Strongly Graded Hereditary Orders

Abstract

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

Crossed product algebras defined by separable extensions

“…Then ν is an A-bimodule map satisfying (10). In fact, if a ∈ A γ , for some γ ∈ C, then, by (17), we get that…”

“…In Sections 4 and 5, we use the results of Sections 2 and 3 to prove Theorems 4-6. For more results concerning group graded rings and modules see [4][5][6][7]10,16].…”

Epsilon-strongly graded rings, separability and semisimplicity

“…A lot of work has been devoted to the question of when ring extensions are separable (see e.g. [1], [14], [2], [3], [7], [9], [10], [15], [16], [19], [21], [23] and [24]). One reason for this intense interest is that some properties of the ground ring R automatically are inherited by S, such as semisimplicity and hereditarity (see e.g.…”