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, Grobner 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.
Recenzijos
"It's all terrific stuff. I hope this book will succeed in bringing many young mathematicians to love cohomology, too, and then to go on from there." MAA Reviews
"This book is an excellent text on local cohomology and complements well the existing sources. It will surely become a standard reference on this theory." Mathematical Reviews
Basic notions
Cohomology
Resolutions and derived functors
Limits
Gradings, filtrations, and Grobner bases
Complexes from a sequence of ring elements
Local cohomology
Auslander-Buchsbaum formula and global dimension
Depth and cohomological dimension
Cohen-Macaulay rings
Gorenstein rings
Connections with sheaf cohomology
Projective varieties
The Hartshorne-Lichtenbaum vanishing theorem
Connectedness
Polyhedral applications
$D$-modules
Local duality revisited
De Rham cohomology
Local cohomology over semigroup rings
The Frobenius endomorphism
Curious examples
Algorithmic aspects of local cohomology
Holonomic rank and hypergeometric systems
Injective modules and Matlis duality
Bibliography
Index
Srikanth B. Iyengar, University of Nebraska, Lincoln, NE.
Graham J. Leuschke, Syracuse University, NY.
Anton Leykin, Institute for Mathematics and Its Applications, Syracuse, NY.
Claudia Miller, Syracuse University, NY.
Ezra Miller, University of Minnesota, Minneapolis, MN.
Anurag K. Singh, University of Utah, Salt Lake City, UT.
Uli Walther, Purdue University, West Lafayette, IN.
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