Crossed product algebras defined by separable extensions

Lundström, Patrik

doi:10.1016/j.jalgebra.2004.09.007

Cited by 16 publications

(16 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [13], the second author of the present article suggested an answer to this question in the case when L/K is a finite separable field extension. In doing so he was naturally lead to structures which no longer, in any natural sense, fitted into the framework of group graded rings, but instead suited well in the context of rings graded by groupoids.…”

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

Object-unital groupoid graded rings, crossed products and separability

Cala

Lundström

Pinedo

2020

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation.

show abstract

Section: Introductionmentioning

confidence: 94%

“…Let L/K be a finite separable (not necessarily normal) field extension. The corresponding crossed product construction from [13] runs as follows. Let N be a normal closure of L/K and let G denote the Galois group of N/K.…”

Section: Introductionmentioning

confidence: 99%

Object-unital groupoid graded rings, crossed products and separability

Cala

Lundström

Pinedo

2020

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…dim R. Equalities hold if R is a separable extension over S. Now we apply the previous results to investigate homological dimensions of crossed products. Similar techniques have been used to explore other properties of crossed product in [17]. Suppose that G is a finite group and let H G be a subgroup.…”

Section: (1) the Induction And Restriction Functors Induce Functors Bmentioning

confidence: 99%

Homological Dimensions of Crossed Products

Li¹

2016

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. In this paper we consider several homological dimensions of crossed products A σ α G, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of A σ α G are classified: global dimension of A σ α G is either infinity or equal to that of A, and finitistic dimension of A σ α G coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S G, we show that A and A σ α G share the same homological dimensions under extra assumptions, extending the main results in [15,16].

show abstract

“…Now we recall some old and define some new notions from the theory of category graded rings (also see e.g. [14], [15], [16], [19], [20] or [21]). Let R be a ring and Γ a category.…”

Section: Category Filtered Ringsmentioning

confidence: 99%

Good Magma Gradings on Rings

Lundström

2014

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

Suppose that G and H are magmas and that R is a strongly G-graded ring. We show that there is a bijection between the set of elementary (nonzero) H-gradings of R and the set of (zero) magma homomorphisms from G to H. Thereby we generalize a result by Dăscălescu, Năstăsescu and Rios Montes from group gradings of matrix rings to strongly magma graded rings. We also show that there is an isomorphism between the preordered set of elementary (nonzero) H-filters on R and the preordered set of (zero) submagmas of G × H. These results are applied to category graded rings and, in particular, to the case when G and H are groupoids. In the latter case, we use this bijection to determine the cardinality of the set of elementary H-gradings on R.

show abstract

Crossed product algebras defined by separable extensions

Cited by 16 publications

References 16 publications

Object-unital groupoid graded rings, crossed products and separability

Object-unital groupoid graded rings, crossed products and separability

Homological Dimensions of Crossed Products

Good Magma Gradings on Rings

Contact Info

Product

Resources

About