Hochschild homology of complete intersections (Q1181470)

Let \(f_ 1,\ldots,f_ r\) be a regular sequence in the polynomial ring \(k[X_ 1,\ldots,X_ n]\) over a commutative ring \(k\) and denote by \(A\) the \(k\)-Algebra \(k[X_ 1,\ldots,X]/(f_ 1,\ldots,f_ r)\). Under the assumption that \(A\) is a flat, respectively projective \(k\)-module, the authors compute the Hochschild homology respectively cohomology of \(A\) with coefficients in an \(A\)-module \(M\) as a function of the homology of the generalized Koszul complex of \(M\). The computation is based on the construction of an \(A^ e\)-free resolution of \(A\) by the exterior algebra over a certain free \(A^ e\)-module which is mapped into the more complicated Hochschild resolution.