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45
Generators and representability of functors in commutative and noncommutative geometry
 MOSC MATH. J
, 2002
"... We give a sufficient condition for an Extfinite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived ca ..."
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Cited by 82 (2 self)
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We give a sufficient condition for an Extfinite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
Model category structures on chain complexes of sheaves
 Trans. Amer. Math. Soc
"... of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently ..."
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Cited by 25 (0 self)
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of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct
Classifying subcategories of modules
 Trans. Amer. Math. Soc
"... A basic problem in mathematics is to classify all objects one is studying up to isomorphism. A lesson this author learned from stable homotopy theory [HS] is that while this is almost always impossible, it is sometimes possible, and very useful, to classify collections of objects, or certain full su ..."
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Cited by 10 (0 self)
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A basic problem in mathematics is to classify all objects one is studying up to isomorphism. A lesson this author learned from stable homotopy theory [HS] is that while this is almost always impossible, it is sometimes possible, and very useful, to classify collections of objects, or certain full subcategories of the category
Pointed and copointed Hopf algebras as cocycle deformations,” arxiv:0709.0120
"... Abstract. We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the applica ..."
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Cited by 8 (2 self)
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Abstract. We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about MoritaTakeuchi equivalence and of Schauenburg about Hopf Galois extensions. The “infinitesimal ” part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered. Finite dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero, particularly when the group of points is abelian, have been studied quite extensively with various methods in [AS, BDG, Gr1, Mu]. The most far reaching results as yet in this area have been obtained in [AS], where a large
L²Invariants from the Algebraic Point of View
, 2008
"... We give a survey on L²invariants such as L²Betti numbers and L²torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and Ktheory. ..."
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Cited by 5 (3 self)
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We give a survey on L²invariants such as L²Betti numbers and L²torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and Ktheory.
L 2 invariants of finite aspherical CWcomplexes
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 152
, 2001
"... Abstract. Let X be a finite aspherical CWcomplex whose fundamental group π1(X) possesses a subnormal series π1(X) ⊲ Gm ⊲... ⊲ G0 with a nontrivial elementary amenable group G0. We investigate the L 2invariants of the universal covering of such a CWcomplex X. We show that the NovikovShubin inva ..."
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Cited by 4 (0 self)
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Abstract. Let X be a finite aspherical CWcomplex whose fundamental group π1(X) possesses a subnormal series π1(X) ⊲ Gm ⊲... ⊲ G0 with a nontrivial elementary amenable group G0. We investigate the L 2invariants of the universal covering of such a CWcomplex X. We show that the NovikovShubin invariants αn ( ˜ X) are positive. We further prove that the L 2torsion ρ (2) ( ˜ X) vanishes if π1(X) has semiintegral determinant. 1.
A homological estimate for the Thurston norm
"... Abstract We establish a new homological lower bound for the Thurston norm on 1cohomology of 3manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1cohomology of 2dimensional CWcomplexes. AMS Classification 57M27, ..."
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Cited by 3 (0 self)
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Abstract We establish a new homological lower bound for the Thurston norm on 1cohomology of 3manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1cohomology of 2dimensional CWcomplexes. AMS Classification 57M27, 57M20, 57M05
Noncommutative deformations of sheaves and presheaves of modules, ArXiv: math.AG/0405234
, 2005
"... We work over an algebraically closed field k, and consider the noncommutative deformation functor DefF of a finite family F of presheaves of modules defined over a presheaf of kalgebras A on a small category c. We develop an obstruction theory for DefF, with certain global Hochschild cohomology gro ..."
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Cited by 2 (1 self)
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We work over an algebraically closed field k, and consider the noncommutative deformation functor DefF of a finite family F of presheaves of modules defined over a presheaf of kalgebras A on a small category c. We develop an obstruction theory for DefF, with certain global Hochschild cohomology groups as the natural cohomology. In particular, we show how to calculate the prorepresenting hull H(DefF) in concrete terms. When (X, A) is a ringed space over k, we also consider the noncommutative deformation functor DefF of a finite family F of quasicoherent sheaves of left Amodules on X. We give conditions for this deformation functor to be isomorphic to the corresponding deformation functor of presheaves on U, where U is some open cover of X considered as a small category, and show that these conditions are satisfied in many interesting examples. In these cases, it follows that we can calculate the prorepresenting hull H(DefF) in concrete terms using presheaf techniques. Finally, we show that nfold extensions in the category of quasicoherent sheaves can be calculated using global Hochschild cohomology in many cases.