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Noncommutative Grassmannian of codimension two has coherent coordinate ring
Cornell University
,
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 provided that the codimension d=n-m of the Grassmanian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmanian. The proof is quite different from the recent proofs of the coherence of some graded 3-dimensional Calabi--Yau algebras and is based on properties of a PBW-basis of the algebra.
Keywords: некоммутативная геометрияnoncommutative geometrycoherent sheavesкогерентные пучкиcoherent ringgraded Calabi-Yau algebraкогерентное кольцоградуированная алгебра Калаби-Яу
Publication based on the results of:
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
Pirkovskii A. Y., Journal of Noncommutative Geometry 2015 Vol. 9 No. 1 P. 215-264
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) complex algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of ...
Added: November 14, 2013
Pirkovskii A. Y., / Cornell University. Series math "arxiv.org". 2013. No. 1304.1991.
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) complex algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of ...
Added: May 2, 2013
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Piontkovski D., La Scala R., Tiwari S., / Cornell University. Series math "arxiv.org". 2019.
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 7, 2019
Pirkovskii A. Y., / Cornell University. Series math "arxiv.org". 2012. No. 1204.4936.
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) C-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG ...
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