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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

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).

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.)
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