Local cohomology. An algebraic introduction with geometric applications (Q2888874)

The book is the second, revised and enlarged edition of the authors' book [Local cohomology. An algebraic introduction with geometric applications, Cambridge University Press (1998; Zbl 0903.13006)]. As it was mentioned in the review on the first edition (see Zbl 1133.13017) the intention of the authors was twofold: (1) There was a challenge for an algebraic introduction to Grothendieck's local cohomology theory originally invented by the aid of scheme theory (see \textit{A. Grothendieck} [Séminaire de géométrie algébrique par Alexander Grothendieck 1962. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Fasc. I. Exposés I à VIII; Fasc. II. Exposés IX à XIII. 3ieme édition, corrigée. Bures-Sur-Yvette (Essonne): Institut des Hautes Études Scientifiques (1962; Zbl 0159.50402)]). (2) To present an introduction designed primarly to graduate students with some basic knowledge on homological and commutative algebra. -- Because of the authors' masterpiece for clear, complete and selfcontained writing the first edition became a well accepted and well studied textbook on the subject. The appearance of the book filled a large gap for textbooks as one might see that it is quoted in more than 320 research papers as a basic reference. For young researchers it opened the way for serious research on the vital field of local cohomology. (Note that during the last decades there are more than 500 research articles with the phrase ``local cohomology'' in the title.) Fifteen years after the first edition the authors decided to reflect on how they could change and extend the material in order to enhance its usefulness to the interested readers.NEWLINENEWLINEAs a reaction of the recent developments the authors decided to include a new chapter devoted to the study the canonical module of a local ring \((R,\mathfrak{m})\). The canonical module is defined in the case of a factor ring of a Gorenstein ring. Its completion is the Matlis Dual of \(H^d_{\mathfrak{m}}(R), d = \dim R,\) for the \(d\)-th local cohomology of \(R\) with respect to the maximal ideal \(\mathfrak{m}\). Note that \(H^d_{\mathfrak{m}}(R)\) is (by Grothendieck's Theorems) the last non-vanishing local cohomology module of the family \(\{H^i_{\mathfrak{m}}(R)\}_{i\in \mathbb{Z}}\). It covers several local information of \(R\). In the first edition the authors were mainly interested in the Artinian \(R\)-module \(H^d_{\mathfrak{m}}(R)\) in particular when \(R\) is a Cohen-Macaulay ring. Here they decided to study the canonical module, which is finitely generated for \(R\). The authors study also the \(S_2\)-fication as the endomorphism ring of the canonical module. A second change, in contrast to the first edition concerns the chapters on gradings and graded local cohomology. Because of the usefulness of \(\mathbb{Z}^n\)-gradings in several applications the authors extend most of their results on \(\mathbb{Z}\)-gradings to the more general case of \(n > 1\). The authors illustrate their new perspective in particular to Stanley-Reisner rings. By the work of Hochster and Huneke characteristic \(p\)-methods in commutative algebra and local cohomology became an important tool during the last decades. As the key point of these methods there is an additional subsection on the so-called Frobenius action on the local cohomology modules \(H^i_I(R)\) for an ideal \(I\) in a ring \(R\) of prime characteristic \(p\). This is used in another additional subsection in order to include Hochster's proof of the Monomial Conjecture in the case of prime characteristic. A further new subsection is on locally free sheaves, where Serre's Cohomological Criterion for Local Freeness, Horrock's splitting Criterion and Grothendieck's Splitting Theorem are proved. The authors omitted also some items that they now consider no longer command sufficiently compelling reasons for inclusion (among them applications of local duality, a priori bounds of diagonal type on Castelnuovo-Mumford regularity). Minor changes concern the treatment of Faltings' Annihilator Theorem, the graded Delingne Isomorphism). The Chapters ``Links with projective varieties'', ``Castelnuovo regularity'', ``Connectivity in algebraic geometry'', ``Links with sheaf cohomology'' built a bridge to algebraic geometry. They are intended to graduate students as a starting point for geometric applications as anounced in the title of the book.NEWLINENEWLINEFor an interested reader the book opens the view towards the beauty of local cohomology not as an isolated subject but as a tool helpful in commutative algebra and algebraic geometry.NEWLINENEWLINEReviewer's remark: As a source for further applications not covered in the book under review like Gröbner bases, algorithmic aspects, \(D\)-modules, DeRham cohomology a.o. one might also see [\textit{S. B. Iyengar} et al., Twenty-four hours of local cohomology. Providence, RI: American Mathematical Society (AMS) (2007; Zbl 1129.13001)]).