Journal ArticleDOI

# On equivariant Serre problem for principal bundles

21 Aug 2018-International Journal of Mathematics (World Scientific)-Vol. 29, Iss: 9, pp 110-116

TL;DR: In this paper, a Γ-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of Γ, where G and Γ are complex linear algebraic groups.

AbstractLet EG be a Γ-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of Γ, where G and Γ are complex linear algebraic groups. Suppose X is contracti...

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TL;DR: In this article, the authors proved Sylvester's law of nullity and exercise, which states that the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullness of A.
Abstract: In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

152 citations

Posted Content
TL;DR: In this paper, the authors define the notion of piecewise linear maps from a fan of a toric vector bundle to the underlying space of the Tits building of a linear algebraic group.
Abstract: We define the notion of a piecewise linear map from a fan $\Sigma$ to $\mathcal{B}(G)$, the underlying space of the Tits building of a linear algebraic group $G$. We show that if $X_\Sigma$ is a toric variety over $\mathbb{C}$ with fan $\Sigma$, the set of integral piecewise linear maps from $\Sigma$ to $\mathcal{B}(G)$ classifies (framed) toric principal $G$-bundles on $X_\Sigma$. In particular, this recovers Klyachko's classification of toric vector bundles on $X_\Sigma$. Moreover, extending the case of line bundles, it is shown that criteria for ampleness and global generation of a toric vector bundle translates to convexity conditions on the associated piecewise linear map.

5 citations

Journal ArticleDOI
TL;DR: In this paper, the notion of compatible ∑-filtered vector space was introduced, where ∑ denotes the fan of a toric variety and G a reductive algebraic group defined over an algebraically closed field.
Abstract: Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.

5 citations

##### References
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Book
30 Mar 1995
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Abstract: Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.

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TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

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TL;DR: In this article, a mini-course is presented to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications, concluding with Stanley's theorem characterizing the number of simplicies in each dimension in a convex simplicial polytope.
Abstract: Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

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01 Jan 2012
TL;DR: A compilation of two sets of notes at the University of Kansas was published in the Spring of 2002 by?? and the other in the spring of 2007 by Branden Stone.
Abstract: 1 A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed

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