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

Author

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  • Anastasis Kratsios

    (Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland)

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 .

Suggested Citation

  • Anastasis Kratsios, 2021. "Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras," Mathematics, MDPI, vol. 9(3), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:251-:d:487903
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