an:01305008
Zbl 0933.13002
Green, Mark L.
Generic initial ideals
EN
Elias, J. (ed.) et al., Six lectures on commutative algebra. Lectures presented at the summer school, Bellaterra, Spain, July 16--26, 1996. Basel: Birkh??user. Prog. Math. 166, 119-186 (1998).
1998
a
13A15 14A05
generic initial ideals
This is a semi-expository article (with many new proofs of known results) on generic initial ideals which began as course notes for a course given at UCLA (University of California, Los Angeles) and was later given as a series of lectures at the Recerca Matem??tica Summer School in Commutative Algebra during the summer of 1996 in Barcelona.
Besides the introduction, the article consists of six chapters: (1) The initial ideal; (2) Regularity and saturation; (3) The Macaulay-Gotzmann estimates on the growth of ideals, (4) Points in \(\mathbb{P}^2\) and curves in \(\mathbb{P}^3\), (5) Gins (``generic initial ideals'') in the exterior algebra; and (6) Lexicographic gins and partial elimination ideals.
Each chapter ends with some short comments for students. According to the author these notes are ``a strange brew of commutative algebra, geometry, and combinatorics, with a little bit of non-commutative algebra thrown in for good measure''. Some of the topics included are monomial ideals, gins, Galligo's theorem, Eliahou-Kervaire's theorem, Hilbert's syzygy theorem, the Bayer-Stillman theorem on the regularity of an ideal I, Macaulay's estimate on growth of ideals, the hyperplane restriction theorem, Gotzmann's persistence theorem, Gotzmann's regularity theorem, results of Ellia-Peskine and Gruson-Peskine, the Hilbert-Burch theorem, Laudal's lemma, Macaulay's bound for exterior ideals, and the Kruskal-Katona theorem.
For the entire collection see [Zbl 0892.00031].
D.D.Anderson (Iowa City)