Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

Anastasis Kratsios
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Anastasis Kratsios: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland

Mathematics, 2021, vol. 9, issue 3, 1-22

Abstract: The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n -forms Ω n ( X , M ) . Further restricting the notion of smoothness, we use our result to show that most k -algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k = C .

Keywords: hochschild cohomology; homological dimension theory; non-commutative geometry; quasi-free algebras; pointcaré duality; higher differential forms (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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