scholarly journals Noncommutative algebraic geometry

2003 ◽  
pp. 509-580 ◽  
Author(s):  
Olav Arnfinn Laudal
Keyword(s):  
Algebraic Geometry ◽  
Noncommutative Algebraic Geometry

scholarly journals VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES

2017 ◽  
Vol 18 (3) ◽  
pp. 619-627
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Algebraic Geometry ◽  
Upper Bound ◽  
Homotopy Theory ◽  
Krull Dimension ◽  
Noncommutative Algebraic Geometry ◽  
Noetherian Scheme ◽  
Quotient Singularities ◽  
Cyclic Quotient Singularities

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

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

2020 ◽  
pp. 1-19
Author(s):  
Masaki Matsuno
Keyword(s):  
Algebraic Geometry ◽  
Complete Classification ◽  
Complete List ◽  
Graded Algebra ◽  
Regular Algebra ◽  
3 Dimensional ◽  
Noncommutative Algebraic Geometry ◽  
Regular Algebras ◽  
Point Scheme

Abstract Classification 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.

scholarly journals Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Héctor Suárez ◽  
Duban Cáceres ◽  
Armando Reyes
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Noncommutative Differential Geometry ◽  
Regular Algebras ◽  
Skew Pbw Extensions ◽  
Finitely Presented

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.

scholarly journals A RECOLLEMENT APPROACH TO GEIGLE–LENZING WEIGHTED PROJECTIVE VARIETIES

2016 ◽  
Vol 226 ◽  
pp. 71-105 ◽  
Author(s):  
BORIS LERNER ◽  
STEFFEN OPPERMANN
Keyword(s):  
Algebraic Geometry ◽  
Abelian Category ◽  
New Method ◽  
Projective Varieties ◽  
Noncommutative Algebraic Geometry ◽  
Coherent Sheaves ◽  
Endomorphism Algebras ◽  
Projective Lines

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

scholarly journals NAKAYAMA AUTOMORPHISMS OF ORE EXTENSIONS OVER POLYNOMIAL ALGEBRAS

2019 ◽  
Vol 62 (3) ◽  
pp. 518-530 ◽  
Author(s):  
LIYU LIU ◽  
WEN MA
Keyword(s):  
Algebraic Geometry ◽  
Cyclic Group ◽  
Invariant Theory ◽  
Polynomial Algebra ◽  
Ore Extension ◽  
Polynomial Algebras ◽  
Noncommutative Algebraic Geometry ◽  
Ore Extensions ◽  
Group Sharing ◽  
Noncommutative Invariant Theory

AbstractNakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

Skew Poincaré–Birkhoff–Witt extensions over weak compatible rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2050225 ◽  
Author(s):  
Armando Reyes ◽  
Héctor Suárez
Keyword(s):  
Algebraic Geometry ◽  
Theoretical Physics ◽  
Quantum Algebras ◽  
Noncommutative Rings ◽  
Noncommutative Algebraic Geometry ◽  
Ore Extensions ◽  
Skew Pbw Extensions ◽  
Weak Notion

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.

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