Twenty-four Hours of Local Cohomology

Joint work with Srikanth Iyengar, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther.

Since the book has now appeared (AMS, Amazon), I don’t have a download link here. You can still view a (very, very) preliminary version at the web page for the conference.

This book is aimed to provide an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. It covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Gröbner bases in the commutative setting as well as for $D$-modules, the Frobenius morphism and characteristic $p$ methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems arising from semigroups.

The book begins with basic notions in geometry, sheaf theory, and homological algebra leading to the definition and basic properties of local cohomology. Then it develops the theory in a number of different directions, and draws connections with topology, geometry, combinatorics, and algorithmic aspects of the subject.

In June 2005, the AMS held a summer school called Local cohomology and its interactions with algebra, geometry, and analysis. Anurag Singh and Uli Walther organized the entire thing, from the 5-day school for graduate students, to the 3-day conference afterwards. The school consisted of 24 one-hour lectures (whence the title) by the seven of us so-called Young Guns, and now, two and a half years later, those lectures have become a book. Each YG had prepared notes for their lectures, but resolving the enormous notational and stylistic differences took a huge amount of work. Everybody pitched in and helped, but it certainly wouldn’t have happened without Anurag and Uli riding herd.

Bibtex code:

 @book {MR2355715,
    AUTHOR = {Iyengar, Srikanth B. and Leuschke, Graham J. 
              and Leykin, Anton and Miller, Claudia and 
              Miller, Ezra and Singh, Anurag K. and
              Walther, Uli},
     TITLE = {Twenty-four hours of local cohomology},
    SERIES = {Graduate Studies in Mathematics},
    VOLUME = {87},
 PUBLISHER = {American Mathematical Society},
   ADDRESS = {Providence, RI},
      YEAR = {2007},
     PAGES = {xviii+282},
      ISBN = {978-0-8218-4126-6},
   MRCLASS = {13D45 (55N30)},
  MRNUMBER = {MR2355715},