Three Lectures on Commutative Algebra
This book contains lecture notes from the Winter School on Commutative Algebra and Applications held in 2006. The first lecture (by H. Brenner) contains a discussion of a geometrical approach to problems concerning tight closures. Problems treated here include the question of characterization of the tight closure of an ideal in a Noetherian domain over a field of positive characteristic and the question of whether the tight closures commute with localizations in these rings. Both questions have negative answers and the counter-example concludes this part of the book. However the core of the lecture is the understanding of tight closure under the properties of vector bundles on corresponding projective curves. Next, a positive answer to the first question is given for some particular cases. The second lecture (by J. Herzog) considers relations of various structures and monomial ideals in commutative free algebras naturally given by these structures. These chapters are devoted to various possibilities of shifting operation of simplicial complexes, discrete polymatroids, a variation of Dirac's theorem on chordal graphs and Cohen-Macaulay graphs.
The last lecture (by Orlando Villamayor) gives a quite detailed exposition of two important results in algebraic geometry proved by Hironaka: the desingularization and embedded principalization of ideals (both theorems are in characteristic zero). The original proofs were existence proofs; the proofs presented here are constructive. All the lectures suppose that the reader has some experience in commutative algebra or algebraic geometry. The first one requires knowledge of vector bundles and cohomology and the second one supposes the reader to be familiar with the basic concepts of commutative algebra including the Stanley-Reisner rings. This lecture, in general, seems to be written for readers working in algebraic geometry. The topics of these lectures are quite attractive and the work contains very recent results of the authors. Any reader with a good background in commutative algebra could find this book interesting.