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

On the Taylor tower of relativeK–theory

“…From the calculational point of view, the fact that for any simplicial R-bimodule M , (0-3) z K.R Ë M / ' z K.RI B:M / ' W .RI B:M / (note that B:M is connected) is useful: for example, in [17] we use this and Lars Hesselholt and Ib Madsen's calculation of W .F p I F p / D TR.F p / to completely calculate z K.F p Ë . L n iD1 F p //p .…”

On the algebraic K-theory of formal power series

“…We are, of course, quotienting this whole picture out byK(R; M ). By following the decomposition of Theorem 2.2 in [LMcC2] on the 0-dimensional part (as we did before), we see that…”

“…Since we assume that M is connected, N ) . For the fiber of ΣK(R; M ⊕ N ) → ΣK(R; M ), we can describe it using [LMcC1] which shows that for connected bimodulesK(R, ) ≃ W (R; ) (that is, for connected bimodules the Taylor tower converges toK(R; )) together with the splitting of Theorem 2.2 in [LMcC2] for W (R; ). We get that for M, N connected,…”

Some recent advances in topological Hochschild homology

“…A basic example would be the case of a square‐zero extension $$k \oplus V$$, for $$k$$ a perfect field and $$V$$ a $$k$$‐vector space, where the $$K$$‐theory is calculated in [92]. This calculation has been extended to perfectoid rings by Riggenbach [109], using the approach to $$\mathrm{TC}$$ of [105].…”