Nonarchimedean Bornologies, Cyclic Homology and Rigid Cohomology

Analytification, localization and homotopy epimorphisms

“…For A a finitely generated commutative algebra over a field of characteristic zero and J an ideal of A and P J denotes the J-adic completion then again using the derived algebraic adic (Weierstrass) localization we recover the results HH * (A J ) ∼ = A J ⊗ A HH * (A) of Proposition 4.2.1 of [11]. Corollary 7.12 ([11, Proposition 4.17]).…”

“…13. In [11,17], the authors define dagger algebras over non-archimedean discrete valuation rings in a slightly different way, using bornological analysis. Let V be a complete discrete valuation ring with uniformiser π.…”

“…by [11,Lemma 3.2.2]. This has the bornology where a subset is bounded if it contains power series as above for a fixed C > 0 and 0 < β < 1.…”

“…The hypotheses of Proposition 5.22 apply to the rationalized dagger completion (see [17] and Remark 4.13) R † := R lg ⊗ Zp Q p of a torsion-free, finite-type commutative Z p -algebra R with the fine bornology. More concretely, by [11,Lemma 4.14], the map R → R † is algebraically flat. Furthermore, a subset in the dagger completion R † is bounded if and only if it is contained in the Z p -submodule generated by…”

“…Remark 7.13. The Hochschild homology base change result in Corollary 7.12 was used in [11] to compare the periodic cyclic homology of A † with the rigid cohomology of A/πA with coefficients in F , in the case where A is smooth over the fraction field F . To see this, we first note that for the smooth algebra A, the Hochschild-Kostant-Rosenberg Theorem implies an isomorphism…”