Projective exterior Koszul homology and decomposition of the Tor functor
In this paper the authors generalize the work of \textit{J. A. Guccione} and \textit{J. J. Guccione} [J. Pure Appl. Algebra 74, No. 2, 159-176 (1991; Zbl 0746.13004)] on Hochschild homology of complete intersections. Let \(A\) be a commutative ring, \(I\) be an ideal generated by a set \({\mathbf t}\), \(B=A/I\), and \(E\) be the Koszul complex on \({\mathbf t}\). One says that \({\mathbf t}\) has the free (respectively flat, projective) exterior Koszul homology property FEKH (FLEKH, PEKH) if (i) \(H_1(E)\) is a free, respectively flat or projective, \(B\)-module, and (ii) the canonical map \(\bigwedge^*H_1(E) \to H_*(E)\) is an isomorphism. The latter properties are independent of the choice of generators of \(I\). If one begins with \(E\) and builds the Tate resolution of \(B\) over \(A\), then the authors show that the second step \(F\) is already acyclic if and only if FEKH holds. In this case the resulting exterior and divided power structure of Tor leads to a decomposition \[ \text{Tor}^A_m(B,B) = \bigoplus^{[m/2]}_{j=0} H^{m-2j} (\text{Kos}^* (H_1(E) \to E_1 \otimes_AB)_{m-j}). \] With only FLEKH one gets \[ H_m(S^{m-j}_B \mathbb{L}_{B/A}) = H^{m-2j} (\text{Kos}^* (H_1(E) \to E_1 \otimes_AB)_{m-j}), \] where \(\mathbb{L}_{B/A}\) is the cotangent complex, and degeneration of the Quillen spectral sequence \[ E^2_{p,q}=H_{p+q}(S^q_B\mathbb{L}_{B/A})\Rightarrow\text{Tor}^A_n(B,B); \] with PEKH, one further gets \(\text{Tor}^A_n(B,B) = \bigoplus_{p+q=n} E^2_{p,q}\). In particular the decomposition depends only on \(A\) and \(B\), and not on choices of generators of \(I\) or \(E\). As application the main results of \textit{A. Lago} and \textit{A. G. Rodicio} [Invent. Math. 107, No. 2, 433-446 (1992; Zbl 0768.14007)] and \textit{J. A. Guccione} and \textit{J. J. Guccione} [J. Pure Appl. Alg. 95, No. 2, 131-150 (1994; Zbl 0821.13004)] can be recovered. The theory can also be developed in the context of simplicial resolutions.
Hochschild homology of complete intersections
scientific article; zbMATH DE number 871766