Hochschild homology of complete intersections and smoothness
Let \(k\) be a commutative ring, \(A\) a smooth \(k\)-algebra, \(\{a_ 1, \dots, a_ m \}\) a regular sequence in \(A\), and put \(B = A/(a_ 1, \dots, a_ m)\). Then, after necessary preliminaries about commutative differential graded algebras and the Hochschild (co)homology, the main result of this paper shows that when \(B\) is \(k\)-flat (resp. \(k\)- projective) and \(M\) is a \(B\)-module, the Hochschild homology \(H_ n (B,M)\) (resp. the Hochschild cohomology \(H^ n (B,M))\) is computed by using a certain commutative differential graded algebra obtained by tensoring the algebra \(\Omega^*_{A/k}\) of differentials with a divided power algebra. As an application, some criteria are given for the smoothness of \(B\) in terms of the Hochschild (co)homology modules.
scientific article; zbMATH DE number 681815