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.

A GEOMETRIC VERSION OF DYSON'S LEMMA FOR FACTORS OF ARBITRARY DIMENSION

2002 ◽  
Vol 13 (01) ◽  
pp. 43-65 ◽  
Author(s):  
MARKUS WESSLER
Keyword(s):  
Algebraic Geometry ◽  
Arbitrary Dimension ◽  
Product Theorem ◽  
Projective Varieties ◽  
Quantitative Version ◽  
Weak Positivity ◽  
Projective Lines ◽  
Smooth Projective Varieties ◽  
Geometric Analogue

This paper generalizes the geometric part of the Esnault–Viehweg paper on Dyson's Lemma for a product of projective lines. Using the method of weak positivity from algebraic geometry, we are able to study products of smooth projective varieties of arbitrary dimension and to prove a geometric analogue of Dyson's Lemma for this case. Our main result is in fact a quantitative version of Faltings' product theorem.

Fourier-Mukai Transforms in Algebraic Geometry

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Algebraic Geometry ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Point Of View ◽  
Basic Knowledge ◽  
Research Directions ◽  
Coherent Sheaves ◽  
Geometric Point ◽  
Singular Cohomology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

scholarly journals A Riemann–Hurwitz Theorem for the Algebraic Euler Characteristic

2017 ◽  
Vol 60 (3) ◽  
pp. 490-509
Author(s):  
Andrew Fiori
Keyword(s):  
Euler Characteristic ◽  
Euler Characteristics ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Irreducible Components ◽  
Arbitrary Dimensions ◽  
Smooth Projective Varieties ◽  
Ramification Locus ◽  
Hurwitz Theorem

AbstractWe prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

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.

Cut cotorsion pairs

2021 ◽  
pp. 1-38
Author(s):  
Mindy Huerta ◽  
Octavio Mendoza ◽  
Marco A. Pérez
Keyword(s):  
General Framework ◽  
Abelian Category ◽  
Triangulated Categories ◽  
Finitistic Dimension ◽  
Coherent Sheaves ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Homological Algebra ◽  
Serre Subcategories ◽  
Dimension Conjecture

Abstract We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

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