A well-known conjecture says that every one-relator group is coherent. We state and partly prove an analogous statement for graded associative algebras. In particular, we show that every Gorenstein algebra $A$ of global dimension 2 is graded coherent.
This allows us to define a noncommutative analogue of the projective line $\PP^1$ as a noncommutative scheme based on the coherent noncommutative spectrum $\cohp A$ of such an algebra $A$, that is, the category of coherent $A$-modules modulo the torsion ones. This category is always abelian Ext-finite hereditary with Serre duality, like the category of coherent sheaves on $\PP^1$. In this way, we obtain a sequence $\PP^1_n $ ($n\ge 2$) of pairwise non-isomorphic noncommutative schemes which generalize the scheme $\PP^1 = \PP^1_2$.
Let $k$ be a field. We describe necessary and sufficient conditions for a $k$-linear abelian category to be a noncommutative projective line, i.e. a noncommutative $\mathbb{P}^{1}$-bundle over a pair of division rings over $k$. As an application, we prove that $\mathbb{P}^{1}_{n}$, Piontkovski's $n$th noncommutative projective line, is the noncommutative projectivization of an $n$-dimensional vector space.
We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of math.QA/0211262 allowing to interpret all the data in terms of autoequivalences of the deriv…
We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along lines of Rickard and Orlov.
Let $k$ be a field, let ${\sf C}$ be a $k$-linear abelian category, let $\underline{\mathcal{L}}:=\{\mathcal{L}_{i}\}_{i \in \mathbb{Z}}$ be a sequence of objects in ${\sf C}$, and let $B_{\underline{\mathcal{L}}}$ be the associated orbit algebra. We describe sufficient conditions on $\underline{\mathcal{L}}$ such that there is a canonical morphism from the noncommutative space ${\sf Proj }B_{\underline{\mathcal{L}}}$ to a noncommutative projective line in the sense of \cite{abstractp1}, generalizing the usual construction of a map from a scheme $X$ to $\mathbb{P}^{1}$ defined by an invertible…
In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics $\operatorname{Proj_{nc}} A$ embedded into Calabi-Yau quantum projective planes. In particular, we give complete classifications of homogeneous coordinate algebras $A$ of noncommutative conics up to isomorphism of graded algebras, and of noncommutive conics $\operatorname{Proj_{nc}} A$ up to isomorphism of noncommutative schemes.
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
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…
Following Artin and Zhang's formulation of noncommutative projective geometry, we classify up a family of skew polynomial quadratic algebras up to graded Morita equivalence and their corresponding noncommutative projective spaces up to birational equivalences. We also study their point varieties and provide examples of non-isomorphic noncommutative projective spaces.
We give an explicit algorithm to compute a projective resolution of a module over the noncommutative ring based on the noncommutative Groebner bases theory.
These notes are an expanded version of the author's lectures at the graduate workshop "Noncommutative Algebraic Geometry" at the Mathematical Sciences Research Institute in June 2012. The main topics discussed are Artin-Schelter regular algebras, point modules, and the noncommutative projective scheme associated to a graded algebra.
We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the alg…
We generalize a theorem of Kapranov by showing that the Hall algebra of the category of coherent sheaves on a weighted projective line (over a finite field) provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the correponding Kac-Moody algebra. In particular this yieds a geometric realization of the quantized enveloping algebra of 2-toroidal (or elliptic) algebras of types D_4, E_6, E_7 or E_8 in terms of weighted projective lines of genus one.
A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different …
We prove that if a positively-graded ring $R$ is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme ${\rm Tails}(R)$ is a Gorenstein category in the sense of \cite{EEG}. Moreover, under this condition, a (right) recollement relating Gorenstein-injective sheaves in ${\rm Tails}(R)$ and (graded) Gorenstein-injective $R$-modules is given.
We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $χ$-conditions of Artin and Zhang and the strong noetherian property have very different behavior on the left and right sides.
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…
I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneou…
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.
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 …
Constructive proofs of fact that a stably free left $S$-module $M$ with rank$(M)\geq$sr$(S)$ is free, where sr$(S)$ denotes the stable rank of an arbitrary ring $S$, were developed in some articles. Additionally, in such papers, are presented algorithmic proofs for calculating projective dimension, and to check whether a left $S$-module $M$ is stably free. Given a left $A$-module $M$, with $A$ a bijective skew $PBW$ extension, we will use these results and Gröbner bases theory, to establish algorithms that allow us to calculate effectively the projective dimension for this module, to check whe…
A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.
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…
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Th…
In an earlier paper (D. S. Keeler, D. Rogalski, and J. T. Stafford, ``Naive noncommutative blowing up,'' Duke Math. J., 126 (2005), 491-546), we defined and investigated the properties of the naive blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naively blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despit…
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.
We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2. The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E < P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic (1-n) provides a sy…
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 prove that the category of representations of the N-Kronecker quiver and that of coherent sheaves on the noncommutative projective scheme of $R=k< X_1,...,X_N >/(\sum^N_{i=1}X_i^2)$ are derived equivalent. This equivalence is easily proved by applying Orlov's Theorem, on the other hand,in our proof,the quadratic relation $\sum_{i=1}^NX_i^2$ naturally arises from Auslander-Reiten Theory.
In this paper, we construct a vast collection of maximal numerically Calabi-Yau orders utilising a noncommutative analogue of the well-known commutative cyclic covering trick. Such orders play an integral role in the Mori program for orders on projective surfaces and although we know a substantial amount about them, there are relatively few known examples.
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…
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 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 make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine noncommutative algebraic geometry. The main example for us is the noncommutative affine space. The algebra of functions on this space is free finitely generated algebra. We propose a general correspondence principle which allows us to give definitions of various differential-geometric structures in the noncommutative setting. In the second part we propose a defini…
The noncommutative (Cohn) localization S^{-1}R of a ring R is defined for any collection S of morphisms of f.g. projective left R-modules. We exhibit S^{-1}R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if S^{-1}R is "stably flat over R" (meaning that Tor^R_i(S^{-1}R,S^{-1}R)=0 for i>0) then every bounded f.g. projective S^{-1}R-module chain complex D with [D] \in im(K_0(R)-->K_0(S^{-1}R)) is chain equivalent to S^{-1}C for a bounded f.g. projective R-module chain complex C, and that there is a localization exact sequence in higher…
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…
