Igor Nikolaev
A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic AF-algebras.
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The "noncommutative geometry" of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C*-algebras T_t={u,v | vu=exp(2πit)uv}, or noncommutative tori. The main result says that under the morphism isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces.
We relate SLOCC equivalence classes of qudit states to moduli spaces of Calabi-Yau manifolds equipped with a collection of line bundles. The cases of 3 qutrits and 4 qubits are also related to noncommutative algebraic geometry.
The first part of these notes gives an introduction to noncommutative projective geometry after Artin--Zhang. The second part provides an overview of the work of Polishchuk that reconciles noncommutative two-tori having real multiplication with the Artin--Zhang setting.
In this paper we study noncommutative plane curves, i.e. non-commutative k-algebras for which the 1-dimensional simple modules form a plane curve. We study extensions of simple modules and we try to enlighten the completion problem, i.e. understanding the connection between simple modules of different dimension.
A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheave…
We desribe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that in other cases the classification of vector bundles over a noncommutative curve is a wild problem.
Degenerations of orbifold curves as noncommutative varieties
本文从非交换代数几何角度研究轨道投影曲线在堆叠点碰撞下的退化特性。
Noncommutative varieties and stability conditions: an overview
本文综述了非交换代数几何中稳定性条件和模空间理论的最新进展及其在超kähler几何和经典代数几何中的应用。
We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
Dimension theory of noncommutative curves
本文研究轨形曲线凝聚层有界导出范畴的多种维数，完成 Reiten-van den Bergh 意义下非交换曲线的维数计算，并构造稳定条件、刻画带遗传倾斜丛的轨形曲线。
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…
Let k be a field. In this paper, we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative P^1-bundle. This result can be considered a noncommutative, one-dimensional version of Tsen's theorem. By specializing this theorem, we show that every arithmetic noncommutative projective line is a noncommutative curve, and conversely we characterize exactly those noncommutative curves of genus zero which are arithmetic. We then use this characterization, together with results regarding arithmetic noncommutative projective lines, to address some p…
We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parametrizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build insta…
We show that pairs $(X,Y)$ of 1-spherical objects in $A_\infty$-categories, such that the morphism space ${\rm Hom}(X,Y)$ is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact, we establish a more precise correspondence at the level of isomorphism of moduli spaces which we show to be affine schemes of finite type over ${\Bbb Z}$.
We establish a new fundamental class of varieties in nonnoetherian algebraic geometry related to the central geometry of dimer algebras. Specifically, given an affine algebraic variety $X$ and a finite collection of non-intersecting positive dimensional algebraic sets $Y_i \subset X$, we construct a nonnoetherian coordinate ring whose variety coincides with $X$ except that each $Y_i$ is identified as a distinct positive dimensional closed point. We then show that the noncommutative blowup of such a singularity is a noncommutative desingularization, in a suitable geometric sense.
We present some methods and results in the application of algebraic geometry and computer algebra to the study of algebraic vector bundles, foliations and zeta functions. A connection of the methods and results with noncommutative geometry will be consider.
This paper studies etale twists of derived categories of schemes and associative algebras. A general method, based on a new construction called the twisted Brauer space, is given for classifying etale twists, and a complete classification is carried out for genus 0 curves, quadrics, and noncommutative projective spaces. A partial classification is given for curves of higher genus. The techniques build upon my recent work with David Gepner on the Brauer groups of commutative ring spectra.
We study an interplay between operator algebras and geometry of rational elliptic curves. Namely, let $\mathcal{O}_B$ be the Cuntz-Krieger algebra given by square matrix $B=(b-1, ~1, ~b-2, ~1)$, where $b$ is an integer greater or equal to two. It is proved, that there exists a dense self-adjoint sub-algebra of $\mathcal{O}_B$, which is isomorphic (modulo an ideal) to a twisted homogeneous coordinate ring of the rational elliptic curve $\mathcal{E}({\Bbb Q})=\{(x,y,z) \in {\Bbb P}^2({\Bbb C}) ~|~ y^2z=x(x-z)(x-{b-2\over b+2}z)\}$.
In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.
We introduce new mathematical aspects of the Bell states using matrix factorizations, nonnoetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial $p$ consists of two matrices $ϕ_1,ϕ_2$ such that $ϕ_1ϕ_2 = ϕ_2ϕ_1 = p \operatorname{id}$. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on nonnoetherian singularities. Moreover, we find that the collapse …
In this paper we introduce a noncommutative analogue of the notion of linear system, which we call a helix $\underline{\mathcal{L}} := (\mathcal{L}_{i})_{i \in \mathbb{Z}}$ in an abelian category ${\sf C}$ over a quadratic $\mathbb{Z}$-indexed algebra $A$. We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from ${\sf Proj }\operatorname{End}(\underline{\mathcal{L}})$ to ${\sf Proj }A$. We construct examples of helices of vector bundles on elliptic curves generalizing the elliptic helices of line bundles constructed by Bondal-Polishchuk, where $A$ is the…
Noncommutative rigidity of the moduli stack of stable pointed curves
本文证明了稳定性标点曲线模栈的第二Hochschild上同调群在大部分(g,n)情况下为零，揭示了其非交换刚性特性。
The Riemann-Roch Theorem on higher dimensional complex noncommutative tori
本文证明了配备复结构的任意维非交换环面上的黎曼-罗赫定理与霍奇定理的类似物，并探讨了区分非交换阿贝尔簇与非代数非交换复环面的理论意义。
We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and transversality hypotheses.
We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus. We develop a sheaf theory using techniques from noncommutative algebraic geometry. The cases of projective varieties are studied in detail, and several explicit examples are worked out, including new noncommutative deformations of Grassmann and flag varieties. Our constructions set up …
We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a finite last residue field the constructed local central extension is isomorphic to a central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov.
