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178
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 203 (4 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.
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
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Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
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Noncommutative curves and noncommutative surfaces
 Bulletin of the American Mathematical Society
"... Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative grad ..."
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Cited by 91 (8 self)
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Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing
Proposed Research
, 1997
"... GKdimension 3 rings can be formed in this fashion, including homogeneous coordinate rings of quantum quadric surfaces, quotients of 4dimensional Sklyanin algebras, and quotients of the homogenized enveloping algebra of the Lie algebra sl 2 . We start with the usual construction of an algebraicall ..."
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Cited by 62 (0 self)
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GKdimension 3 rings can be formed in this fashion, including homogeneous coordinate rings of quantum quadric surfaces, quotients of 4dimensional Sklyanin algebras, and quotients of the homogenized enveloping algebra of the Lie algebra sl 2 . We start with the usual construction of an algebraically ruled surface S over a smooth projective curve X, namely S = P(E), where E is a rank 2 vector bundle over X. This construction describes S as a P 1 bundle over X. Specifically, P(E) = Proj S(E) where S(E) is the sheaf of symmetric algebras of E. Van den Bergh has proposed the idea of constructing a noncommutative ruled surface by taking E to be, instead of a v
Noncommutative instantons and twistor transform
 Commun. Math. Phys
"... Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of R4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one ..."
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Cited by 54 (4 self)
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Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of R4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit onetoone correspondences between three classes of objects: framed bundles on a noncommutative P2, certain complexes of sheaves on a noncommutative P3, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative P2 has a natural hyperkähler metric and is isomorphic as a hyperkähler manifold to the moduli space of framed torsion free sheaves on the commutative P2. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R4 than the one considered by Nekrasov and Schwarz (a q – deformed R4).
Twisted graded algebras and equivalences of graded categories
 Proc. London Math. Soc
, 1996
"... Let A = 0n S,o^n be a connected graded /ralgebra and let GrA denote the category of graded right /4modules with morphisms being graded homomorphisms of degree 0. If {r,,  n e 1} is a set of graded A:linear bijections of degree 0 from A to itself satisfying for all I, m, n e Z and all y e Am, z ..."
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Cited by 52 (10 self)
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Let A = 0n S,o^n be a connected graded /ralgebra and let GrA denote the category of graded right /4modules with morphisms being graded homomorphisms of degree 0. If {r,,  n e 1} is a set of graded A:linear bijections of degree 0 from A to itself satisfying for all I, m, n e Z and all y e Am, z e Ah we define a new graded associative multiplication * on the underlying graded Avector space 0 n s> o ^ n b> ' y*z=yrm(z) for all y e Am, z s A,. The graded algebra with the new multiplication * is called a twisted algebra of A. THEOREM. Let A and B be two connected graded algebras generated in degree 1. Then the categories GxA and GxB are equivalent if and only if A is isomorphic to a twisted algebra of B. If algebras are noetherian, then GelfandKirillov dimension, global dimension, injective dimension, Krull dimension, and uniform dimension are preserved under twisting. Moreover, we prove the following: THEOREM. The following properties are preserved under twisting for connected graded noetherian algebras: (a) ArtinSchelter Gorenstein (or ArtinSchelter regular); (b) Auslander Gorenstein (or Auslander regular) and CohenMacaulay. Some of these results are also generalized to certain semigroupgraded algebras. 1.
Derived Categories of Quadric Fibrations and Intersections of Quadrics
, 2005
"... We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algeb ..."
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Cited by 51 (12 self)
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We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov’s description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety (P(S 2 W ∗), B0), where B0 is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space P(W) in the double Veronese embedding P(W) → P(S 2 W). Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.
Rings with Auslander Dualizing Complexes
, 1998
"... A ring with an Auslander dualizing complex is a generalization of an AuslanderGorenstein ring. We show that many results which hold for AuslanderGorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show th ..."
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Cited by 42 (27 self)
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A ring with an Auslander dualizing complex is a generalization of an AuslanderGorenstein ring. We show that many results which hold for AuslanderGorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to nongraded algebras. We also prove some results of a categorical nature, most notably the functoriality