Finiteness conditions for the Hochschild homology algebra of a commutative algebra (Q1270115)

Let \(A\) be a commutative algebra over a commutative ring \(k\) and let \((C_*(A), b)\) be the Hochschild complex endowed with the shuffle product, so that it is a commutative differential graded algebra. The homology of this complex, denoted \(HH_*(A)\), is by definition the Hochschild homology of \(A\); it inherits the structure of a commutative graded algebra over \(HH_0 (A)=A\). The author proves the following theorems. Theorem A: Let \(A\) be a commutative algebra of finite type over a field of characteristic zero. Assume that \(A\) is isomorphic to the quotient of a polynomial algebra \(k[X_1, \dots, X_m]\) by an ideal generated by a regular sequence \((f_1, \dots, f_r)\), where the \(X_l\) have a strictly positive weight and the \(f_j\) are homogeneous for the induced graduation. Then \(HH_*(A)\) is not finitely generated as an \(A\)-algebra. Theorem B: Let \(A\) be a commutative algebra satisfying the hypothesis of theorem \(A\). Then the product of \(m+1\) elements of odd degrees in \(HH_*(A)\) is always zero.