Let k be a field. We show that all homogeneous noncommutative curves of genus zero over k are noncommutative P^1-bundles over a (possibly) noncommutative base. Using this result, we compute complete isomorphism invariants of homogeneous noncommutative curves of genus zero, allowing us to generalize a theorem of Witt.
In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten's conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.
Twisted tensor product, smooth DG algebras, and noncommutative resolutions of singular curves
研究利用扭曲张量积构造并刻画含两个单模的代数与光滑 DG 代数，证明其有限整体维数，并为奇异有理曲线给出光滑极小非交换消解。
We study the noncommutative modular curve (which was already studied by Connes, Manin and Marcolli), and the space of geodesics on the usual modular curve, from the viewpoint of algebraic groups, linear algebra and class field theory. This allows us, first, to understand differently some aspects of Manin's real multiplication program, and secondly, to study higher dimensional analogs of the noncommutative modular curve, called irrational or noncommutative boundaries of Shimura varieties. These are double cosets spaces $L\backslash G(\mathbb{R})/P(K)$ where $G$ is a connected reductive algebrai…
A ringed-space-like structure on coalgebras for noncommutative algebraic geometry
受Reyes非交换谱理论启发，研究者引入可作为域$k$上概形非交换推广的环化余代数，证明代数闭域下相关函子的满忠实性与交换情形的同构性，并将两类模/层范畴嵌入对应环化余代数的模范畴。
Admissible subcategories of noncommutative curves
该研究针对光滑非交换曲线导出范畴的容许子范畴展开，证明其仍为光滑非交换曲线的导出范畴，据此分类半正交分解、排除幻影范畴存在性，并将Bondal-Orlov重建定理推广到轨形曲线。
The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises is attached. Our notes are intended for the graduate students and faculty with interests in noncommutative geometry; they can be read by non-experts in the field.
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…
We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid, and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra is finite dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
The title refers to the nilcommutative or $NC$-schemes introduced by M. Kapranov in math.AG/9802041. The latter are noncommutative nilpotent thickenings of commutative schemes. We consider also the parallel theory of nil-Poisson or $NP$-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for $NC$- and $NP$-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck. It turns out that ea…
In this article, we propose a way of seeing the noncommutative tori in the category of noncommutative motives. As an algebra, the noncommutative torus is lack the smoothness property required to define a noncomutative motive. Thus, instead of working with the algebra, we work with the category of holomorphic bundles. It is known that these are related to the coherent sheaves of an elliptic curve. We describe the cyclic homology of the category of holomorphic bundle on a noncommutative torus. We then introduce a notion of (weak) t-structure in dg categories. By applying the t-structure to a non…
Let $\mathcal{H}$ be a noncommutative regular projective curve over a perfect field $k$. We study global and local properties of the Auslander-Reiten translation $τ$ and give an explicit description of the complete local rings, with the involvement of $τ$. We introduce the $τ$-multiplicity $e_τ(x)$, the order of $τ$ as a functor restricted to the tube concentrated in $x$. We obtain a local-global principle for the (global) skewness $s(\mathcal{H})$, defined as the square root of the dimension of the function (skew-) field over its centre. In the case of genus zero we show how the ghost group, …
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…
This paper gives a short and historical survey on the theory of pure motives in algebraic geometry and reviews some of the recent developments of this theory in noncommutative geometry. The second part of the paper outlines the new theory of endomotives and some of its relevant applications in number-theory.
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the…
Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we classify all large (that is, non-coherent) tilting sheaves and the corresponding resolving classes. In particular we show that in the elliptic and in the tubular cases every large tilting sheaf has a well-defined slope.
We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $\infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in $\operatorname{Qcoh}\mathbb{X}$.
Noncommutative Gelfand Duality: the algebraic case
该研究借助导出代数几何技术定义了非交换Gelfand对偶，证明环范畴与预环化位点子范畴反等价，提出的新谱的相对精细化版本在复有限生成交换代数情形与Grothendieck谱一致，为量子时空几何研究奠基。
In this article one extends the classical theory of (intermediate) Jacobians to the "noncommutative world". Concretely, one constructs a Q-linear additive Jacobian functor J(-) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf(X)) (associated to the derived dg category perf(X) of a smooth projective scheme X) identifies with the un…
Noncommutative Quillen-Lichtenbaum Conjecture
本文通过非交换几何视角，扩展了经典Quillen-利文萨姆猜想至有限型分离复方案，并建立了代数与拓扑K群比较映射的同构范围。
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.
Central curves on noncommutative surfaces
该研究拓展了非交换代数中序与代数几何中栈曲线/曲面的对应词典，研究驯顺序在曲线的限制与纤维积的对应，将其拓展至非遗传序与奇异栈曲线，并应用于非交换二次曲线和斜三次曲线的分类。
In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.
Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.
We study non-commutative real algebraic geometry for a unital associative *-algebra A viewing the points as pairs (π,v) where π is an unbounded *-representation of A on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A. Finally, we compare our results with their analogues in the usual (i.e. Schmüdgen's) non-commutative real algebraic geometry where the points are unbounded *-representation of A.
This survey is based on lectures given by the authors during the program "Noncommutative algebraic geometry and representation theory" at the MSRI, Berkeley, in the spring of 2013. It covers the recent work of the authors on noncommutative motives and their applications, and is intended for a broad mathematical audience.
In this note we endow Kontsevich's category KMM of noncommutative mixed motives with a non-degenerate weight structure in the sense of Bondarko. As an application we obtain a convergent weight spectral sequence for every additive invariant (e.g. algebraic K-theory, cyclic homology, topological Hochschild homology, etc.), and a ring isomorphism between the Grothendieck ring of KMM and the Grothendieck ring of the category of noncommutative Chow motives.
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