Generalized crossed products applied to maximal orders, Brauer groups and related exact sequences

In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C1 ⊆ C2 could be related by a tidy exact sequenceIt was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].

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Finite generalized crossed products over tame and maximal orders

Journal of Algebra ◽

10.1016/0021-8693(86)90096-7 ◽

1986 ◽

Vol 101 (1) ◽

pp. 61-68 ◽

Cited By ~ 11

Author(s):

E Nauwelaerts ◽

F Van Oystaeyen

Keyword(s):

Crossed Products ◽

Maximal Orders

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Six-term exact sequences for smooth generalized crossed products

Journal of Noncommutative Geometry ◽

10.4171/jncg/124 ◽

2013 ◽

Vol 7 (2) ◽

pp. 499-524 ◽

Cited By ~ 1

Author(s):

Olivier Gabriel ◽

Martin Grensing

Keyword(s):

Crossed Products ◽

Exact Sequences

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Exact Sequences for the Kasparov Groups of Graded Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-013-x ◽

1985 ◽

Vol 37 (2) ◽

pp. 193-216 ◽

Cited By ~ 13

Author(s):

George Skandalis

Keyword(s):

Exact Sequence ◽

Free Groups ◽

Crossed Products ◽

Graded Algebras ◽

Completely Positive ◽

Exact Sequences

In [11] G. G. Kasparov defined the “operator K-functor” KK(A, B) associated with the graded C*-algebras A and B. If the algebras A and B are trivially graded and A is nuclear he proves six term exact sequence theorems. He asks whether this extends to the graded case.Here we prove such “six-term exact sequence” results in the graded case. Our proof does not use nuclearity of the algebra A. This condition is replaced by a completely positive lifting condition (Theorem 1.1).Using our result we may extend the results by M. Pimsner and D. Voiculescu on the K groups of crossed products by free groups to KK groups [15]. We give however a different way of computing these groups using the equivariant KK-theory developed by G. G. Kasparov in [12]. This method also allows us to compute the KK groups of crossed products by PSL2(Z).

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Maximal orders containing local crossed-products

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(88)90100-4 ◽

1988 ◽

Vol 50 (3) ◽

pp. 211-222 ◽

Cited By ~ 2

Author(s):

A. Chalatsis ◽

Th. Theohari-Apostolidi

Keyword(s):

Crossed Products ◽

Maximal Orders

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Exact sequences for relative Brauer groups and Picard groups

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(83)90076-2 ◽

1983 ◽

Vol 28 (1) ◽

pp. 93-108 ◽

Cited By ~ 8

Author(s):

A. Verschoren

Keyword(s):

Brauer Groups ◽

Picard Groups ◽

Exact Sequences

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Equivalence Classes of Maximal Orders

Nagoya Mathematical Journal ◽

10.1017/s002776300001271x ◽

1968 ◽

Vol 31 ◽

pp. 131-171 ◽

Cited By ~ 1

Author(s):

Susan Williamson

Keyword(s):

Valuation Ring ◽

Equivalence Classes ◽

Brauer Group ◽

Crossed Products ◽

Quotient Field ◽

Maximal Orders ◽

Rank One

Let k denote the quotient field of a complete discrete rank one valuation ring R. The purpose of this paper is to establish a relationship between the Brauer group of k and the set of maximal orders over R which are equivalent to crossed products over tamely ramified extensions of R.

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