Flat Covers in the Category of Quasi-coherent Sheaves Over the Projective Line

2004 ◽  
Vol 32 (4) ◽  
pp. 1497-1508 ◽  
Author(s):  
Edgar Enochs ◽  
S. Estrada ◽  
J. R. García Rozas ◽  
L. Oyonarte
Keyword(s):  
Projective Line ◽  
Coherent Sheaves

scholarly journals THE GROUP OF COVERING AUTOMORPHISMS OF A QUASI-COHERENT SHEAF ON $\bm{P}^1(K)$

2007 ◽  
Vol 50 (2) ◽  
pp. 325-341
Author(s):  
E. Enochs ◽  
S. Estrada ◽  
J. R. García Rozas ◽  
L. Oyonarte
Keyword(s):  
Commutative Ring ◽  
Wide Class ◽  
Vector Bundles ◽  
Universal Covering ◽  
Coherent Sheaf ◽  
Projective Line ◽  
Topological Properties ◽  
Coherent Sheaves ◽  
Flat Cover ◽  
Natural Way

AbstractCoGalois groups appear in a natural way in the study of covers. They generalize the well-known group of covering automorphisms associated with universal covering spaces. Recently, it has been proved that each quasi-coherent sheaf over the projective line $\bm{P}^1(R)$ ($R$ is a commutative ring) admits a flat cover and so we have the associated coGalois group of the cover. In general the problem of computing coGalois groups is difficult. We study a wide class of quasi-coherent sheaves whose associated coGalois groups admit a very accurate description in terms of topological properties. This class includes finitely generated and cogenerated sheaves and therefore, in particular, vector bundles.

scholarly journals Applications of Mutations in the Derived Categories of Weighted Projective Lines to Lie and Quantum Algebras

2018 ◽  
Vol 2020 (19) ◽  
pp. 5814-5871
Author(s):  
Bangming Deng ◽  
Shiquan Ruan ◽  
Jie Xiao
Keyword(s):  
Projective Line ◽  
Derived Categories ◽  
Line Bundles ◽  
Hall Algebra ◽  
Moody Algebra ◽  
Weighted Projective Line ◽  
Coherent Sheaves ◽  
Enveloping Algebra ◽  
The Right ◽  
Projective Lines

Abstract Let $\textrm{coh}\ \mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\textrm{coh}\ \mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\textrm{coh}\ \mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver ${Q}$ associated with $\mathbb{X}$. By further dealing with the Ringel–Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra ${\mathfrak g}_{Q}$ associated with ${Q}$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of ${\mathfrak g}_{Q}$.

The composition Hall algebra of a weighted projective line

2013 ◽  
Vol 2013 (679) ◽  
pp. 75-124 ◽  
Author(s):  
Igor Burban ◽  
Olivier Schiffmann
Keyword(s):  
Lie Algebras ◽  
Projective Line ◽  
Affine Lie Algebras ◽  
Hall Algebra ◽  
Weighted Projective Line ◽  
Drinfeld Double ◽  
Coherent Sheaves ◽  
Enveloping Algebras ◽  
Quantized Enveloping Algebras

Abstract In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of the quantized enveloping algebras of affine Lie algebras of simply-laced types as well as some new embeddings between them. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types D4(1, 1), E6(1, 1), E7(1, 1) and E8(1, 1). In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.

GENERALIZED QUASI-COHERENT SHEAVES

2003 ◽  
Vol 02 (01) ◽  
pp. 63-83 ◽  
Author(s):  
E. ENOCHS ◽  
S. ESTRADA ◽  
J. R. GARCÍA-ROZAS ◽  
L. OYONARTE
Keyword(s):  
Projective Line ◽  
Coherent Sheaves ◽  
Projective Representations ◽  
Line P ◽  
Torsion Free

The category of quasi-coherent sheaves on the projective line P1(k) (k is a field) is equivalent to certain representations of the quiver • → • ← •. Many of the techniques which are used to study these sheaves apply to more general categories. We give the definitions of these more general categories and then consider one particular such category in depth. In this particular category we prove that there are no (nonzero) projective representations but that the category admits flat covers (or, equivalently in this situation, torsion free covers) and cotorsion envelopes.

scholarly journals The category of coherent sheaves over a weighted projective line and loop algebras

2018 ◽  
Vol 48 (11) ◽  
pp. 1551
Author(s):  
Dou Rujing ◽  
Xu Fan ◽  
Xiao Jie ◽  
Ruan Shiquan
Keyword(s):  
Projective Line ◽  
Weighted Projective Line ◽  
Coherent Sheaves ◽  
Loop Algebras

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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