Local cohomology. An algebraic introduction with geometric applications.

*(English)*Zbl 0903.13006
Cambridge Studies in Advanced Mathematics. 60. Cambridge: Cambridge University Press. xv, 416 p. (1998).

At the time when A. Grothendieck established his ingenious ideas about schemes as the foundation of algebraic geometry there grew out a deep interest in the study of section functors and its right derived functors. Motivated by questions about fundamental groups, Lefschetz theorems a.o., A.Grothendieck [see “Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux”, Sémin. géométrie algébrique 2 (SGA2) (1962; Zbl 0159.50402), see also the enlarged edition (Amsterdam 1968; Zbl 0197.47202)] developed the local cohomology theory in algebraic geometry. In his work as well as in R. Hartshorne’s notes about A. Grothendieck’s ideas [see R. Hartshorne, Notes in the book by A. Grothendieck: “Local cohomology”, Lect. Notes Math. 41 (1967; Zbl 0185.49202)] it turned out that the power of these techniques is – at least – two-fold. Firstly it allows to transform the homological techniques invented by J.-P. Serre in algebraic geometry [see J.-P. Serre, “Faisceaux algébriques cohérents”, Ann. Math., II. Ser. 61, 197-278 (1955; Zbl 0067.16201)] to commutative ring theory. Secondly it opens a wide field for research activities on local cohomology in commutative algebra, as one might see by the literature over the last decades. In the mean time there are several approaches to local cohomology, among them those by W. Bruns and J. Herzog [“Cohen-Macaulay rings” (revised edition 1998; see also the first edition 1993; Zbl 0788.13005)], D. Eisenbud [“Commutative algebra. With a view toward algebraic geometry” (1995; Zbl 0819.13001)], M. Herrmann, S. Ikeda and U. Orbanz [“Equimultiplicity and blowing up. An algebraic study” (1988; Zbl 0649.13011)], M. Hochster [“Notes on local cohomology”, Lect. Univ. Michigan, Ann Arbor], P. Schenzel [“On the use of local cohomology in algebra and geometry”, Lect. Summer School Commutative Algebra and Algebraic Geometry, Bellaterra 1996 (Birkhäuser 1998)], and C. Weibel [“An introduction to homological algebra” (1994; Zbl 0797.18001)].

The basic motivations for the introduction under review are the following: 1. The authors feel a challenge for an algebraic introduction to Grothendieck’s local cohomology theory originally invented by the aid of scheme theory and in that form not available yet. – 2. The introduction is designed primarily to graduate students who have some experience of basic commutative and homological algebra. So the approach is homologically based on the fundamental ‘\(\delta\)-functor’ technique. – 3. A large part of the investigations follows algebraic properties of local cohomology, most of them for the first time available in a textbook, e.g., local duality, secondary representations of local cohomology modules, annihilator and finiteness results, graded local cohomology, Hilbert polynomials etc. – 4. From an algebraic point of view the authors illustrate the geometric significance of various aspects of local cohomology, in particular by applications of local cohomology to connectivity, Castelnuovo-Mumford regularity, sheaf cohomology.

The authors expect that the interested reader should be familiar with the basic sections of the books by H. Matsumura [“Commutative ring theory” (1986; Zbl 0603.13001)] and J. J. Rotman [“An introdution to homological algebra” (1979; Zbl 0441.18018)]. Consequently they included expositions about Matlis duality, the indecomposable injective modules, Hilbert polynomials, foundations about \(\mathbb Z\)-graded rings and modules for the use in graded local cohomology theory. Besides of the authors’ approach to the fundamental vanishing theorems on local cohomology, the Lichtenbaum-Hartshorne vanishing theorem a.o. there are very interesting chapters about the annihilation and finiteness theorems on local cohomology, Castelnuovo-Mumford regularity in geometry, connectivity in algebraic geometry, where research results are provided in a textbook form for the first time.

