Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7. Regular rings 8. Flatness revisited 9. Derivations 10. I-smoothness 11. Applications of complete local rings Appendices References Index.

Commutative Ring Theory

This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Foundations of Module and Ring Theory : A Handbook for Study and Research

This book provides a careful and detailed algebraic introduction to Grothendieck’s local cohomology theory, and provides many illustrations of applications of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo–Mumford regularity, the Fulton–Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.

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Local Cohomology: An Algebraic Introduction with Geometric Applications

Introduction. If P is a ring and M a left P-module, then homological algebra attaches three dimensions to M, projective, weak, and injective(1)By taking the supremum of one of these dimensions as M ranges over all left P-modules, one obtains one of the left \"global\" dimensions of R. Auslander and Buchsbaum [3] and, subsequently, Serre [14], found it relevant and fruitful, in the study of commutative Noetherian rings, to introduce a \"finitistic global\" dimension defined by restricting the supremum of projective dimensions to finitely generated modules of finite projective dimension. The impressive theory developed by these authors prompted Kaplansky to consider, for general commutative rings, a similar finitistic dimension (waiving the restriction, in the above, to finitely generated modules), and, in a seminar at the University of Chicago (1958), he proved the theorem below, characterizing commutative rings R for which this dimension vanishes. This result is the origin of the present paper. Definition. If N is an ideal in a ring P, we say that N is left T-nilpotent i\"T\" ior transfinite) if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0. iRight T-nilpotence requires instead that a„ • • • oi = 0.)

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Finitistic dimension and a homological generalization of semi-primary rings

1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.

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Representation Theory of Artin Algebras

The notions of torsion and torsion freeness have played a very important role in module theory--particularly in the study of modules over integral domains. Furthermore, the use of homological techniques in this connection has been well established. It is the aim of this paper to extend these techniques and to show that this extension leads naturally to several new concepts (e.g. k-torsion freeness and Gorenstein dimension) which are useful in the classification of modules and rings.

Stable module theory

Applications of contravariantly finite subcategories

Introduction. This paper contains the foundations of a general theory of separable algebras over arbitrary commutative rings. Of the various equivalent conditions for separability in the classical theory of algebras over a field, there is one which is most suitable for generalization; we say that an algebra A over a commutative ring £ is separable if A is a projective module over its enveloping algebra Ae = A®fiA°. The basic properties of separable algebras are developed in the first three sections. The results obtained show that a considerable portion of the classical theory is preserved in our generalization. For example, it is proved that separability is maintained under tensor products as well as under the formation of factor rings. Furthermore, an £-algebra A is separable over £ if, and only if, A is separable over its center C and C is separable over £. This fact shows that the study of separability can be split into two parts: commutative algebras and central algebras. The purely commutative situation has been studied to some extent by Auslander and Buchsbaum in [l]. The present investigation is largely concerned with central algebras. In the classical case, an algebra which is separable over a field K, and has K for its center, is simple. One cann'ot expect this if the center is not a field; however, if A is central separable over £, then the two-sided ideals of A are all generated by ideals of £. In the fourth section we consider a different aspect of the subject, one which is more analogous to ramification theory. If A is an algebra over a ring £, the homological different §(A/£) is an ideal in the center C of A which essentially describes the circumstances under which A®rRv is separable over £(>, when p is a prime ideal of £. (Suitable finiteness conditions must be imposed in the statement of these theorems.) The general question of ramification in noncommutative algebras is only touched on in the present paper; various arithmetic applications will be treated in another publication. In the classical theory of central simple algebras, the full matrix algebras have a special significance. The proper analogue of the full matrix algebra in the present context is the endomorphism ring Homje(£, £) of a finitely generated projective £-module £. It is easy to show for such a module £ that rlomR(E, E) is separable over £, and central if £ is faithful. By analogy with the classical theory, we introduce an equivalence relation between central separable algebras over a ring £, under which the equivalence classes form

https://www.jstor.org/stable/pdfplus/1993378.pdf

Maurice Auslander

Papers

Representation Theory of Artin Algebras

Stable module theory

Applications of contravariantly finite subcategories

The Brauer group of a commutative ring

Preprojective modules over artin algebras