We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of Thomason-Trobaugh’s [51] and Yao’s [58] localization theorems in K-theory (cf. also [55]).

These notes are based on lectures given at the Workshop on Structured ring spectra and

Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.

Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)-module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kG-module M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.

These notes are based on a series of five lectures given during the

Abstract. A spectral sequence is constructed whose nonzero E1-terms are the Witt groups of the residue fields of a regular scheme X, arranged in Gersten-Witt complexes, and whose limit is the four global Witt groups of X. There are several immediate consequences concerning purity for Witt groups of low-dimensional schemes. The Witt groups of punctured spectra of regular local rings are also computed. Let X be a regular integral separated noetherian scheme in which 2 is everywhere invertible. (We will maintain these hypotheses throughout the introduction.) It is now known to the experts that the Witt groups of the residue fields of X form a nonexact cochain complex

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is

Abstract. A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHomA(C, C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.

Abstract not found

. A method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations which is either homogeneous or admissable (with some additional finiteness restrictions in the latter case). In particular, it applies to any finite dimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras and Koszul algebras, with the aim of unifying existing treatments of these in the literature. 1991 Mathematics Subject Classification. Primary: 16E99, 18G10. Secondary: 16D20, 16E40, 16G20, 16W50. 1. Introduction A projective resolution of an algebra , considered as a bimodule over itself, is fundamental in governing the homological properties of the algebra. Such a resolution may be used to compute Hochschild homology and cohomology, to ...