EAN: 9783319755649

COMMUTATIVE ALGEBRA AND ITS INTERACTIONS TO ALG...

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.

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