## 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[ε])_{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}) ≃ K̃(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}))* 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].

Original language | English (US) |
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Pages (from-to) | 577-625 |

Number of pages | 49 |

Journal | Indiana University Mathematics Journal |

Volume | 57 |

Issue number | 2 |

DOIs | |

State | Published - 2008 |

## Keywords

- Algebraic K-theory
- Square-zero extension
- Topological Hochschild homology
- Topological Witt vectors

## ASJC Scopus subject areas

- Mathematics(all)