Projective exterior Koszul homology and decomposition of the Tor functor
type: Item
instance of: scholarly article
title: Projective exterior Koszul homology and decomposition of the Tor functor
zbMATH Open document ID: 0856.13009
DOI: 10.1007/BF01232370
author: Amalia Blanco
author: Antonio G. Rodicio
author: Javier Majadas
published in: Inventiones Mathematicae
publication date: 1996-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/144339
review text: 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.
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 18G15
zbMATH DE Number: 871766
zbMATH Keywords: Koszul homology
zbMATH Keywords: Hochschild homology of complete intersections
zbMATH Keywords: Tate resolution
zbMATH Keywords: simplicial resolutions
MaRDI profile type: Publication
cites work: Q4771408
cites work: Produit tensoriel et complexe cotangent. (Tensor product and cotangent complex)
cites work: Hochschild homology of complete intersections
cites work: Hochschild homology of complete intersections and smoothness
cites work: A Sequence of Complexes Associated with a Matrix
cites work: Generalized Koszul complexes and Hochschild (co-)homology of complete intersections
cites work: Homotopical algebra
cites work: On the rigidity of the generalized Koszul complexes with applications to Hochschild homology
cites work: Commutative augmented algebras with two vanishing homology modules
cites work: Flat exterior Tor algebras and cotangent complexes
cites work: Minimal models and derived functors
cites work: Homology of Noetherian rings and local rings
OpenAlex ID: W2024457572
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description: scientific article; zbMATH DE number 871766