Localization and Flatness in Quantale Theory

A commutative ring A is said to be algebraically closed if every finite system of polynomial equations and inequations in one or more variables with coefficients in A which has a solution in some (commutative) extension of A already has a solution in A. Abraham Robinson's study of model-theoretic forcing has provided powerful new tools for the study of algebraically closed structures in general, and will be applied here to the study of algebraically closed commutative rings. Familiarity with the model-theoretic notions connected with the study of algebraically closed structures is assumed; for background consult [1], [2], and [3].Our main results are the following:1. The theory of commutative rings with identity has no model companion in the sense of Robinson.2. The Hilbert Nullstellensatz, suitably formulated for the class of algebraically closed commutative rings, holds for finitely generated polynomial ideals but fails for certain infinitely generated polynomial ideals.3. If A is algebraically closed, then A/rad A need not be algebraically closed as a semiprime ring: If A is finitely generic then A/rad A is algebraically closed as a semiprime ring, but if A is infinitely generic then A/rad A is not algebraically closed as a semiprime ring.

The Brauer group Br(R) of a commutative ring was introduced by Auslander and Goldman in their 1960 paper The Brauer Group of a Commutative Ring, building on earlier work of Azumaya. This group coincides with the “classical” Brauer group (cf. Chapter 4) in the case when R is a field. One of the points of extending the theory to rings is that one can relate Brauer groups of fields to Brauer groups of related rings in exact sequences; one then hopes that this will help compute the classical Brauer group. The Brauer group of a commutative ring is also part of a Galois theory of commutative rings. For more on these matters, the reader may consult Galois Theory and Cohomology of Commutative Rings by Chase, Harrison and Rosenberg, The Brauer Group of Commutative Rings by Orzech and Small, Separable Algebras Over Commutative Rings by DeMeyer and Ingraham, or the paper of Auslander and Goldman quoted above.

The notion of content is used to solve certain problems. In the first part, we show that the structural morphism of a content algebra (see the paper of D.E. Rush [22] ) is spectrally open, under mild hypothesis. We show also that a flat module is universally content if and only if it is a Mittag-Leffler’s module in the sense of f2ll . In the second part, using content, we exhibit a kind of localization of a commutative ring A, attached to eyery subset X of Spec(A) i.e. a flat morphism A→ X(A). We can thus show that every quasi-compact, stable under generization subset of a spectra is a spectral image under a flat morphism, in a canonical way. We can also give in certain cases an elementary construction of the maximal flat injective epimor-phism of a ring. Suppose that A is a Noetherian ring and consider pro-perties of Noetherian rings such as factoriality, normality and so on. Let X be the set of prime ideals of A at which A has the property. If X is stable under generization, the flat morphism A→ X(A) verifies hin general the ring X(A) has lornlly the property and a prime ideal P of A has a prime ideal lying over in X(A) if and only if pthe ring has the property at P.

It is proved that <TEX>$k[X_1,{\ldots},X_v ]$</TEX> localized at the ideal (<TEX>$X_1,{\ldots},X_v$</TEX> ), where k is a field and <TEX>$X_1,{\ldots},X_v$</TEX> indeterminates, is not weakly quasi-complete for <TEX>$v{\geq}2$</TEX>, thus proving a conjecture of D. D. Anderson and solving a problem from "Open Problems in Commutative Ring Theory" by Cahen, Fontana, Frisch, and Glaz.

The theory of Noetherian rings may be said to have begun with Goldie's 1958 paper. Since then the theory has steadily gathered strength. By now, various methods and results from the theory of commutative Noetherian rings have been adapted to non-commutative Noetherian rings. The theory has also contributed to and benefitted from developments in three nearby areas: P.I. rings, enveloping algebras of Lie algebras, and group rings of polycyclic-by-finite groups.

For commutative algebras there are three important homology theories, Harrison homology, Andre-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quillen equivalent to the model category of pointed simplicial commutative monoids in symmetric sequences.

It is proven that each commutative arithmetical ring $R$ has a finitistic\nweak dimension $\\leq 2$. More precisely, this dimension is 0 if $R$ is locally\nIF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.\n

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Principal ideal theorem for Noetherian P. I. rings

On certain problems of the general theory of commutative normed rings

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider We examine these annihilators for an arbitrary local ring when i is the cohomological dimension of I. When the cohomological dimension coincides with the dimension of the ring, we provide an explicit description of In general, we examine its dimension over R and conjecture that for i > 0.

Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative N-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative N-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.

The model completion of the theory of commutative regular rings

Introduction.Given a field k and a fc-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a [[/]]-algebra structure on the module A [[/]].One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual A-[[r]]-algebra structure on .4[[*]].Such algebras are said to be rigid.If A is a commutative /c-algebra, we may restrict our attention to those deformations of A which are commutative.This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11].The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k:(1) H2(A, M)s = 0 for all ^-modules M.(2) H\A,f=0.(3) A is a rigid /r-algebra in the commutative deformation theory.(4) The module of differentials of A over k is a projective /4-module.(5) A is a finite product of (not necessarily algebraic) separable extension fields of k.We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65].Condition (1) may be expressed by saying that A is a formally smooth fc-algebra [9, 19.4.4].When A is a field, Gerstenhaber has shown that (1), (2), and ( 5) are equivalent and he conjectured that these conditions were equivalent to (3).We show that (1), ( 2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5).Moreover, if A is artinian, then (1) and ( 5) are equivalent (Proposition 5.9).In another direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a).If in addition A is an integral domain with a separable quotient field, then (1), ( 2), (3), and (4) are equivalent (Theorem 6.8c).In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p.In addition, we consider the following condition on G and k:(6) G has no element of order p = char (k).

Localization and ideal theory in iterated differential operator rings