On the Dimension of Modules and Algebras, I

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

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“…This theorem may also be deduced from a standard spectral sequence (see (5) on p. 346 of [4], or Proposition 2 of [5]). 2.…”

Section: )-»272(g U(s))->©(£) ->(B(s) Is Exact (U(s) Is the Group Omentioning

confidence: 90%

“…We associate to/ the element ££HomÄ (5,5) given by 2?= 5(£/(cr)w(r). Explicitly, £(x) = £/(cr)cr(x), for xQS.…”

Section: Propositionmentioning

confidence: 99%

The Brauer group of a commutative ring

Auslander¹,

Goldman²

1960

Trans. Amer. Math. Soc.

520

323

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“…Our lemma (as well as Proposition 2 below) follows readily from the consideration of "minimal resolution" (i.e. a projective resolution consisting of "minimal homomorphisms") (Eilenberg-Nakayama [4], Eilenberg [2]). But, since we are dealing here with a very simple situation, we shall give a direct replaced by an isomorphic one, the sequence remains connected.…”

mentioning

confidence: 99%

On the Dimension of Modules and Algebras, VII. Algebras with Finite-Dimensional Residue-Algebras

Jans¹,

Nakayama²

1957

Nagoya Mathematical Journal

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It was shown in Eilenberg-Nagao-Nakayama [3] (Theorem 8 and § 4) that if Ω is an algebra (with unit element) over a field K with (Ω : K) < °° and if the cohomolgical dimension of Ω } dim 42, is ^ 1, then every residue-algebra of Ω has a finite cohomological dimension. In the present note we prove a theorem of converse type, which gives, when combined with the cited result, a rather complete general picture of algebras whose residue-algebras are all of finite cohomological dimension. Namely, if A is an algebra over a field K withwhere N is the radical of Λ, then A is a homomorphic image of an algebra Ω over K with (Ω : K) < oo such that dim 42 ^ 1.We may further impose the conditionwhere M is the radical of Ω f and with this additional condition the algebra Ω and the homomorphism Ω -> A are determined uniquely up to an isomorphism.Thus, algebras with cohomological dimension ^ 1 are in a sense "prototypes" for algebras with finite-dimensional residue-algebras. The construction of Ω and the homomorphism Ω -» A is essentially what was employed by Hochschild [5,6] in connection with his notion of "maximal algebra" and by Jans[3] as free algebras.We shall start with semi-primary rings (in the sense in [3]). For them and for their global dimensions we shall prove a theorem which is quite similar as above but which assumes an additional condition on "splitting".

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“…By induction on t. If t=0, then since A is left and right noetherian A satisfies the minimum condition on left ideals and on right ideals, by [9,Theorem 18]. Therefore Rad A is nilpotent.…”

Section: -^S-+e'-+e'is^0mentioning

confidence: 99%

Maximal Orders over Regular Local Rings of Dimension Two

Ramras¹

1969

Transactions of the American Mathematical Society

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MARK RAMRASOIntroduction. Auslander and Goldman [4] have studied maximal orders over discrete valuation rings. In this paper, relying heavily on their results, we investigate maximal orders over regular local rings of dimension two. The two theories are rather different. We begin §5 by listing three theorems from [4]. Then we prove a partial generalization of two of them in dimension two (Theorem 5.4) and exhibit two examples which show that the remaining statements do not generalize.In [4] a structure theorem is proved for maximal orders over discrete valuation rings which is the analogue of the Wedderburn structure theorem for simple artin rings. In §6 we extend this theorem, in a weakened form, to maximal orders over an arbitrary integrally closed noetherian domain R. We sharpen this somewhat when R is regular local of dimension two and the maximal order is well behaved. Various other structure theorems are given in this section.The first four sections are devoted to building up homological machinery, most of which is applied to orders in the last two sections. The setting is fairly general : (R, m) is a commutative noetherian local ring with maximal ideal m, and A is an /¿-algebra which is finitely generated as an /¿-module. The best results (Theorems 1.10 and 2.16) are obtained when A is quasi-local (i.e., A/Rad A is a simple artin ring, where Rad A is the Jacobson radical of A). Such rings behave very much like commutative local rings.The author wishes to express his gratitude to Professor Maurice Auslander for his many helpful suggestions and his patient supervision of the research for this paper, which is the major portion of the author's doctoral dissertation. Many thanks also to Professor David Buchsbaum and Dr. Silvio Greco for useful and stimulating conversations.Notations and conventions. Throughout this paper all rings have units and all modules are unitary. We use the abbreviations pd, inj dim, and gl dim for proj'ective, injective, and global dimension, respectively. Only when there is a possible leftright ambiguity will we write l.pd or r.inj dim, etc.R is a commutative noetherian ring and A is an /¿-algebra which is finitely generated as an /¿-module. For the first four sections R is local with maximal ideal m. By R we will mean the completion of R in the ra-adic topology, and if M is an

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On the Dimension of Modules and Algebras, I

Abstract: In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension <=n. In a subsequent paper C4U Eilenberg gave an

Cited by 56 publications

References 4 publications

The Brauer group of a commutative ring

The Brauer group of a commutative ring

On the Dimension of Modules and Algebras, VII. Algebras with Finite-Dimensional Residue-Algebras

Maximal Orders over Regular Local Rings of Dimension Two

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