Hiroki Abe

Let R be a commutative Gorenstein ring. We call a Noether R-algebra Λ a Gorenstein R-order provided Λ has Gorenstein dimension zero as an R-module and Λ ≅ Hom _R (Λ, R) as Λ-bimodules. Let Λ be a Gorenstein R-order. We provide several ways to extend Λ to a Gorenstein R-order Δ containing an idempotent e such that Λ ≅ eΔe. In particular, for a finitely generated right Λ-module M having Gorenstein dimension zero as an R-module, setting Δ = End_Λ(M ⊕ Λ), we show that End_Λ(Ω ⁿ M ⊕ Λ) is derived equivalent to Δ for all n ∈ ℤ, and hence if Δ is a Gorenstein R-order then so is every End_Λ(Ω ⁿ M ⊕ Λ), and that Δ is a Gorenstein R-order if M

Let R be a commutative noetherian ring and A a noetherian R-algebra. Take a minimal injective resolution R → I ^• and set . We deal with the case where V ^• is a dualizing complex for A. We show that A itself is a dualizing complex for A if and only if V ^• is isomorphic to a tilting complex in 𝒟(\textMod-A) and that if A is a dualizing complex for A, then induces a self-equivalence of 𝒟 ^b(\textmod-A).

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^• ∊ K ^b(𝒫 _A ) with Hom _K(Mod−A)(P ^•,P ^•[i]) = 0 for i ≠ 0 and add(P ^•) = add(νP ^•) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.