods. In particular they have been used in the computations of the relative algebraic K -theory groups K * (Z[x]/(x m ), (x)) and K * (Z[x, y]/(xy), (x, y)) up to extensions (see [2,1] respectively).…”