# The Brauer group of commutative rings

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M. Dekker , New York
Brauer groups., Commutative r
Classifications The Physical Object Statement Morris Orzech, Charles Small. Series Lecture notes in pure and applied mathematics ; v. 11 Contributions Small, Charles, 1943- joint author. LC Classifications QA251.3 .O78 Pagination ix, 183 p. ; Open Library OL5060544M ISBN 10 0824762614 LC Control Number 74024669

Buy The Brauer Group of Commutative Rings (Lecture notes in pure and applied mathematics ; v. 11) on FREE SHIPPING on qualified orders The Brauer Group of Commutative Rings (Lecture notes in pure and applied mathematics ; v.

11): Orzech, Morris, Small, Charles: : BooksCited by: 1 The Brauer Group of a Local Ring Throughout this section, Rwill be a commutative Noetherian local ring with maximal ideal m and Aa not necessarily commutative algebra over R. We assume that Ahas an identity element and that the map R!A, r7!r1, identiﬁes Rwith a subring of the center of A.

The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph.

Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for Brand: Springer Netherlands.

Additional Physical Format: Online version: Orzech, Morris. Brauer group of commutative rings.

### Description The Brauer group of commutative rings PDF

New York: M. Dekker, [] (OCoLC) Material Type. Abstract. The Brauer group Br(R) of a commutative ring was introduced by Auslander and Goldman in their paper The Brauer Group of a Commutative Ring, building on earlier work of 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 Author: Benson Farb, R. Keith Dennis. ] THE BRAUER GROUP OF A COMMUTATIVE RING From now on R will denote generically a commutative ring. A ring A together with a (ring) homomorphism of R into the center of A will be called an R-algebra.

We denote by AO the opposite ring of A (i.e. AO consists of a set in one-to-one correspondence with A, the correspondence being indicated by. ] THE BRAUER GROUP OF A COMMUTATIVE RING From now on £ will denote generically a commutative ring. A ring A together with a (ring) homomorphism of £ into the center of A will be called an R-algebra.

We denote by A0 the opposite ring of A (i.e. A0 consists of a set. Azumaya algebras are in many ways near to commutative rings; their explicit structure is on most occasions still mysterious. When trying to make a classification of Azumaya algebras over a given commutative ring, the step to the consideration of the Brauer group and its possible cohomological descriptions is a natural and a fruitful one.

Relative Brauer groups and Picard groups have been introduced in [19] resp. [17]. In both papers we restricted atteention to the Brauer group resp. Picard group of a single couple (R,σ) consisting of a commutative ring R and an indempotent kernel funktor σ in R-mod.

(For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group.) Numerous other instances of a wedding of ring theory and category (albeit a.

abelian extension abelian groups admissible T-isomorphism algebra algebra homomorphism Als,T Amer apply arbitrary automorphism Azumaya Brauer group called canonical clear Clearly commutative R-algebras commutative ring completes the proof complex condition consists Corollary corresponding covariant functor defined definition denote diagram.

Chapter II addresses the Brauer group of a commutative ring, and automorphisms of separable algebras. Chapter III surveys the principal theorems of the Galois theory for commutative rings.

In Chapter IV the authors present a direct derivation of the first six terms of the seven-term exact sequence for Galois cohomology. The Brauer group of a commutative ring is an important invariant of a commutative ring, a common journeyman to the group of units and the Picard group.

Burnside rings of finite groups play an important role in representation theory, and their groups of units and Picard groups have been studied extensively. In this short note, we completely determine the Brauer groups of Burnside rings: they. The Brauer group was generalized from fields to commutative rings by Auslander and Goldman.

Grothendieck went further by defining the Brauer group of any scheme. There are two ways of defining the Brauer group of a scheme X, using either Azumaya algebras over X or projective bundles over X.

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Destination page number Search scope Search Text Search scope Search Text. The Brauer Group of Commutative Rings: Orzech, Morris, Small, Charles: Books - or: Morris Orzech, Charles Small. This book introduces various notions defined in graded terms extending the notions most frequently used as basic ingredients in the theory of Azumaya algebras: separability and Galois extensions of commutative rings, crossed products and Galois cohomology, Picard groups, and the Brauer group.

Buy Separable Algebras over Commutative Rings / Edition 1 by Frank De Meyer, Edward Ingraham | at Barnes & Noble. Books, Toys, Games and much more. Central separable algebras and the brauer group.- Galois theory.- The six-term exact sequence.- Applications and : $Synopsis This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and etale cohomology, and Author: Stefaan Caenepeel. The author, coauthor, editor, or coeditor of over articles, proceedings, book chapters, and books, including Brauer Groups and the Cohomology of Graded Rings, Commutative Algebra and Algebraic Geometry, A Primer of Algebraic Geometry, and Interactions Between Ring Theory and Representations of Algebras (all titles, Marcel Dekker, Inc.), he.$\begingroup\$ I won't insist there is no way to compute the Brauer group of a local field without orders, but "Associative Algebras" by Pierce has this result (section ) and on a cursory check he does not seem to use orders; no orders or maximal orders explicitly in the table of contents, the index, or Chapter I might be overlooking.

The later chapters discuss primitive rings, some representation theory of finite groups, dimension theory of rings, and the Brauer group of a commutative ring.

I have not gone through this later material in detail (yet), so I cannot give comment on s: 2. tion of Brauer spectra for commutative rings, such that their homotopy groups are given by the Brauer group, the Picard group and the group of units.

Then, in the context of structured ring spectra, the same idea leads to two-fold non-connected deloopings of the spectra of units. Lectures on Topics in Algebraic K Theory (PDF P) This note covers the following topics: The exact sequence of algebraic K-theory, Categories of modules and their equivalences, Brauer group of a commutative ring, Brauer-Wall group of graded Azumaya.

The Brauer group for commutative rings in general is recalled in Section 4, while the ﬁnal Section 5 contains the main result and its proof. The following Section 1 reviews the most important means to study the Burnside ring of a ﬁnite group: the ghost map.

1 The ghost map Arguably the most important means to study the Burnside ring A(G) of. putational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two Key Words Commutative ring spectra, derived algebraic geometry, moduli spaces, Azumaya algebras, and Brauer.

The Brauer group of a eld F, classifying central simple algebras over F, plays a critical role in class eld theory. The de nition was generalized by Auslander and Goldman [1] to the case of a commutative ring: the Brauer group of R consists of Morita equivalence classes of Azumaya algebras over R.

modules over PIDs. A discussion of graded k-algebras, for k a commutative ring, leads to tensor algebras, central simple algebras and the Brauer group, exterior algebra (including Grassmann algebras and the binomial theorem), determinants, differential forms, and an.

I’ll call the particular 3-group we get from a commutative ring k k its Brauer 3-group, and I’ll denote it as Br (k) \mathbf{Br}(k). It’s discussed on the n n Lab: there it’s called the Picard 3-group of k k but denoted as Br (k) \mathbf{Br}(k).

A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension ≥ combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic (0, p).We establish the remaining cases by using the.

We attach to any commutative ring R a subgroup of the Brauer group of R, called the Brauer-Galois group of R. An illustration of an open book.

Books. An illustration of two cells of a film strip. Video. An illustration of an audio speaker.