The Algebraic K-theory of Extensions of a Ring by Direct Sums of Itself

Abstract: We calculate K(A×(A⊕k))∧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 ⊗ S1) ' K̃(A×M). We give a sum formula for W (A;M1 ⊕ · · · ⊕Mn), 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 E1 spectral sequence converging to pi∗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 K̃(Z×Q) different than Goodwillie’s original one in [7]