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This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matemàtica, Institut d’Estudis Catalans, July 15–26, 1996.
A very powerful technique in commutative algebra was introduced by Macaulay, who realized that studying the initial terms of elements of an ideal gives one great insight into the algebra and combinatorics of the ideal. The initial ideal depends on the choice of coordinates, but there is an object, the initial ideal in generic coordinates, which is coordinate-independent. Generic initial ideals appeared...
These notes are based on five lectures given at the Summer School on Commutative Algebra held at the CRM in Barcelona during July, 1996. I would like to thank the organizers J. Elias, J. M. Giral, R. M. Miró-Roig, and S. Zarzuela for the excellent job they did. The great success of the Summer School was due mainly to their efforts.
Local cohomology is a useful tool in several branches of commutative algebra and algebraic geometry. The main aim of this series of lectures is to illustrate a few of these techniques. The material presented in the sequel needs some basic knowledge about commutative resp. homological algebra. The basic chapters of the textbooks [9], [28], and [48] are a recommended reading for the preparation. The...
Since a projective variety $$ V = \mathcal{Z}(I) \subseteq P^n $$ is an intersection of hypersurfaces, one of the most basic problems we can pose in relation to V is to describe the hypersurfaces containing it. In particular, one would like to know the maximal number of linearly independent hypersurfaces of each degree containing V, that is to know the dimension of Id...
By a degree of a module M we mean a numerical measure of information carried by M. It must serve the purposes of allowing comparisons between modules and to exhibit flexible calculus rules that track the degree under some basic constructions in module theory.
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