Algebraic \(K\)-theory of the first Morava \(K\)-theory
The main result of the paper is the calculation of the \(V(1)\)-homology of the \(K\)-spectrum of the first Morava \(K\)-theory \({\ell}/p\), where \(p \geq 5\) is a prime. The calculation is of the homology as a module over \(P(v_2)\), the polynomial algebra over \(\mathbb{F}_p\). Here \(V(1)\) is the Smith-Toda complex which is a commutative ring spectrum. A similar calculation is given for the first periodic Morava \(K\)-theory \(L/p\).NEWLINENEWLINEThe calculations are done by means of the cyclotomic trace. So, along the way, the \(V(1)\)-homology of the topological Hochschild homology spectrum and of the topological cyclic homology spectrum of \({\ell}/p\) and \(L/p\) are computed. For the homology of the \(THH\)-spectrum, the authors use the Tate homology spectral sequences to compute the homology of the \(C_{p^n}\)-fixed point and homotopy fixed point sets of the spectrum. So, the homology of the \(S^1\)-fixed points and homotopy fixed points are computed. The next step is the calculation of the \(V(1)\)-homology of the topological cyclic homology spectrum. The result follows from the connectivity properties of the reduction map from \({\ell}/p\) to \(\mathbb{Z}/p\).NEWLINENEWLINEIn an earlier paper [``Algebraic K-theory of topological K-theory'', Acta Math. 188, No. 1, 1--39 (2002; Zbl 1019.18008)] the authors calculated the \(V(1)\)-homology of the spectrum \({\ell}_p\), the \(p\)-complete Adams summand of the connective complex \(K\)-theory. Their work here extends their earlier results. After all, \(K({\ell}/p)\) is a \(K({\ell}_p)\)-module. The paper under review is one of a series of papers that have as goal the understanding of the \(K\)-theory of \({\ell}_p\) and other \(S\)-algebras using Galois decent.
scientific article; zbMATH DE number 6058742