We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rin... more We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rings B that are graded equivalent to A: that is, the categories gr-A and gr-B are equivalent. This includes some surprising examples: in particular, we show that A is graded equivalent to an idealizer in a localization of A. We obtain this classification as an application of a general Morita-type characterization of equivalences of graded module categories. Given a Z-graded ring R, an autoequivalence F of gr-R, and a finitely generated graded projective right R-module P , we show how to construct a twisted endomorphism ring End F R (P) and prove: Theorem. The Z-graded rings R and S are graded equivalent if and only if there are an autoequivalence F of gr-R and a finitely generated graded projective right R-module P such that the modules {F n P } generate gr-R and S ∼ = End F R (P). Contents 1. Introduction 1 2. Graded equivalences and twisted endomorphism rings 4 3. The Picard group of a graded module category 11 4. Graded modules over the Weyl algebra 15 5. The Picard group of gr-A 20 6. Classifying rings graded equivalent to the Weyl algebra 26 7. The graded K-theory of the Weyl algebra 35 References 37
Given a projective scheme X over a field k, an automorphism σ : X → X, and a σ-ample invertible s... more Given a projective scheme X over a field k, an automorphism σ : X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.
Given Z-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivale... more Given Z-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivalent. Using Z-algebras, we relate the Morita-type results ofÁhn-Márki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem. If A and B are Z-graded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the Z-algebras A = L i,j∈Z A j−i and B = L i,j∈Z B j−i are isomorphic. (2) If A and B are connected graded with A 1 = 0, then gr-A ≃ gr-B if and only if A and B are isomorphic. This simplifies and extends Zhang's results.
We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rin... more We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rings B that are graded equivalent to A: that is, the categories gr-A and gr-B are equivalent. This includes some surprising examples: in particular, we show that A is graded equivalent to an idealizer in a localization of A. We obtain this classification as an application of a general Morita-type characterization of equivalences of graded module categories. Given a Z-graded ring R, an autoequivalence F of gr-R, and a finitely generated graded projective right R-module P , we show how to construct a twisted endomorphism ring End F R (P) and prove: Theorem. The Z-graded rings R and S are graded equivalent if and only if there are an autoequivalence F of gr-R and a finitely generated graded projective right R-module P such that the modules {F n P } generate gr-R and S ∼ = End F R (P). Contents 1. Introduction 1 2. Graded equivalences and twisted endomorphism rings 4 3. The Picard group of a graded module category 11 4. Graded modules over the Weyl algebra 15 5. The Picard group of gr-A 20 6. Classifying rings graded equivalent to the Weyl algebra 26 7. The graded K-theory of the Weyl algebra 35 References 37
Given a projective scheme X over a field k, an automorphism σ : X → X, and a σ-ample invertible s... more Given a projective scheme X over a field k, an automorphism σ : X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.
Given Z-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivale... more Given Z-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivalent. Using Z-algebras, we relate the Morita-type results ofÁhn-Márki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem. If A and B are Z-graded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the Z-algebras A = L i,j∈Z A j−i and B = L i,j∈Z B j−i are isomorphic. (2) If A and B are connected graded with A 1 = 0, then gr-A ≃ gr-B if and only if A and B are isomorphic. This simplifies and extends Zhang's results.
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Papers by Susan Sierra