Noncommutative Algebraic Geometry Research Articles

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew PBW extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.

On weak annihilators and nilpotent associated primes of skew PBW extensions

We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier (Weak annihilator over extension rings, 2012) in the setting of skew PBW extensions. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research.

Representation Theory of Geigle-Lenzing Complete Intersections

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings R R graded by abelian groups L \mathbb {L} of rank 1 1 , which we call Geigle-Lenzing complete intersections. We study the stable category C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R of Cohen-Macaulay representations, which coincides with the singularity category D s g L ( R ) \mathsf {D}^{\mathbb {L}}_{\mathrm {sg}}(R) . We show that C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R is triangle equivalent to D b ( m o d A C M ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{\mathrm {CM}}) for a finite dimensional algebra A C M A^{\mathrm {CM}} , which we call the CM-canonical algebra. As an application, we classify the ( R , L ) (R,\mathbb {L}) that are Cohen-Macaulay finite. We also give sufficient conditions for ( R , L ) (R,\mathbb {L}) to be d d -Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category c o h X = m o d L R / m o d 0 L R \mathsf {coh}\mathbb {X}=\mathsf {mod}^{\mathbb {L}}R/\mathsf {mod}^{\mathbb {L}}_0R of coherent sheaves on the Geigle-Lenzing projective space X \mathbb {X} . Geometrically this is the quotient stack X = [ X / G ] \mathbb {X}=[X/G] for X = S p e c R ∖ { R + } X={\mathrm {Spec}}\,R\setminus \{R_+\} and G = S p e c k [ L ] G={\mathrm {Spec}}\,k[\mathbb {L}] . We show that D b ( c o h X ) \mathsf {D}^{\mathrm {b}}(\mathsf {coh}\mathbb {X}) is triangle equivalent to D b ( m o d A c a ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{ca}) for a finite dimensional algebra A c a A^{\mathrm {ca}} , which we call a d d -canonical algebra. We study when X \mathbb {X} is d d -vector bundle finite, and when X \mathbb {X} is derived equivale’nt to a d d -representation infinite algebra in the sense of higher Auslander-Reiten theory. Our d d -canonical algebras provide a rich source of d d -Fano and d d -anti-Fano algebras in non-commutative algebraic geometry. We also observe Orlov-type semiorthogonal decompositions of D s g L ( R ) \mathsf {D}_{\mathrm {sg}}^{\mathbb {L}}(R) and D b ( c o h X ) \mathsf {D}^{b}(\mathsf {coh}\mathbb {X}) .

Interactions between Algebraic Geometry and Noncommutative Algebra

This workshop was on the interactions between noncommutative algebra, representation theory and algebraic geometry. The major objective was to bring together researchers from those areas with the focus on topics and problems where geometric methods are prevalent.

The structure of matrix polynomial algebras

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

Noncommutative conics in Calabi-Yau quantum projective planes

In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics ProjncA 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 ProjncA up to isomorphism of noncommutative schemes.

Gluing approximable triangulated categories

AbstractGiven a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful.Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category isapproximableif this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that$\mathsf {D}_{\mathsf {qc}}( X )$is approximable whenXis a quasi compact, separated scheme led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier.In this article, we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors.

The Connes embedding problem: A guided tour

The Connes embedding problem (CEP) is a problem in the theory of tracial von Neumann algebras and asks whether or not every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II 1 _1 factor. The CEP has had interactions with a wide variety of areas of mathematics, including C ∗ \mathrm {C}^* -algebra theory, geometric group theory, free probability, and noncommutative real algebraic geometry, to name a few. After remaining open for over 40 years, a negative solution was recently obtained as a corollary of a landmark result in quantum complexity theory known as MIP ∗ = RE \operatorname {MIP}^*=\operatorname {RE} . In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from MIP ∗ = RE \operatorname {MIP}^*=\operatorname {RE} . In fact, we outline two such proofs, one following the “traditional” route that goes via Kirchberg’s QWEP problem in C ∗ \mathrm {C}^* -algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic.

Noncommutative Knörrer’s periodicity theorem and noncommutative quadric hypersurfaces

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Kn\"orrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory since it reduces the number of variables in computing the stable category $\underline{\operatorname{CM}}(A)$ of maximal Cohen-Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Kn\"orrer's periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ of graded maximal Cohen-Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under high rank property defined in this paper, we also show that computing ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a noncommutative smooth quadric hypersurface $A$ in up to six variables can be reduced to one or two variables cases. In addition, we give a complete classification of ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a smooth quadric hypersurface $A$ in a skew $\mathbb P^{n-1}$, where $n \leq 6$, without high rank property using graphical methods.

