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Twenty-four hours of local cohomology. (English) Zbl 1129.13001
Graduate Studies in Mathematics 87. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4126-6/hbk). xviii, 282 p. (2007).
This book is on local cohomology. It is explained how local cohomology is related to algebra, geometry, topology, combinatorics and computational algebra. The book consists of 24 chapters which are called lectures; each lecture has its own topic. The first few lectures present general (i. e., quite independent from local cohomology) material from Commutative Algebra and Algebraic Geometry (to name a few, there are lectures on Krull dimension, tangent spaces, limits, Cohen-Macaulay rings, Gorenstein rings). An important subject which is also explained is the relation between local cohomology and the minimal number of defining equations of an algebraic set (the so-called arithmetic rank). In the subsequent lectures more sophisticated concepts are explained (e. g. connectedness questions, \(D\)-modules, de Rham cohomology, duality theorems, relations between various notions of cohomology, but also polyhedra and algorithmic aspects of local cohomology).

MSC:
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13D45 Local cohomology and commutative rings
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13N10 Commutative rings of differential operators and their modules
14B15 Local cohomology and algebraic geometry
13H05 Regular local rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
14F40 de Rham cohomology and algebraic geometry
55N30 Sheaf cohomology in algebraic topology
Software:
Macaulay2
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