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On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
On Gorenstein projective, injective and flat dimensions  a functorial description with applications
, 2004
"... For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Moritalike equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial descr ..."
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Cited by 98 (22 self)
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For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Moritalike equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial description meets the expectations and delivers a series of new results, which allows us to establish a wellrounded theory for Gorenstein dimensions. For any pair of adjoint functors, C
Morita theory in abelian, derived and stable model categories
 LONDON MATH. SOC. LECTURE NOTE SER
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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These notes are based on lectures given at the Workshop on Structured ring spectra and
RECOLLEMENT FOR DIFFERENTIAL GRADED ALGEBRAS
, 2005
"... Abstract. A recollement of triangulated categories describes one such category as being “glued together ” from two others. This paper gives a precise criterion for the existence of a recollement ..."
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Abstract. A recollement of triangulated categories describes one such category as being “glued together ” from two others. This paper gives a precise criterion for the existence of a recollement
Topological Hochschild cohomology and generalized Morita equivalence, Algebraic & Geometric Topology 4
, 2004
"... Abstract. We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case ..."
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Abstract. We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer & Greenlees and of Schwede & Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH ∗ (A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground Salgebra R is an EilenbergMacLane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya Ralgebra is the endomorphism Ralgebra FR(M, M) of a finite cell Rmodule. We show that the spectrum of mod 2 topological Ktheory KU/2 is a nontrivial topological Azumaya algebra over the 2adic completion of the Ktheory spectrum ̂ KU2. This leads to the determination of THH(KU/2, KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A, A) for a noncommutative Salgebra A.
Quasicoherent sheaves on the moduli stack of formal groups
"... For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, onedimensional formal groups. Specifically, I would say that the ..."
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For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, onedimensional formal groups. Specifically, I would say that the height filtration of Mfg dictates a canonical and natural decomposition of a quasicoherent sheaf on Mfg, and this decomposition predicts and controls the chromatic decomposition of a finite spectrum. This sounds well, and is even true, but there is no single place in the literature where I could send anyone in order for him or her to get a clear, detailed, unified, and linear rendition of this story. This document is an attempt to set that right. Before going on to state in detail what I actually hope to accomplish here, I should quickly acknowledge that the opening sentences of this introduction and, indeed, this whole point of view is not original with me. I have already mentioned Mike Hopkins, and just about everything I’m going to say here is encapsulated in the table in section 2 of [15] and can be gleaned from the notes
Homological epimorphisms of differential graded algebras
"... Abstract. Let R and S be differential graded algebras. In this paper we give a characterisation of when a differential graded RSbimodule M induces a full embedding of derived categories ..."
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Abstract. Let R and S be differential graded algebras. In this paper we give a characterisation of when a differential graded RSbimodule M induces a full embedding of derived categories