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Projective exterior Koszul homology and decomposition of the Tor functor. (English) Zbl 0856.13009

In this paper the authors generalize the work of J. A. Guccione and 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 A. Lago and A. G. Rodicio [Invent. Math. 107, No. 2, 433-446 (1992; Zbl 0768.14007)] and J. A. Guccione and 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.

MSC:

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13C40 Linkage, complete intersections and determinantal ideals
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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References:

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