Twenty-Four Hours of Local Cohomology
This book contains 24 lectures by a group of authors on the topic of local cohomology. The notion of local cohomology was originally introduced by A. Grothendieck in the realm of algebraic geometry. Nowadays, the subject has a lot of relations to various other fields of mathematics. The book contains a revised set of lectures notes on the theme of local cohomology presented at the summer school organised at Snowbird, Utah, in 2005. Quite a few first lectures in the series cover the basic prerequisites needed later from geometry, sheaf theory and homological algebra (the Krull dimension of a ring, the dimension of an algebraic set and the dimension of a module; sheaves and Čech cohomology; complexes, resolutions and derived functors, and projective dimension; gradings, filtrations and Gröbner bases; and the Koszul complex and depth). It makes it possible to introduce the local cohomology functors and its first properties (depth and cohomological dimension). A further two lectures discuss properties of Cohen-Macaulay and Gorenstein rings. Further lectures treat relations to commutative algebra, algebraic geometry, topology and combinatorics. A few lectures also cover computational aspects (algorithms related to Gröbner bases, Weyl algebras and D-modules).