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Homology of perfect complexes. (English) Zbl 1186.13006

Adv. Math. 223, No. 5, 1731-1781 (2010); corrigendum ibid. 225, No. 6, 3576-3578 (2010).
Let \((R,\mathfrak{m},k)\) be a local ring. The Loewy length of a finetely generated \(R\)-module \(M\) is defined to be the number \({\ell \ell}_R\, M := \text{inf}\, \{i\geq 0\, | \, \mathfrak{m}^iM = 0\}\). In the paper under review, the authors provide uniform lower bounds on Loewy lengths of modules of finite projective dimension in terms of invariants depending only on \(R\). In Section 1, they show that if \(R\) is Gorenstein and if its associated graded ring \(\text{gr}_{\mathfrak{m}}(R)\) is Cohen-Macaulay then, for each non-zero \(R\)-module \(M\) of finite projective dimension, one has: \[ {\ell \ell}_R\, M \geq \text{reg}_P(\text{gr}_{\mathfrak{m}}(R)) + 1 \] where \(P = k[X_1, \dots ,X_e]\) is a polynomial ring in \(e = \text{edim}\, R := \text{dim}_k\, \mathfrak{m}/\mathfrak{m}^2\) indeterminates and “reg” denotes the Castelnuovo-Mumford regularity. The proof of this result uses invariants of Gorenstein local rings defined by M. Auslander and studied by S. Ding [Proc. Am. Math. Soc. 120, No. 4, 1029–1033 (1994; Zbl 0796.13021)].
A second estimate asserts that if \(R = Q/\mathfrak{p}Q\), where \((P,\mathfrak{p}) \rightarrow (Q,\mathfrak{q})\) is a flat local morphism then, for every non-zero \(R\)-module \(M\) of finite projective dimension, one has: \[ {\ell \ell}_R\, M \geq \text{edim}\, P - \text{edim}\, Q + \text{edim}\, R + 1\, . \] This estimate is derived from a more general result asserting that, for every finite free complex \(F\) of \(R\)-modules with \(\text{H}(F) \neq 0\), and for every presentation of the completion \(\widehat R\) of \(R\) as a quotient \(Q/I\) where \((Q,\mathfrak{q})\) is a regular local ring and \(I \subseteq \mathfrak{q}^2\), one has: \[ \sum_{n\in {\mathbb Z}}\, {\ell \ell}_R\, \text{H}_n(F) \geq \text{f-rank}_{\widehat R}\, (I/I^2) + 1\, . \] Here \(\text{f-rank}_{\widehat R}\, (I/I^2)\) denotes the maximal rank of a free direct summand of the conormal \(\widehat R\)-module \(I/I^2\). The proof of the last result is based on techniques borrowed from differential graded homological algebra.
The first step is to replace \(F\) by \(K\otimes_RF\), where \(K\) is the Koszul complex associated to a minimal system of generators of \(\mathfrak{m}\). \(K\otimes_RF\) is a DG module over the DG algebra \(K\). Then comes the “crucial new idea” of the paper which is to bound the Loewy length in terms of a new numerical invariant of objects in a derived category called level. More precisely, let \(\text{D}(A)\) be the derived category of DG modules over a DG algebra \(A\) and \(\Sigma\) the suspension functor of \(\text{D}(A)\) (the authors use chain complexes and \(\Sigma\) decreases the degrees by 1). For every \(X\) in \(\text{D}(A)\) let \(\text{thick}_A^0(X) := \{0\}\), let \(\text{thick}_A^1(X)\) be the full subcategory of \(\text{D}(A)\) consisting of retracts of finite direct sums of iterated suspensions of \(X\), and, for \(n \geq 2\), let \(\text{thick}_A^n(X)\) be the full subcategory of \(\text{D}(A)\) consisting of retracts of those \(U \in \text{D}(A)\) that appear in some distingushed triangle \(U^{\prime} \rightarrow U \rightarrow U^{\prime \prime} \rightarrow \Sigma U^{\prime}\), with \(U^{\prime} \in \text{thick}_A^{n-1}(X)\) and \(U^{\prime \prime} \in \text{thick}_A^1(X)\). Then, for \(U \in \text{D}(A)\): \[ \text{level}_A^X(U) := \text{inf}\, \{n\in {\mathbb N}\, | \, U\in \text{thick}_A^n(X)\}\, . \] If \(\phi : A \rightarrow B\) is a morphism of DG algebras then the exact functors \(B\otimes_A^{\text{L}}- : \text{D}(A) \rightarrow \text{D}(B)\) and \({\phi}_{\ast} : \text{D}(B) \rightarrow \text{D}(A)\) (restriction of scalars) decrease the level, and preserve it if \(\phi\) is a quasi-isomorphism. Moreover, the authors show that if \(M\) is a complex of \(R\)-modules with \(\text{H}(M)\) of finite length then \(\sum_{n\in {\mathbb Z}}\, {\ell \ell}_R\, \text{H}_n(M) \geq \text{level}_R^k(M)\).
Now, using an idea of M. André [Comment. Math. Helv. 57, 648–675 (1982; Zbl 0509.13007)] the authors prove that there exist quasi-isomorphisms of DG algebras linking \(K\) and \(\Lambda \otimes_kB\), where \(\Lambda\) is the exterior algebra \(\bigwedge_k(\Sigma k^c)\), \(c = \text{f-rank}_{\widehat R} (I/I^2)\), and \(B\) is a DG algebra with \(B_0 = k\) and \(\text{rank}_k\, B < \infty\). Then they prove a variant of the Bernstein-Gelfand-Gelfand equivalence asserting that the derived category \(\text{D}(\Lambda )\) of the DG algebra \(\Lambda\) (with the differential \(= 0\)) is equivalent to \(\text{D}(S)\), where \(S\) is a polynomial ring in \(c\) indeterminates of degree \(-2\). From the BGG equivalence and from a sharper version of the New Intersection Theorem for commutative algebras over fields, the authors deduce that for every perfect DG \(\Lambda\)-module \(N\) with \(\text{H}(N) \neq 0\) one has: \[ \text{level}_{\Lambda}^k(N) = c+1 \leq \text{card}\, \{n\in {\mathbb Z}\, | \, \text{H}_n(N) \neq 0\}\, . \] Putting all of these steps toghether one gets a proof of the main result of the paper stated above.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
16E45 Differential graded algebras and applications (associative algebraic aspects)
13D09 Derived categories and commutative rings
13D22 Homological conjectures (intersection theorems) in commutative ring theory
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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