IUMJ

Title: The algebraic K-theory of extensions of a ring by direct sums of itself

Authors: Ayelet Lindenstrauss and Randy McCarthy

Issue: Volume 57 (2008), Issue 2, 577-626

Abstract:

We calculate $K (A \ltimes (A^{\oplus k}))_p^{\wedge}$ when $A$ is a perfect field of characteristic $p > 0$, generalizing the $k = 1$ case $K(A[\epsilon])_p^{\wedge}$ which was calculated by Hesselholt and Madsen by a different method in \cite{HM}. We use $W(A;M)$, a construction which can be thought of as topological Witt vectors with coefficients in a bimodule. For a ring or more generally an FSP $A$, $W(A;M \otimes S^1) \simeq \tilde K(A \ltimes M)$. We give a sum formula for $W(A;M_1 \oplus \cdots \oplus M_n)$, and a splitting of $W(A;M)_p^{\wedge}$ analogous to the splitting of the algebraic Witt vectors into a product of $p$-typical Witt vectors after completion at $p$. We construct an $E^1$ spectral sequence converging to $\pi_* W^{(p)}(A;M \otimes X)$, where $W^{(p)}$ is the topological version of $p$-typical Witt vectors with coefficients. This enables us to complete the calculation of $K (A \ltimes (A^{\oplus k}))_p^{\wedge$}}$ in terms of $W^{(p)}(A;A)$ if the homotopy of the latter is concentratedin dimension $0$; for perfect fields of characteristic $p > 0$, Hesselholt and Madsen showed in \cite{HM} that this condition holds. Using our methods we also give a complete calculation of $W(A;M)$ where $A$ is a commutative ring and $M$ a symmetric, flat $A$-bimodule whose homotopy groups are vector spacesover $\mathbb{Q}$, and a way of calculating $\tilde K(\Z \ltimes \mathbb{Q})$ different than Goodwillie\'s original one in \cite{Good}.