Coherent algebras and noncommutative projective lines

International Journal of Mathematics and Mathematical Sciences, 2005

We show that every finitely generated left R-module in the Auslander class over an nperfect ring R having a dualizing module and admitting a Matlis dualizing module has a Gorenstein projective cover.

2001

To any finite group Γ ⊂ SL2(C) and each element τ in the center of the group algebra of Γ, we associate a category, Coh(P 2 Γ,τ , P 1). It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh(P 2 Γ,τ , P 1) should be thought of as the category of coherent sheaves on a 'noncommutative projective space', P 2 Γ,τ , equipped with a framing at P 1 , the line at infinity. We establish an isomorphism between the moduli space of torsion free objects of Coh(P 2 Γ,τ , P 1) and the Nakajima quiver variety arising from Γ via the McKay correspodence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of 'rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for Γ = {1}, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A1, by points of the Calogero-Moser space, due to Cannings-Holland and Berest-Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on P 2 Γ,τ. It is totally different from the one used by Berest-Wilson, involving τ-functions. Γ,τ 4. Interpretation of Nakajima varieties 5. Proof of generalized Crawley-Boevey and Holland Conjecture 6. Appendix A: Graded preprojective algebra 7. Appendix B: Algebraic generalities 8. Appendix C: Minuscule classes

International Journal of Algebra, 2014

In this paper we continue the study of non connected graded Gorenstein algebras initiated in [13], the main result is the proof of a version of the Local Cohomology formula.

EMS Series of Congress Reports, 2008

Given a commutative ring R (respectively a positively graded commutative ring A = ⊕ j 0 A j which is finitely generated as an A 0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y ⊆ Spec R (respectively Y ⊆ Proj A) of the form Y = S i∈Ω Y i , with Spec R \ Y i (respectively Proj A \ Y i) quasi-compact and open for all i ∈ Ω, is established. Using these bijections, there are constructed isomorphisms of ringed spaces (Spec R, O R) ∼ −→ (Spec(Mod R), O Mod R) and (Proj A, O Proj A) ∼ −→ (Spec(QGr A), O QGr A), where (Spec(Mod R), O Mod R) and (Spec(QGr A), O QGr A) are ringed spaces associated to the lattices Ltor(Mod R) and Ltor(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and the torsion classes of finite type in Mod R is established. Contents 1. Introduction 1 2. Torsion classes of finite type 3 3. The fg-topology 4 4. Torsion classes and thick subcategories 7 5. Graded rings and modules 9 6. Torsion modules and the category QGr A 12 7. The prime spectrum of an ideal lattice 14 8. Reconstructing affine and projective schemes 15 9. Appendix 16 References 17

2009

We consider Γ-graded commutative algebras, where Γ is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on the subject. We then give a recent classification result and formulate an open problem.

Selecta Mathematica, 2016

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the Gerstenhaber-Schack (GS) complex to explicitly parameterize the first order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(A), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have Qch(O) ∼ = Qch(X)). Under a natural identification of Gerstenhaber-Schack cohomology of O and Hochschild cohomology of X, our construction is shown to be equivalent to Toda's construction from [48] in the smooth case.

Advances in Mathematics, 2022

In this paper, we study equivalences between the categories of quasi-coherent sheaves on non-commutative noetherian schemes. In particular, give a new proof of Caldararu's conjecture about Morita equivalences of Azumaya algebras on noetherian schemes. Moreover, we derive necessary and sufficient condition for two reduced noncommutative curves to be Morita equivalent.

2003

We study associative graded algebras which have a ``complete flag'' of cyclic modules with linear free resolutions, i.e., algebras over which there is a cyclic Koszul module with every admissible number of relations (from zero up to the number of generators of the algebra). Commutative algebras with the same property has been studied in several papers by A. Conca and others. Here we present a non-commutative version. We introduce the concept of Koszul filtration in non-commutative algebras and study its connections with Koszul algebras and algebras with quadratic Groebner bases. Also, here are considered several examples, such as ``Groebner flags'', generic algebras, and algebras with one relation. A generalization of the concept Koszul filtration (generalized Koszul, or rate, filtration) leads to algebras with finite Backelin's rate and to coherent algebras. One of our main results is that every algebra with Koszul filtration (or with finite rate filtration...

Mathematische Nachrichten, 2008

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov-Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category Qco(X) of quasi-coherent sheaves on $X$ is such a category and so has these features.

Journal of Pure and Applied Algebra, 1991

Let d,,. , d, be positive integers. We show that there are only finitely many Hilbert functions for rings k[X,.. .. . X,,]l(f,,. , f,). where f; is a form of degree ri,. Furthermore, there is a Zariski-open dense subset of the space of coefficients of the forms. where the Hilbert functions are equal. Both these statements are false in the noncommutativc case.