We study a class of noncommutative surfaces and their higher dimensional analogues which provide answers to several open questions in noncommutative projective geometry. Specifically, we give the first known graded algebras which are noetherian but not strongly noetherian, answering a question of Artin, Small, and Zhang. In addition, these examples are maximal orders and satisfy the $χ_1$ condition but not $χ_i$ for $i \geq 2$, answering a questions of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that these algebras have finite cohomological dimension.
We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn together'' to form Spec R. In particular, we construct a bimodule-determined functor Mod Z -> Mod R, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continous surjection Spec Z -> Spec R. Algebras satisfying the given axiomatic framework include PI algebras finitely generated over fields, noetherian PI algebras, envelo…
A version of Auslander theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite dimensional Lie algebras, and (c) noetherian graded down-up algebras.
Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras which are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy dimer algebras to being NCCRs? To address this question, we introduce a generalization of NCCRs to nonnoetherian tiled matrix rings. We show that if a noetherian dimer algebra is obtained from a nonnoetherian homotopy dimer algebra $A$ by contracting each arrow whose head has indegree 1, then $A$ is a noncommutative desingularization of its nonnoetherian center.…
In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra R) as well as noncommutative analogues of P^1 x P^1 and of the Hirzebruch surface F_2 (arising from Van den Bergh's quadrics R) satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maxi…
One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo's classic theorem …
It is shown that Golod-Shaferevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators, and that these algebras can be homomorphically mapped onto prime, Noetherian algebras with linear growth. It is also shown that Golod-Shafarevich algebras of a reduced number of relations cannot be nil.
For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an application, we prove a very general twisted homology stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.
Residue complexes were introduced by Grothendieck in algebraic geometry. These are canonical complexes of injective modules that enjoy remarkable functorial properties (traces). In this paper we study residue complexes over noncommutative rings. These objects are even more complicated than in the commutative case, since they are complexes of bimodules. We develop methods to prove uniqueness, existence and functoriality of residue complexes. For a noetherian affine PI algebra over a field (admitting a noetherian connected filtration) we prove existence of the residue complex and describe its st…
Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X > 1. Assume that c in X and s in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R=R(X,c,L,s) with surprising properties. In particular: (1) R is always noetherian but never strongly noetherian. (2) If R is generated in degree one then the images of the R-point modules in qgr(R) are naturally in (1-1) correspondence wit…
In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain $S$-Noetherian modules with the following consequence in commutative algebra: If a module $M$ over a commutative ring is $S$-Noetherian with respect to a multiplicative set $S$ that contains no zero-divisors for $…
We call a graded connected algebra $R$ effectively coherent, if for every linear equation over $R$ with homogeneous coefficients of degrees at most $d$, the degrees of generators of its module of solutions are bounded by some function $D(d)$. For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative algebras, including the most common class of rings in noncommutative projective geometry, that is, strongly Noetherian rings, which includes Noetherian PI algebras and Sklyanin algebras. We extensive…
We prove that an abelian category equipped with an ample sequence of objects is equivalent to the quotient of the category of coherent modules over the corresponding algebra by the subcategory of finite-dimensional modules. In the Noetherian case a similar result was proved by Artin and Zhang.
Let $X$ be a projective integral scheme with endomorphism $σ$, where $σ$ is finite, but not an automorphism. We examine noncommutative ampleness of bimodules defined by $σ$. In contrast to the automorphism case, one-sided ampleness is possible. We also find that rings and bimodule algebras associated with $σ$ are not noetherian.
In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\cal D}^b(\operatorname{mod} \nabla S*G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S, G}$ such that ${\mathcal D}^b(\…
Let A=k+A_1+A_2.... be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A)=k(Y)[t,t^{-1},sigma], where sigma is an automorphism of the integral projective surface Y. Then we prove that A can be written as a naive blowup algebra of a projective surface X birational to Y. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme…
One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem. Specifically, let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative P^2.) Let A be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S^(3n). Then we give a reasonably complete description of the structure of A. This is most sat…
We study rational actions of an algebraic torus G by automorphisms on an associative algebra R. The G-action on R induces a stratification of the prime spectrum of R which was introduced by Goodearl and Letzter. For a noetherian algebra R, Goodearl and Letzter showed that the strata of the spectrum of R are isomorphic to the spectra of certain commutative Laurent polynomial algebras. The purpose of this note is to give a new proof of this result which works for arbitrary algebras R.
The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then anoncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given non-commutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a give…
This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (non…
In the past 15 years a study of ``noncommutative projective geometry'' has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or ``noncommutative curves,'' have now been classified by geometric data and these rings must be noetherian. Rings of cubic growth, or ``noncommutative surfaces,'' are not yet classified, but a rich theory is currently forming. In this survey, we describe some…
Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. Old results are reviewed and a number of new results are proved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are given offering evidence in support of this proposal.
We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation $Q\to R$ by a Noetherian algebra $Q$ which is of finite global dimension. We show that hypersurface support defines a support theory for the big singularity category $Sing(R)$, and that the support of an object in $Sing(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz' support theory for (commutative) local complete intersection…
In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided …
Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-graded algebra $E(θ)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. It turns out that the bounded derived category $D^b(\text{gr}_{\mathbb Z_2}E(θ))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(θ)$ is a semisimple …
In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.
Let A be a noncommutative noetherian with dualizing complex R. We study the multiplicities of indecomposable injectives in a minimal injective resolution of R. In particular when A is a Gorenstein ring we get information on minimal injective resolutions of A.
