Description
Commutative Algebra and its Interactions to Algebraic Geometry, 1st ed. 2018
VIASM 2013–2014
Lecture Notes in Mathematics Series, Vol. 2210
Coordinators: Tu CUONG Nguyen, Tuan HOA Le, Viet TRUNG Ngo
Language: EnglishSubject for Commutative Algebra and its Interactions to Algebraic...:
Publication date: 08-2018
Support: Print on demand
Support: Print on demand
Description
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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.
1. Notes on Weyl Algebras and D-modules.- 2. Inverse Systems of Local Rings.- 3. Lectures on the Representation Type of a Projective Variety.- 4. Simplicial Toric Varieties which are set-theoretic Complete Intersections.
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
© 2024 LAVOISIER S.A.S.
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