No abstract

We define a dagger algebra as a bornological algebra over a discrete valuation ring with three properties that are typical of Monsky-Washnitzer algebras, namely, completeness, bornological torsion-freeness and a certain spectral radius condition. We study inheritance properties of the three properties that define a dagger algebra. We describe dagger completions of bornological algebras in general and compute some noncommutative examples.

Let R be a Banach ring. We prove that the category of chain complexes of complete bornological R-modules (and several related categories) is a derived algebraic context in the sense of Raksit. We then use the framework of derived algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which relates the circle action on the Hochschild algebra to the de Rham-differential-enriched-de Rham algebra of a simplicial, commutative, complete bornological algebra. This has a geometric interpretation in the language of derived analytic geometry, namely, the derived loop stack of a derived analytic stack is equivalent to the shifted tangent stack. Using this geometric interpretation we extend our results to derived schemes.

No abstract

Let V be a complete discrete valuation ring with residue field F. We define a cyclic homology theory for algebras over F, by lifting them to free algebras over V , which we enlarge to tube algebras and complete suitably. We show that this theory may be computed using any pro-dagger algebra lifting of an F-algebra. We show that our theory is polynomially homotopy invariant, excisive, and matricially stable.