Preface

Summary

One can take the view that local cohomology is an algebraic child of geometric parents. J.-R Serre's fundamental paper ‘Faisceaux algébriques cohérents’ [56] represents a cornerstone of the development of cohomology as a tool in algebraic geometry: it foreshadowed many crucial ideas of modern sheaf cohomology. Serre's paper, published in 1955, also has many hints of themes which are central in local cohomology theory, and yet it was not until 1967 that the publication of R. Hartshorne's ‘Local cohomology’ Lecture Notes [18] (on A. Grothendieck's 1961 Harvard University seminar) confirmed the effectiveness of local cohomology as a tool in local algebra.

Since the appearance of the Grothendieck-Hartshorne notes, local cohomology has become indispensable for many mathematicians working in the theory of commutative Noetherian rings. But the Grothendieck-Hartshorne notes certainly take a geometric viewpoint at the outset: they begin with the cohomology groups of a topological space X with coefficients in an Abelian sheaf on X and supports in a locally closed subspace.

In the light of this, we feel that there is a need for an algebraic introduction to Grothendieck's local cohomology theory, and this book is intended to meet that need. Our book is designed primarily for graduate students who have some experience of basic commutative algebra and homological algebra; for definiteness, we have assumed that our readers are familiar with many of the basic sections of H. Matsumura's [35] and J. J. Rotman's [52]. Our approach is based on the fundamental ‘δ-functor’ techniques of homological algebra pioneered by Grothendieck, although we shall use the ‘connected sequence’ terminology of Rotman (see [52, pp. 212–214]).