Commutative Algebra and its Interactions to Algebraic Geometry
Title | Commutative Algebra and its Interactions to Algebraic Geometry PDF eBook |
Author | Nguyen Tu CUONG |
Publisher | Springer |
Pages | 265 |
Release | 2018-08-02 |
Genre | Mathematics |
ISBN | 331975565X |
This book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.
Introduction to Commutative Algebra and Algebraic Geometry
Title | Introduction to Commutative Algebra and Algebraic Geometry PDF eBook |
Author | Ernst Kunz |
Publisher | Springer Science & Business Media |
Pages | 253 |
Release | 2012-11-06 |
Genre | Mathematics |
ISBN | 1461459877 |
Originally published in 1985, this classic textbook is an English translation of Einführung in die kommutative Algebra und algebraische Geometrie. As part of the Modern Birkhäuser Classics series, the publisher is proud to make Introduction to Commutative Algebra and Algebraic Geometry available to a wider audience. Aimed at students who have taken a basic course in algebra, the goal of the text is to present important results concerning the representation of algebraic varieties as intersections of the least possible number of hypersurfaces and—a closely related problem—with the most economical generation of ideals in Noetherian rings. Along the way, one encounters many basic concepts of commutative algebra and algebraic geometry and proves many facts which can then serve as a basic stock for a deeper study of these subjects.
Special Issue on Commutative Algebra and Its Interaction with Algebraic Geometry and Combinatorics
Title | Special Issue on Commutative Algebra and Its Interaction with Algebraic Geometry and Combinatorics PDF eBook |
Author | |
Publisher | |
Pages | 203 |
Release | 2015 |
Genre | |
ISBN |
Trends in Commutative Algebra
Title | Trends in Commutative Algebra PDF eBook |
Author | Luchezar L. Avramov |
Publisher | Cambridge University Press |
Pages | 7 |
Release | 2004-12-13 |
Genre | Mathematics |
ISBN | 0521831954 |
This book describes the interaction of commutative algebra with other areas of mathematics, including algebraic geometry, group cohomology, and combinatorics.
Progress in Commutative Algebra 1
Title | Progress in Commutative Algebra 1 PDF eBook |
Author | Christopher Francisco |
Publisher | Walter de Gruyter |
Pages | 377 |
Release | 2012-04-26 |
Genre | Mathematics |
ISBN | 3110250403 |
This is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on Boij-Söderburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals. Homological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles the editors have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both noetherian and non-noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke.
Commutative Algebra
Title | Commutative Algebra PDF eBook |
Author | Luchezar L. Avramov |
Publisher | |
Pages | 358 |
Release | 2003 |
Genre | Commutative algebra |
ISBN | 9780821832332 |
Commutative Algebra and Noncommutative Algebraic Geometry
Title | Commutative Algebra and Noncommutative Algebraic Geometry PDF eBook |
Author | David Eisenbud |
Publisher | Cambridge University Press |
Pages | 303 |
Release | 2015-11-19 |
Genre | Mathematics |
ISBN | 110714972X |
This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 2 focuses on the most recent research.