Dundas, Bjørn Ian; Morrow, Matthew Finite generation and continuity of topological Hochschild and cyclic homology. (La génération finie et continuité en homologies de Hochschild et cyclique.) (English. French summary) Zbl 1372.19002 Ann. Sci. Éc. Norm. Supér. (4) 50, No. 1, 201-238 (2017). Authors’ abstract: Let \(A\) be a commutative \(\mathbb{Z}_{(p)}\)-algebra. Assume \(A\) is \(F\)-finite, i.e. \(A/pA\) is finitely generated over its subring of \(p^{\text{th}}\) powers. Set \[ \widehat{A_p} := \lim A/p^s. \]The goal of the paper is to establish various properties of the Hochschild, topological Hochschild, and topological cyclic homologies of \(A\). The first main result is that \(HH_*(A ; \mathbb{Z}_{(p)})\) and \(THH_*(A; \mathbb{Z}_{(p)})\) are finitely generated over \(\widehat{A_p}\), and the homotopy groups of the fixed point spectrum \(TR^r(A;p, \mathbb{Z}_{(p)})\) are finitely generated modules over the \(p\)-typical Witt ring \(W_r(\widehat{A_p})\) (see Theorem 1.2). They next prove what the authors call a “degree-wise continuity” result for the above theories: that is, given an ideal \(I\) of \(A\), they consider whether the natural map \[ \{HH_n(A) \otimes_A A/I^s\}_s \to \{ HH_n(A/I^s) \}_s \] of pro \(A\)-modules is an isomorphism. Their results concerning finite generation and continuity are extended to schemes in Section 6 of the paper.As a consequence of the above results, the authors prove a “pro-version” of Hesselholt’s Hochschild-Kostant-Rosenberg theorem for \(THH\) and \(TR\) (Theorem 1.6). Reviewer: Michael Brown (Bonn) Cited in 1 ReviewCited in 6 Documents MSC: 19D55 \(K\)-theory and homology; cyclic homology and cohomology 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Keywords:\(K\)-theory; topological cyclic homology PDFBibTeX XMLCite \textit{B. I. Dundas} and \textit{M. Morrow}, Ann. Sci. Éc. Norm. Supér. (4) 50, No. 1, 201--238 (2017; Zbl 1372.19002) Full Text: arXiv Link