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33
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.
Homology of perfect complexes
"... Abstract. It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed ..."
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Abstract. It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of
RIGID COMPLEXES VIA DG ALGEBRAS
, 2008
"... Let A be a commutative ring, B a commutative Aalgebra and M acomplexofBmodules. We begin by constructing the square SqB/A M, which is also a complex of Bmodules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an is ..."
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Let A be a commutative ring, B a commutative Aalgebra and M acomplexofBmodules. We begin by constructing the square SqB/A M, which is also a complex of Bmodules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism ρ: M ≃ − → SqB/A M, then the pair (M, ρ) is called a rigid complex over B relative to A (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes. We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks f ♭ (M, ρ) (resp. f ♯ (M,ρ)) along a finite (resp. essentially smooth) ring homomorphism f ∗ : B → C. In the subsequent paper, Rigid Dualizing Complexes over Commutative Rings, weconsiderrigid dualizing complexes over commutative rings, building
Teleman C., Openclosed field theories, string topology, and Hochschild homology
 in Alpine Perspectives on Algebraic Topology, Editors
"... and Hochschild homology ..."
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Commutative Algebra in the Cohomology of Groups
"... Abstract. Commutative algebra is used extensively in the cohomology of groups. In this series of lectures, I concentrate on finite groups, but I also discuss the cohomology of finite group schemes, compact Lie groups, pcompact groups, infinite discrete groups and profinite groups. I describe the ro ..."
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Abstract. Commutative algebra is used extensively in the cohomology of groups. In this series of lectures, I concentrate on finite groups, but I also discuss the cohomology of finite group schemes, compact Lie groups, pcompact groups, infinite discrete groups and profinite groups. I describe the role of various concepts from commutative algebra, including finite generation, Krull dimension, depth, associated primes, the Cohen–Macaulay and Gorenstein conditions, local cohomology, Grothendieck’s local duality, and Castelnuovo–Mumford regularity.
Local Cohomology In Equivariant Topology
"... The article describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather di ffren ..."
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The article describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather di ffrent families of examples. Both topology and commutative algebra benefit from the connection, and many interesting questions remain open.