We calculate K(A ⋉ (A⊕k))p^ when A is a perfect field of characteristic p > 0, generalizing the k = 1 case K(A[ε])p^ which was calculated by Hesselholt and Madsen by a different method in . 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 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))* 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  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 .
- Algebraic K-theory
- Square-zero extension
- Topological Hochschild homology
- Topological Witt vectors
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