Quantum projective planes finite over their centers

AbstractFor a three-dimensional quantum polynomial algebra $A=\mathcal {A}(E,\sigma )$ , Artin, Tate, and Van den Bergh showed that A is finite over its center if and only if $|\sigma |<\infty $ . Moreover, Artin showed that if A is finite over its center and $E\neq \mathbb P^{2}$ , then A has a fat point module, which plays an important role in noncommutative algebraic geometry; however, the converse is not true in general. In this paper, we will show that if $E\neq \mathbb P^{2}$ , then A has a fat point module if and only if the quantum projective plane ${\sf Proj}_{\text {nc}} A$ is finite over its center in the sense of this paper if and only if $|\nu ^{*}\sigma ^{3}|<\infty $ where $\nu $ is the Nakayama automorphism of A. In particular, we will show that if the second Hessian of E is zero, then A has no fat point module.

Some Open Problems in the Context of Skew PBW Extensions and Semi-graded Rings

In this paper, we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre’s conjecture will be discussed for a particular class of skew PBW extensions. The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.

Noncommutative matrix factorizations with an application to skew exterior algebras

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.

AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

AbstractIn noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

Some special types of determinants in graded skew PBW extensions

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

QUANTUM TORIC DEGENERATION OF QUANTUM FLAG AND SCHUBERT VARIETIES

We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen–Macaulay, can be tested by passing to associated graded rings. In the spirit of noncommutative algebraic geometry, this can be seen as an analogue of the classical result that, in a flat family of varieties over the affine line, regularity properties of the exceptional fiber extend to all fibers. We then show that quantized coordinate rings of flag varieties and Schubert varieties can be filtered so that the associated graded rings are twisted semigroup rings in the sense of [RZ12]. This is a noncommutative version of the result due to Caldero [C02] stating that flag and Schubert varieties degenerate into toric varieties, and implies that quantized coordinate rings of flag and Schubert varieties are AS-Cohen–Macaulay.

$G$-valuations and $G$-valuation rings

We aim to construct a non-commutative algebraic geometry in the style of Chevalley by using generalised valuations. To this end, we introduce groupoid valuation rings and associate suitable value functions to them. We show that many results from classical valuation theory can be generalised in a natural way to this context and give several examples. In the final section, we give a very concrete example of what a non-commutative curve would look like in this new setting.

A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type

In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

AbstractClassification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

A geometric invariant of 6-dimensional subspaces of 4×4 matrices

Let R R denote a 6-dimensional subspace of the ring M 4 ( k ) M_4(\Bbbk ) of 4 × 4 4 \times 4 matrices over an algebraically closed field k \Bbbk . Fix a vector space isomorphism M 4 ( k ) ≅ k 4 ⊗ k 4 M_4(\Bbbk ) \cong \Bbbk ^4 \otimes \Bbbk ^4 . We associate to R R a closed subscheme X R {\mathbf X}_R of the Grassmannian of 2-dimensional subspaces of k 4 \Bbbk ^4 , where the reduced subscheme of X R {\mathbf X}_R is the set of 2-dimensional subspaces Q ⊆ k 4 Q \subseteq \Bbbk ^4 such that ( Q ⊗ k 4 ) ∩ R ≠ { 0 } (Q \otimes \Bbbk ^4) \cap R \ne \{ 0\} . Our main result is that if X R {\mathbf X}_R has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P 5 \mathbb {P}^5 via the Plücker embedding. We present several examples of X R \mathbf X_R that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, X R {\mathbf X}_R is a P 1 \mathbb {P}^1 -bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme X R \mathbf X_R appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L \mathcal {L} of a certain graded algebra determined by R R . In that context, it has been an open question for several years to describe such L \mathcal {L} of minimal dimension, i.e., those L \mathcal {L} of dimension one. Our main result implies that if dim ⁡ ( L ) = 1 \dim (\mathcal {L}) = 1 , then, as a subscheme of P 5 \mathbb {P}^5 under the Plücker embedding, deg ⁡ ( L ) = 20 \deg (\mathcal {L}) = 20 .

Skew Poincaré–Birkhoff–Witt extensions over weak compatible rings

In this paper, we introduce weak [Formula: see text]-compatible rings and study skew Poincaré–Birkhoff–Witt extensions over these rings. We characterize the weak notion of compatibility for several noncommutative rings appearing in noncommutative algebraic geometry and some quantum algebras of theoretical physics. As a consequence of our treatment, we unify and extend results in the literature about Ore extensions and skew PBW extensions over compatible rings.