The text is carefully and clearly written. Very often it is completed by examples and easy exercises. They make it easier to a beginner to learn the subject. The connectivity results are – at least for the reviewer – the highlights of the book. The authors’ use of local cohomology leads to proofs of major results involving connectivity, such as Grothendieck’s connectedness theorem, the Bertini-Grothendieck connectivity theorem, the connectedness theorem for projective varieties due to W. Barth, to W. Fulton and W. Hansen, and to G. Faltings, as well as a ring theoretic version of Zariski’s main theorem. By the authors’ intention the characteristic \(p\) methods in local cohomology – introduced by C. Peskine and L. Szpiro [Inst. Hautes Étud. Sci., Publ. Math. 42 (1972), 47-119 (1973; Zbl 0268.13008)] and further developed by M. Hochster and C. Huneke in the notion of tight closure and related research as well as results about big Cohen-Macaulay modules and the rôle of local cohomology in intersection theorems are beyond the scope of the book. For an introduction to this subject the interested reader, prepared by the basics of that book, might and should consult the third part of the book by W. Bruns and J. Herzog, “Cohen-Macaulay rings”, cited above.

The value of the book under review consists in its consequent introductory nature, which may be welcome to the beginners of the subject. By the aid of illuminating examples and exercises the authors shead different colours on several subjects of commutative algebra and algebraic geometry. The interested reader will be guided to research problems connected to local cohomology as well as the power of the methods in algebraic geometry.

The basic motivations for the introduction under review are the following: 1. The authors feel a challenge for an algebraic introduction to Grothendieck’s local cohomology theory originally invented by the aid of scheme theory and in that form not available yet. – 2. The introduction is designed primarily to graduate students who have some experience of basic commutative and homological algebra. So the approach is homologically based on the fundamental ‘\(\delta\)-functor’ technique. – 3. A large part of the investigations follows algebraic properties of local cohomology, most of them for the first time available in a textbook, e.g., local duality, secondary representations of local cohomology modules, annihilator and finiteness results, graded local cohomology, Hilbert polynomials etc. – 4. From an algebraic point of view the authors illustrate the geometric significance of various aspects of local cohomology, in particular by applications of local cohomology to connectivity, Castelnuovo-Mumford regularity, sheaf cohomology.

The authors expect that the interested reader should be familiar with the basic sections of the books by H. Matsumura [“Commutative ring theory” (1986; Zbl 0603.13001)] and J. J. Rotman [“An introdution to homological algebra” (1979; Zbl 0441.18018)]. Consequently they included expositions about Matlis duality, the indecomposable injective modules, Hilbert polynomials, foundations about \(\mathbb Z\)-graded rings and modules for the use in graded local cohomology theory. Besides of the authors’ approach to the fundamental vanishing theorems on local cohomology, the Lichtenbaum-Hartshorne vanishing theorem a.o. there are very interesting chapters about the annihilation and finiteness theorems on local cohomology, Castelnuovo-Mumford regularity in geometry, connectivity in algebraic geometry, where research results are provided in a textbook form for the first time.

The text is carefully and clearly written. Very often it is completed by examples and easy exercises. They make it easier to a beginner to learn the subject. The connectivity results are – at least for the reviewer – the highlights of the book. The authors’ use of local cohomology leads to proofs of major results involving connectivity, such as Grothendieck’s connectedness theorem, the Bertini-Grothendieck connectivity theorem, the connectedness theorem for projective varieties due to W. Barth, to W. Fulton and W. Hansen, and to G. Faltings, as well as a ring theoretic version of Zariski’s main theorem. By the authors’ intention the characteristic \(p\) methods in local cohomology – introduced by C. Peskine and L. Szpiro [Inst. Hautes Étud. Sci., Publ. Math. 42 (1972), 47-119 (1973; Zbl 0268.13008)] and further developed by M. Hochster and C. Huneke in the notion of tight closure and related research as well as results about big Cohen-Macaulay modules and the rôle of local cohomology in intersection theorems are beyond the scope of the book. For an introduction to this subject the interested reader, prepared by the basics of that book, might and should consult the third part of the book by W. Bruns and J. Herzog, “Cohen-Macaulay rings”, cited above.

The value of the book under review consists in its consequent introductory nature, which may be welcome to the beginners of the subject. By the aid of illuminating examples and exercises the authors shead different colours on several subjects of commutative algebra and algebraic geometry. The interested reader will be guided to research problems connected to local cohomology as well as the power of the methods in algebraic geometry.

Reviewer: P.Schenzel (Halle)

##### MSC:

13D45 | Local cohomology and commutative rings |

14B15 | Local cohomology and algebraic geometry |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13Dxx | Homological methods in commutative ring theory |