Graded equivalence theory with applications

Haefner, Jeremy

doi:10.1016/s0021-8693(05)80009-2

Cited by 18 publications

(18 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This also shows that for 46) where e i j (x) is a matrix with x in the i j-position and zero elsewhere.…”

Section: Denoting the Graded Matrix Ringmentioning

confidence: 60%

“…Theorem 2.3.8 shows the equivalence Gr-A ≈ gr Gr-B induces an equivalence Mod-A ≈ Mod-B. Haefner in [46] observed that Gr-A ≈ gr Gr-B induces other equivalences between different "layers" of grading. We briefly recount this result here.…”

Section: Remark 2312 Grmentioning

confidence: 99%

“…In [46] Haefner shows that, for any two Γ-graded equivalent rings A and B and for any subgroup Ω of Γ, there are equivalences between the categories Gr Γ/Ω -A and Gr Γ/Ω -B. In fact, Haefner works with an arbitrary (non-abelian) group Γ and any subgroup Ω.…”

Section: Remark 2312 Grmentioning

confidence: 99%

“…The decomposition is called an Γ/Ω-grading of M. With the abuse of notation we denote the category of Γ/Ω-graded right A-modules with Gr Γ/Ω -A. Then in [46] it was shown that, Gr…”

Section: Remark 2312 Grmentioning

confidence: 99%

See 3 more Smart Citations

Graded Rings and Graded Grothendieck Groups

Hazrat

2016

129

185

View full text Add to dashboard Cite

This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

show abstract

“…This also shows that for 46) where e i j (x) is a matrix with x in the i j-position and zero elsewhere.…”

Section: Denoting the Graded Matrix Ringmentioning

confidence: 60%

Section: Remark 2312 Grmentioning

confidence: 99%

Section: Remark 2312 Grmentioning

confidence: 99%

“…The decomposition is called an Γ/Ω-grading of M. With the abuse of notation we denote the category of Γ/Ω-graded right A-modules with Gr Γ/Ω -A. Then in [46] it was shown that, Gr…”

Section: Remark 2312 Grmentioning

confidence: 99%

See 2 more Smart Citations

Graded Rings and Graded Grothendieck Groups

Hazrat

2016

129

185

View full text Add to dashboard Cite

show abstract

“…In addition, this equivalence preserves, in a certain sense, the grading of the two orders. This is an example of what is called a graded equivalence; see [2] for more information on graded equivalences.…”

Section: Proofmentioning

confidence: 99%

Strongly Graded Hereditary Orders

Haefner¹,

Pappacena²

2001

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

Abstract. Let R be a Dedekind domain with global quotient field K. The purpose of this note is to provide a characterization of when a strongly graded R-order with semiprime 1-component is hereditary. This generalizes earlier work by the first author and G. Janusz in (J. Haefner and G. Janusz, Hereditary crossed products, Trans. Amer. Math. Soc. 352 (2000), 3381-3410).Recall that, for a Dedekind domain R with quotient field K, an R-order in a separable K-algebra A is a module-finite R-algebra Λ, contained in A, such that KΛ = A. For a group G, we say that the R-order Λ is strongly G-graded provided there is a decomposition Λ = ⊕ g∈G Λ g with Λ g Λ h = Λ gh for all g, h ∈ G. If 1 denotes the identity element of G, then Λ 1 is a subring of Λ, which we denote by ∆. We write Λ = ∆(G) to indicate that Λ is strongly G-graded with identity component ∆. In this note, we consider the following problem:The hereditary problem for strongly graded orders: Determine necessary and sufficient conditions on G, the grading imposed by G, and ∆ to ensure that Λ is hereditary. Our general solution to this problem appears in Theorem 7. The idea of the proof is to use Morita theory to reduce to the case where Λ is a crossed product and then apply a result of [4], which we describe next.Recall that a strongly G-graded R-order Λ = ∆(G) is a crossed product order provided for each g ∈ G, Λ g ∼ = ∆ as left ∆-modules. In this case, there exist u g ∈ Λ * (the unit group of Λ) such that Λ g = ∆u g for all g ∈ G. Moreover, there exist a group homomorphism α : G → Aut R (∆) (the "action of G on ∆") and a cocycle τ ∈ Z 2 (G, R * ) (the "twisting of the action of G") such that the multiplication in Λ is given by u g δ = α(g)(δ)u g for δ ∈ ∆, and u g u h = τ (g, h)u gh . (See [6] for more details on this construction.)If ∆(G) is a crossed product order with action α, we say that a subgroup H of G acts as central outer automorphims of ∆ provided α(H) ∩ Inn(∆) = 1. The main result of [4] is that, if Λ = ∆(G) is a crossed product order, then Λ is hereditary if and only if ∆ is hereditary and, for each maximal ideal m of R containing a prime divisor p of |G|, any p-Sylow subgroup of G acts as central outer automorphisms of ∆ m . Definition 1. Given a strongly G-graded ring Λ = ∆(G) and g ∈ G, we say g is inner on ∆ provided Λ g ∼ = ∆ as ∆-bimodules. Otherwise, g is outer on ∆. For a Date: December 9, 2017. 1991

show abstract

Reduction Techniques for Strongly Graded Rings and Finite Representation Type

Haefner

1997

Journal of Algebra

View full text Add to dashboard Cite

DEDICATED TO PROFESSOR IRVING REINER WHOSE RESEARCH, BOOKS, AND STUDENTS HAVE INSPIRED SO MANY OF USWe present reduction techniques for studying the category of lattices over strongly graded orders. In particular, we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring.ᮊ 1997 Academic Press

show abstract

Graded equivalence theory with applications

Cited by 18 publications

References 17 publications

Graded Rings and Graded Grothendieck Groups

Graded Rings and Graded Grothendieck Groups

Strongly Graded Hereditary Orders

Reduction Techniques for Strongly Graded Rings and Finite Representation Type

Contact Info

Product

Resources

About