We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical AuslanderBuchsbaum formula for the depth of finitely generated modules of finite projective dimension. We extend also ...

We prove a version of Koszul duality and the induced derived equivalence for Adams connected A∞-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernšte˘ ın-Gel'fand-Gel'fand correspondence for Adams connected A∞-algebras. We give various applications. For example, a connected graded algebra A is Artin-Schel...

We investigate the (unbounded) derived category of a diierential Z-graded category (=DG category). As a rst application, we deduce a 'triangulatedanaloguè (4.3) of a theorem of Freyd's 5, Ex. 5.3 H] and Gabriel's 6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem`(8.2) generalizing results of 19] and 2...

We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad O, generalizing the construction already known for the associative operad. This is done by defining a colored operad b O, which describes modules over O with invariant inner products. We show that b O satisfies Koszulness and identify algebras over a resolution of b O in terms of derivations and module map...

We define a poset of partitions associated to an operad. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is given by the Koszul dual operad. On the other hand, we get new methods for proving that an operad is Koszul.

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (gr...

It is shown that Morita equivalence preserves quasi-Koszulity, and a finite-dimensional K-algebra A is quasi-Koszul if and only if the skew group algebra A * G is, where G is a finite group satisfying charK ∤ |G|. It follows from these results that a finite-dimensional K-algebra A is quasi-Koszul if and only if the smash product A#G * is, where G is a finite group satisfying charK ∤ |G|. Furthe...

Let M be a module over Noetherian local ring A. We study M-independent sequences of elements the maximal ideal mA in sense Lech and Hanes. The main tool is new characterization M-independence sequence terms associated Koszul complex. As applications, we give result linkage theory, freeness criterion for existence strongly length equal to embedding dimension edim(A) A, another inspired from patc...

LET %’ be a highest weight category [CPSl] with finitely many simple objects and such that every object has finite length. Then Ce = mod S for a finite dimensional quasi-hereditary algebra S. In a series of papers, [CPS4,5,6,7], we have studied various Kazhdan-Lusztig theories for %. This concept arises naturally in, for example, the modular representation theory of semisimple algebraic groups....