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Commutative Algebra and its Interactions to Algebraic Geometry

VIASM 2013–2014

  • Book
  • © 2018

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Editors:
  1. Nguyen Tu CUONG

    1. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

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  2. Le Tuan HOA

    1. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

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  3. Ngo Viet TRUNG

    1. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

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  • Suitable for graduate courses, requiring only a basic background in commutative algebra
  • Includes many interesting open problems and ideas for further investigation
  • Describes recent research in commutative algebra and its applications to algebraic geometry

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2210)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-ix
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  2. Notes on Weyl Algebra and D-Modules

    • Markus Brodmann
    Pages 1-117

  3. Inverse Systems of Local Rings

    • Juan Elias
    Pages 119-163

  4. Lectures on the Representation Type of a Projective Variety

    • Rosa M. Miró-Roig
    Pages 165-216

  5. Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections

    • Marcel Morales
    Pages 217-256

  6. Back Matter

    Pages 257-258
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About this book

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.

Editors and Affiliations

  • Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Nguyen Tu CUONG, Le Tuan HOA, Ngo Viet TRUNG

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