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From algebras to varieties
Kavli Institute for the Physics and Mathematics of the Universe News. 2011. Vol. 14. P. 4-9.
Alexey Bondal
Bondal Alexey, Rosly Alexei, / IPMU. Series IPMU11-0117 "IPMU11-0117". 2011.
We construct a twist-closed enhancement of the derived category of coherent sheaves on a smooth compact complex-analytic manifold by means of DG-category of dbar-superconnections. ...
Added: October 30, 2013
Piontkovski D., / Cornell University. Series math "arxiv.org". 2014. No. arXiv:1401.6549.
A noncommutative Grassmanian 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 ...
Added: May 13, 2014
Piontkovski D., Journal of Noncommutative Geometry 2016
A noncommutative Grassmanian 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 ...
Added: April 14, 2016
Antipov M., Звонарёва А. О., Journal of Mathematical Sciences 2014 Vol. 202 No. 3 P. 333-345
In this paper, all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 are described, using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application, all two-term tilting complexes over a Brauer star algebra are described and their endomorphism rings are ...
Added: December 25, 2018
Lu L., Piontkovski D., International Mathematics Research Notices 2023 Vol. 2023 No. 3 P. 2446-2473
Let A be a finitely presented associative monomial algebra. We study the category qgr(A) which is a quotient of the category of graded finitely presented A-modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category. We calculate ...
Added: April 28, 2023
Katzarkov L. V., Karzhemanov I., / Cornell University. Series math "arxiv.org". 2023. No. 2310.13319.
Added: October 31, 2023
Tikhomirov A. S., Markushevich D., Verbitsky M., Central European Journal of Mathematics 2012 Vol. 10 No. 4 P. 1185-1187
In this preface we give a short description of the current issue of the Central European Journal of Mathematics containing 22 papers which spin around the topics of the conference “Instantons in complex geometry”, held on March 14–18, 2011 in Moscow. The main goal of the conference was to bring together specialists in complex algebraic ...
Added: October 21, 2014
Kaledin D., Труды Математического института им. В.А. Стеклова РАН 2015 Т. 290 С. 43-60
Строится некоммутативное обобщение изоморфизма Картье для любой ассоциативной унитальной алгебры над совершенным полем k нечетной положительной характеристики. Роль дифференциальных форм играют классы гомологий Хохшильда, а дифференциал де Рама заменяется на дифференциал Конна–Цыгана. ...
Added: April 10, 2017
Antipov M., Zvonareva A., Mathematische Zeitschrift 2022 Vol. 301 No. 2 P. 1963-1981
In this paper the class of Brauer graph algebras is proved to be closed
under derived equivalence. For that we use the rank of the maximal torus of the identity
component of the group of outer automorphisms Out0(A) of a symmetric stably biserial
algebra A. ...
Added: November 1, 2019
Piontkovski D., / Cornell University. Series math "arxiv.org". 2014. No. arXiv:1412.8601.
We show that a direct limit of surjections of (weak) Golod--Shafarevich algebras is a weak Golod--Shafarevich algebra as well. This holds both for graded and for filtered algebras provided that the filtrations are induced by the filtration of the first entry of the sequence. It follows that the limit is an algebra of exponential growth. ...
Added: February 2, 2015
Elagin A. D., Лунц В. А., Математический сборник 2018 Т. 209 № 12 С. 87-116
Предположим, что R – коммутативное нётерово кольцо и схема X = SpecR связна. Мы доказываем, что категория Db(cohX) не содержит нетривиальных полных триангулированных подкатегорий, обладающих сильным генератором. Также мы ограничиваем снизу размерность Рукье триангулированной категории T при условии, что существует триангулированный функтор T→Db(cohX), обладающий определенными свойствами. Полученные результаты применяются для изучения когомологического аннулятора кольца R и точечных объектов в категории T. ...
Added: November 29, 2018
Guseva L., / Cornell University. Series math "arxiv.org". 2018.
We construct a full exceptional collection of vector bundles in the bounded derived category of
coherent sheaves on the Grassmannian IGr(3,8) of isotropic 3-dimensional subspaces in a symplectic vector
space of dimension 8. ...
Added: October 19, 2018
Bondal A. I., Rosly A. A., Известия РАН. Серия математическая 2023 Т. 87 № 3 С. 23-55
We construct a twist-closed enhancement of the category Dbcoh(X), the bounded derived category of complexes of OX-modules with coherent cohomology, by means of the DG-category of ∂¯-superconnections. Then we apply the techniques of ∂¯-superconnections to dene Chern classes and Bott–Chern classes of objects in the category, in particular, of coherent sheaves. ...
Added: December 10, 2022
Elagin A. D., Lunts V., Schnürer O., Moscow Mathematical Journal 2020 Vol. 20 No. 2 P. 277-309
We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, thereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated ...
Added: May 11, 2020
Elagin Alexey, Lunts Valery, Schnürer O., / Cornell University. Series arXiv "math". 2018.
We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated ...
Added: December 1, 2018
Fonarev A., Успехи математических наук 2014 Т. 69 № 4(418) С. 189-190
В работе приведена конструкция некоторых исключительных векторных расслоений на грассманианах, а также построено семейство полуортогональных разложений их ограниченных производных категорий. ...
Added: September 19, 2014
Bondal A. I., Bodzenta-Skibinska A., Advances in Mathematics 2018 Vol. 323 P. 226-278
Given a relatively projective birational morphism f : X → Y
of smooth algebraic spaces with dimension of fibers bounded
by 1, we construct tilting relative (over Y) generators TX,f
and S_X,f in D^b(X). We develop a piece of general theory of
strict admissible lattice filtrations in triangulated categories
and show that D^b(X) has such a filtration L where the ...
Added: May 2, 2018
Finkelberg M. V., Braverman A., / Cornell University. Series arXiv "math". 2018.
In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr_GL(n)) of (all) D-modules on the affine Grassmannian of the group GL(n) as the category of ind-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a ...
Added: December 3, 2018
Tikhomirov A. S., Markushevich D., Trautmann G., Central European Journal of Mathematics 2012 Vol. 19 No. 4 P. 1331-1355
We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundled connections an in differential geometry. The new moduli spaces are algebraic spaces arising as quotients ...
Added: October 21, 2014
Piontkovski D., La Scala R., Springer INdAM Series 2021 Vol. 44 P. 279-289
In this paper, homological methods together with the theory of formal languages of theoretical computer science are proved to be effective tools to determine the growth and the Hilbert series of an associative algebra. Namely, we construct a class of finitely presented associative algebras related to a family of context-free languages. This allows us to ...
Added: April 3, 2021
Zürich : European Mathematical Society Publishing house, 2012
The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic ...
Added: October 14, 2013
La Scala R., Piontkovski D., Tiwari S., Journal of symbolic computation 2020 Vol. 101 No. November–December P. 28-50
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class ...
Added: October 26, 2020
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021