One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). In a companion paper the authors described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surf…
All iterated skew polynomial extensions arising from quantized universal enveloping algebras of Kac-Moody algebras are special examples of a very large, axiomatically defined class of algebras, called CGL extensions. For the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra R, and classification of the automorphisms of R, one needs embeddings of R into quantum tori T which have the property that R contains the corresponding quantum affine space algebra A. We explicitly construct such an embedding A \subseteq R \subset T for each CGL extension R usin…
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 g of degree 1, T/gT 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 st…
The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface $Σ$. This is a noncommutative algebra ${\mathcal A}_Σ$ generated by "noncommutative geodesics" between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Plücker relations. It turns out that the algebra ${\mathcal A}_Σ$ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of $Σ$, which confirms its "cluster nature". As a surprising byproduct, we obtain a new topological invariant of $Σ$, which is a free or a 1-relator group easily c…
We construct a family of connected graded domains of GK-dimension 4 that are birational to P2, and show that the general member of this family is noetherian. This disproves a conjecture of the first author and Stafford. 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 our family. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on …
We study finite dimensional semisimple Hopf algebra actions on noetherian connected graded Artin-Schelter regular algebras, and introduce definitions of the Jacobian, the reflection arrangement and the discriminant in a noncommutative setting.
Recently the Euler forms on numerical Grothendieck groups of rank 4 whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number $m$, and suggests the existence of noncommutative surfaces which up to that point had not been considered for $m\geq 2$. These have been constructed for $m=2$ using noncommutative $\mathbb{P}^1$-bundles, and for all $m\geq 2$ by a different construction using maximal orders on $\mathrm{Bl}_x\mathbb{P}^2$. In this article we compare the constructions for $m=2$, i.e. we compare the c…
An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$
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…
To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fitting ideal of $M$ over $R$. This is of interest because it is always contained in the $R$-annihilator $\mathrm{Ann}_{R}(M)$ of $M$, but is often much easier to compute. This notion has recently been generalised to that of so-called `Fitting invariants' over certain noncommutative rings; the present author considered the case in which $R$ is an $\mathfrak{o}$-order $Λ$ in a finite dimensional separable algebra, where $\mathfrak{o}$ is an integr…
A major current goal of noncommutative geometry is the classification of noncommutative projective surfaces. The generic case is to understand algebras birational to the Sklyanin algebra. In this work we complete a considerable component of this problem. Let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field, and assume that S is not a finite module over its centre. In earlier work Rogalski, Sierra and Stafford classified the maximal orders inside the 3-Veronese of S. We complete and extend their work and classify all maximal orders inside S. As in Rogalski, Sierra …
To each finitely presented module M over a commutative ring R one can associate an R-ideal Fit_R(M) which is called the (zeroth) Fitting ideal of M over R and which is always contained in the R-annihilator of M. In an earlier article, the second author generalised this notion by replacing R with a (not necessarily commutative) o-order Lambda in a finite dimensional separable algebra, where o is an integrally closed complete commutative noetherian local domain. To obtain annihilators, one has to multiply the Fitting invariant of a (left) Lambda-module M by a certain ideal H(Lambda) of the centr…
Let X be a projective surface, let σbe an automorphism of X, and let L be a σ-ample invertible sheaf on X. We study the properties of a family of subrings, parameterized by geometric data, of the twisted homogeneous coordinate ring B(X, L, σ). In particular, we find necessary and sufficient conditions for these subrings to be noetherian. We also study their homological properties, their associated noncommutative projective schemes, and when they are maximal orders. In the process, we produce new examples of maximal orders; these are graded and have the property that no Veronese subring is gene…
Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $f\in A_2$ be a central regular element of $A$. The quotient algebra $A/(f)$ is usually called a (noncommutative) quadric hypersurface. In this paper, we use the Clifford deformation to study the quadric hypersurfaces obtained from the tensor products. We introduce a notion of simple graded isolated singularity and proved that, if $B/(g)$ is a simple graded isolated singularity of 0-type, then there is an equivalence of triangulated categories $\underline{\text{mcm}}\,A/(f)\cong\underline{\text{mcm}}\,(A\otimes B)/(f+g)$ of…
A noncommutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin-Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example, we formulate an infinitesimal version of the conjecture, and provide some evidence in the case of smooth projective surfaces.
We implement a noncommutative analogue of the well-known commutative cyclic covering trick and implement it to explicitly construct a vast collection of numerically Calabi-Yau orders, noncommutative analogues of surfaces of Kodaira dimension 0. We construct maximal orders, noncommutative analogues of normal schemes, on rational surfaces and ruled surfaces. We also use Ogg-Shafarevich theory to construct Azumaya algebras and, more generally, maximal orders on elliptically fibred surfaces.
The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type E_r,r=6,7,8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra C[x,y,z] to a noncommutative algebra with generators x,y,z and the following 3 relations (where [u,v]_t = uv- t.vu): [x,y]_t=F_1(z), [y,z]_t=F_2…
Noncommutative surfaces finite over their centres can be realised as orders over surfaces. The aim of this paper is to present a noncommutative generalisation of rational singularities, which we call numerical rationality, for such orders. We show that numerical rationality is independent of the choice of resolution. Our main result is that the log terminal orders arising from the noncommutative minimal model program, in particular, canonical orders are numerically rational. Both of these generalise well known facts about rational singularities in commutative algebraic geometry.
We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, n…
We study a new class of quiver algebras on surfaces, called 'geodesic ghor algebras'. These algebras generalize cancellative dimer algebras on a torus to higher genus surfaces, where the relations come from perfect matchings rather than a potential. Although cancellative dimer algebras on a torus are noncommutative crepant resolutions, the center of any dimer algebra on a higher genus surface is just the polynomial ring in one variable, and so the center and surface are unrelated. In contrast, we establish a rich interplay between the central geometry of geodesic ghor algebras and the topology…
