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Title: Some problems in commutative algebra
Author: Sharp, Rodney Y.
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1969
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In this thesis, the word "ring" will always mean "commutative, (associative,) Noetherian ring with a non-zero multiplicative identity". An arbitrary module M over a ring A determines in a natural, but mildly complicated, fashion a certain complex:-

d-1 d0 dn
C(M): 0 M M0 M1 andhellip; Mn andhellip;

of A-modules and -homomorphisms. This thesis is concerned with the construction of C(M), with some of its properties and uses, and with some natural extensions of this theory. I have called C(M) the Cousin Complex for M: this seems appropriate because it is, in fact, the commutative algebra analogue of the Cousin Complex of andsect;2 of Chapter IV of Hartshorne (12). One can, of course, regard A as a module over itself and construct C(A). The power of the Cousin Complex stems mainly from simple characterizations of Cohen-Macaulay rings and Gorenstein rings in terms of the Cousin Complex. A ring A is Cohen-Macaulay if and only if C(A) is exact, and A is Gorenstein if and only if C(A) provides an injective (resp. minimal infective) resolution for A. Indeed, it was while looking for some sort of natural minimal injective resolution for a Gorenstein ring that I stumbled upon the existence of C(A) for an arbitrary ring A, and it was only later (considerably later, in fact, for my ability to work with algebraic geometry in general and schemes in particular is limited) that I realised C(A) was, in fact, the analogue of the Cousin Complex of the Affine scheme of the ring A as defined in Hartshorne (12). While familiarity with the fundamental ideas of commutative algebra and homological algebra is assumed, a survey of those basic standard results and definitions which are important for the subsequent work is given in the first two Chapters: Chapter 1 is devoted to commutative algebra and Chapter 2 to homological algebra. The main purposes of this are to establish notation and to attempt to leave the way open for comparatively unimpeded progress through the discussion of the Cousin Complex and its applications in Chapters 3-12. The experienced reader may like to begin at Chapter 3 and refer to Chapters 1 and 2 when necessary. The construction and properties of the Cousin Complex provide the subject matter of Chapter 3, and Chapters 4 and 5 contain the characterizations of Cohen-Macaulay rings and Gorenstein rings in terms of the Cousin Complex. I have, of course, made considerable use of the existing literature on Cohen-Macaulay rings and Gorenstein rings, and two papers to which I have frequently referred are Rees (24) and Bass(5). Chapter 4 actually examines the Cousin Complex for Cohen-Macaulay modules: the generalization from rings to modules is, in this case, fairly straightforward, and I have given the theory in the more general case. While several authors have studied Cohen-Macaulay modules (for example, see Chap. IV of Serre (26)), I have found no mention of "Gorenstein module" in the literature, and I have defined Gorenstein modules (over a ring A) as those non-zero, finitely-generated A-modules M for which C(M) provides a minimal injective resolution. In Chapter 6, I have examined some of the properties of Gorenstein modules and developed various characterizations of them: I have looked for characterizations which reduce to some of Bass' characterisations of Gorenstein rings in the particular case when m = A. (See Fundamental Theorem of andsect;1 of Bass (5).) This has not been as easy as eight have been expected, mainly for the following reason. Suppose for the moment that A is a local ring. In (3.3) of (5), Bass showed that, if M is a non-zero, finitely-generated A-module of finite injective dimension, then inj.dim.AM = codhAA (not necessarily codhAM). This result has particularly pleasant consequences when M = A: one is able to deduce that a local ring of finite injective dimension (as a module over itself) must be Cohen-Macaulay. However, in the situation of Bass' result mentioned above, it is not necessarily true that M is Cohen-Macaulay. Consequently, while Bass was able to characterize Gorenstein local rings as those local rings A for which inj.dim.AA is finite, there are non-zero, finitely-generated modules M over a local ring A which are not Gorenstein modules but for which inj.dim.AM is finite. The generalization of this characterization is more complicated and less obvious. Difficulties also arise when one attempts to generalize some of Bass' other characterizations of Gorenstein rings. Bass himself conjectured (also in (5)) that, for a local ring A, there exist non-zero, finitely-generated A-modules M of finite injective dimension only if A is Cohen-Macaulay. I have perhaps added some weight to this conjecture by showing that, if A is local and there exists a Gorenstein A-module, then A has to be a Cohen-Macaulay ring. The remarkably simple characterizations of Cohen-Macaulay rings and Gorenstein rings in terms of the Cousin Complex seem adequate reason for suspecting that the Cousin Complex may help in attempts to classify (commutative Noetherian) rings which are not necessarily Cohen-Macaulay. Chapter 7 contains a few brief skirmishes in this direction. As it seemed appropriate to examine the positions of Gorenstein and Cohen-Macaulay local rings in the general hierarchy of local rings, I have included a brief survey of some well-known special classes of local rings (Regular, Complete Intersections, Gorenstein, and Cohen-Macaulay) together with examples to illustrate the distinctions between these classes. Consequently, only a small part of the work in Chapter 7 is original. Chapter 8 is an illustration of the use of the Cousin Complex. The situation studied in (15) by Iversen is ideally suited for Cousin Complex arguments, and it turns out that the theory of the Cousin Complex yields more information than Iversen was able to obtain. As mentioned above, the construction of the Cousin Complex is mildly complicated. In certain situations it becomes desirable to compare two Cousin Complexes, or variations thereof, and these comparisons become very complicated. I have tried to describe these comparisons as simply and clearly as possible, and I feel that the results justify these somewhat unpleasant computations. One such comparison occurs in Chapter 9. Suppose A is a ring. In Chapter 6 it gradually becomes evident that it would be helpful to compare, for a Gorenstein A-module M and an x andisin; A which is such that xandsdot;M andne; M and x is not a zero-divisor on M, minimal injective resolutions for the A-module M and the A/(x)-module M/xM, and that the existing machinery is inadequate for this purpose. This problem is overcome in Chapter 9 by comparing the Cousin Complexes involved in a more general situation, and then using the fact the the Cousin Complex provides a minimal injective resolution for a Gorenstein module.

Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available