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

Abstract

We calculate K(A & (A xk )) ^ p when A is a perfect field of characteristic p > 0, generalizing the k = 1 case K(A[∈]) ^ p which was calculated by Hesselholt and Madsen by a different method in [6]. 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 ⊗ S 1 ) ≃ K(A & M). We give a sum formula for W(A;M 1 ⊕··· ⊕ M n ), and a splitting of W(A; M) ^ p 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 π* w (p) (A;M ⊗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 & (A ⊕k )) ^ p in terms of W (p) (A; A) if the homotopy of the latter is concentrated in dimension 0; for perfect fields of characteristic p > 0, Hesselholt and Madsen showed in [6] 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 spaces over Q, and a way of calculating K(Z & Q) different than Goodwillie's original one in [7].