Twenty-Four Hours of Local Cohomology

Where it came from

In June 2005 the seven of us gave twenty-four lectures on local cohomology over one week at a summer school in Snowbird, Utah. I gave lectures 3, 5, 7 and 12; my notes for those are on the talks page, and they end with the line “at some point there will probably be a book.” There was.

Abstract

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.

The twenty-four hours

1. Basic notions
2. Cohomology
3. Resolutions and derived functors
4. Limits
5. Gradings, filtrations, and Gröbner bases
6. Complexes from a sequence of ring elements
7. Local cohomology
8. Auslander–Buchsbaum formula and global dimension
9. Depth and cohomological dimension
10. Cohen–Macaulay rings
11. Gorenstein rings
12. Connections with sheaf cohomology
13. Projective varieties
14. The Hartshorne–Lichtenbaum vanishing theorem
15. Connectedness
16. Polyhedral applications
17. D-modules
18. Local duality revisited
19. De Rham cohomology
20. Local cohomology over semigroup rings
21. The Frobenius endomorphism
22. Curious examples
23. Algorithmic aspects of local cohomology
24. Holonomic rank and hypergeometric systems

Plus an appendix on injective modules and Matlis duality.

BibTeX

@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 = {13-01 (13D45 14B15 55N30)},
  MRNUMBER = {2355715},
}

The 2007 hardcover ISBN is above; the book is now in print as a softcover, 978-1-4704-7159-0.