Homological questions in local algebra. (English) Zbl 0786.13008

London Mathematical Society Lecture Note Series. 145. Cambridge: Cambridge University Press. xiv, 308 p. (1990).
The homological questions of the title originated in a celebrated set of lecture notes by Serre, where he formulated an abstract version of the classical notion of intersection multiplicity. Some of the questions raised remain to this day unresolved. One of its side-effects was the vision – shared by pioneers such as Auslander, Bass, Buchsbaum, Kaplansky, and others – that many of the properties of modules over regular local rings should hold true for all local rings, so long as the modules have finite projective resolutions. As a group of problems, they became a central focus in commutative algebra and probably were the subject of the most sustained effort ever in the field. With the thesis of Peskine and Szpiro, with its dazzling plethora of new methods, grounded in local cohomology, several of these conjectures were solved in finite characteristics. Not much later, Hochster brought to bear entirely new devices of his creation, culminating in the construction of big Cohen-Macaulay modules. More conjectures fell, but others were spawned left and right.
This book is a presentation of these exciting developments. The contagious enthusiasm of the author for his topic is evident on almost every page. To deal with the required technical background and delicate constructions, he chose to present a detailed exposition of core homological algebra. This occupies about half of the text but is well worth the effort. The overall exposition is crisp, and its detailed bibliographical references should be a boon to all.
But the saga does not end. The solution by Roberts and Gillet and Soulé of one of Serre’s original questions, and of a full intersection conjecture by the former, are mentioned but not fully discussed. Furthermore, the freshly-minted theory of tight closure, due to Hochster and Huneke, in addition to having its effect on these questions, is beginning to show homological features of rings of which we were previously unaware. It is to be hoped that the author, in his inimitable style, returns to this topic to inform us of these new happenings.


13Hxx Local rings and semilocal rings
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13C14 Cohen-Macaulay modules
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra