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The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t-1; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element ∈ T, T/ g T is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T( d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T( d) is quite rigid. Our results generalise those of the first author in Rogalski (Advances Math. 226, 1433-1473, ). In the companion paper Rogalski et al. (), we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra. [ABSTRACT FROM AUTHOR]

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property. [ABSTRACT FROM AUTHOR]

We study the existence of free subalgebras in division algebras, and prove the following general result: if $$A$$ is a noetherian domain which is countably generated over an uncountable algebraically closed field $$k$$ of characteristic $$0$$ , then either the quotient division algebra of $$A$$ contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if $$k$$ is an uncountable algebraically closed field and $$A$$ is a finitely generated $$k$$ -algebra that is a domain of GK-dimension strictly less than $$3$$ , then either $$A$$ satisfies a polynomial identity, or the quotient division algebra of $$A$$ contains a free $$k$$ -algebra on two generators. [ABSTRACT FROM AUTHOR]

We construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ2, and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ1×ℙ1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques. [ABSTRACT FROM AUTHOR]

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 the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2. [ABSTRACT FROM AUTHOR]