2025/04/24 by Fernando Peña Vázquez, Vázquez, Fernando Peña
Mathematics · #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2504.17167
openalex publication_date 2025/04/24 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Let X be a smooth p-adic Stein space with free tangent sheaf. We use the notion of Hochschild cohomology for sheaves of Ind-Banach algebras developed in our previous work to study the Hochschild cohomology of the algebra of infinite order differential operators DX-cap. In particular, we show that the Hochschild cohomology complex of DX-cap is a strict complex of nuclear Fréchet spaces which is quasi-isomorphic to the de Rham complex of X. We then use this to compare the first Hochschild cohomology group of DX-cap with a wide array of Ext functors. Finally, we investigate the relation of the Hochschild cohomology of DX-cap with the deformation theory of DX(X)-cap. Assuming some finiteness conditions on the de Rham cohomology of X, we define explicit isomorphisms between the first Hochschild cohomology group of DX-cap and the space of bounded outer derivations of DX(X)-cap, and between the second Hochschild cohomology group of DX-cap and the space of infinitesimal deformations of DX(X)-cap.