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).

Reviewer: Michael Hellus (Leipzig)

##### 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 | GrĂ¶bner 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 |