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Frobenius bimodules between noncommutative spaces
"... Abstract. In this paper we study Frobenius bimodules between noncommutative spaces (quasischemes), developing some of their basic properties. If X and Y are spaces, we study those Frobenius X, Ybimodules XMY satisfying properties that are natural in the context of noncommutative algebraic geometry ..."
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Abstract. In this paper we study Frobenius bimodules between noncommutative spaces (quasischemes), developing some of their basic properties. If X and Y are spaces, we study those Frobenius X, Ybimodules XMY satisfying properties that are natural in the context of noncommutative algebraic geometry, focusing in particular on cartain “local ” conditions on M. As applications, we prove decomposition and gluing theorems for those Frobenius bimodules which have good local properties. Additionally, when X and Y are schemes we relate Frobenius X, Ybimodules to the sheaf X, Ybimodules introduced by Van den Bergh in [33].
Maps between noncommutative spaces
, 2000
"... Abstract. Let J be a graded ideal in a not necessarily commutative graded kalgebra A = A0 ⊕A1 ⊕ · · · in which dimk Ai < ∞ for all i. We show that the map A → A/J induces a closed immersion i: Proj nc A/J → Proj nc A between the noncommutative projective spaces with homogeneous coordinate rings ..."
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Abstract. Let J be a graded ideal in a not necessarily commutative graded kalgebra A = A0 ⊕A1 ⊕ · · · in which dimk Ai < ∞ for all i. We show that the map A → A/J induces a closed immersion i: Proj nc A/J → Proj nc A between the noncommutative projective spaces with homogeneous coordinate rings A and A/J. We also examine two other kinds of maps between noncommutative spaces. First, a homomorphism φ: A → B between not necessarily commutative Ngraded rings, induces an affine map Proj nc B ⊃ U → Proj nc A from a nonempty open subspace U ⊂ Proj nc B. Second, if A is a right noetherian connected graded algebra (not necessarily generated in degree one), and A (n) is a Veronese subalgebra of A, there is a map Proj nc A → Proj nc A (n) ; we identify open subspaces on which this map is an isomorphism. Applying these general results when A is (a quotient of) a weighted polynomial ring produces a noncommutative resolution of (a closed subscheme of) a weighted projective space.
NOETHERIANITY OF THE SPACE OF IRREDUCIBLE REPRESENTATIONS
, 2001
"... Abstract. Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left Rmodules (or, more generally, simple objects in a complete abelian category). Under this topology the points are cl ..."
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Abstract. Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left Rmodules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and when R is left noetherian the corresponding topological space is noetherian. If R is commutative (or PI, or FBN) the topology is equivalent to the Zariski topology, and when R is the first Weyl algebra (in characteristic zero) we obtain a onedimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg’s spectra are briefly noted. 1.
A∞MODULES AND CALOGEROMOSER SPACES
, 2007
"... The Hilbert schemes Hilbn(C 2) of points on C 2 have a rich geometric structure with many interesting links to representation theory, combinatorics and integrable systems. One reason for this is perhaps that the points of Hilbn(C 2) admit a few different algebraic incarnations ..."
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The Hilbert schemes Hilbn(C 2) of points on C 2 have a rich geometric structure with many interesting links to representation theory, combinatorics and integrable systems. One reason for this is perhaps that the points of Hilbn(C 2) admit a few different algebraic incarnations
COMPLETE COVERINGS AND GALOIS CORINGS
, 2005
"... Abstract. It is shown that any finite complete covering of a noncommutative algebra in the sense of Calow and Matthes (J. Geom. Phys. 32 (2000), 114–165) gives rise to a Galois coring. 1. ..."
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Abstract. It is shown that any finite complete covering of a noncommutative algebra in the sense of Calow and Matthes (J. Geom. Phys. 32 (2000), 114–165) gives rise to a Galois coring. 1.
A REMARK ON CLOSED NONCOMMUTATIVE SUBSPACES
, 2006
"... Given an abelian category with arbitrary products, arbitrary coproducts, and a generator, we show that the closed subspaces (in the sense of A. L. Rosenberg) are parameterized by a suitably defined poset of ideals in the generator. In particular, the collection of closed subspaces is itself a small ..."
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Given an abelian category with arbitrary products, arbitrary coproducts, and a generator, we show that the closed subspaces (in the sense of A. L. Rosenberg) are parameterized by a suitably defined poset of ideals in the generator. In particular, the collection of closed subspaces is itself a small poset.
IDEAL CLASSES OF THREE DIMENSIONAL ARTINSCHELTER REGULAR ALGEBRAS
, 2005
"... Abstract. We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional ArtinSchelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of tor ..."
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Abstract. We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional ArtinSchelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a noncommutative P 2 is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford [30]. For the Weyl algebra
THE BGQ SPECTRAL SEQUENCE FOR NONCOMMUTATIVE SPACES
, 2002
"... Abstract. We prove an analogue of the BrownGerstenQuillen (BGQ) spectral sequence for noncommutative spaces. As applications, we consider this spectral sequence over affine and projective spaces associated to right fully bounded noetherian (FBN) rings. 1. ..."
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Abstract. We prove an analogue of the BrownGerstenQuillen (BGQ) spectral sequence for noncommutative spaces. As applications, we consider this spectral sequence over affine and projective spaces associated to right fully bounded noetherian (FBN) rings. 1.