Mitsuo Hoshino

On t-structures and torsion theories induced by compact objects

Abstract First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P · of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P · to be a tilting complex. Finally, in the case of artin algebras, we give a one-to-one correspondence between tilting complexes of term length two and torsion theories with certain conditions.

On Derived Equivalences for Selfinjective Algebras

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.

Tilting complexes associated with a sequence of idempotents

First, we show that a certain sequence of idempotents e0,e1,…,el in a ring A defines a tilting complex P• for A of term length l+1 and that there exists a sequence of rings B0=A,B1,…,Bl=EndK(Mod-A)(P•) such that for any 0⩽i<l, Bi+1 is the endomorphism ring of a tilting complex for Bi of term length two defined by an idempotent. Next, in the case of A being a finite dimensional algebra over a field, we provide a construction of a two-sided tilting complex corresponding to P•. Simultaneously, we provide a sufficient condition for an algebra B containing A as a subalgebra to be derived equivalent to A.

An elementary construction of tilting complexes

Abstract Let A be an artin algebra and e ∈ A an idempotent with add( eA A )=add( D ( A Ae )). Then a projective resolution of Ae eAe gives rise to tilting complexes {P(l) • } l⩾1 for A , where P ( l ) • is of term length l +1. In particular, if A is self-injective, then End K ( Mod- A) (P(l) • ) is self-injective and has the same Nakayama permutation as A . In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes { T (2 l ) • } l ⩾1 for A , where T (2 l ) • is of term length 2 l +1. In particular, if eAe is of Loewy length two, then we get tilting complexes { T ( l ) • } l ⩾1 for A , where T ( l ) • is of term length l +1.

Strongly quasi-Frobenius rings

Recall that a ring A is called quasi-Frobenius if it is artinian and self-injective on both sides (see e.g. 17, 81, [2], [3] and [6]). In this note, following Yukimoto [ i l l , we call a ring A strongly quasi-Frobenius if End,(P) is a quasi-Frobenius ring for every finitely generated non-zero projective right A-module P. Our aim is to establish a structure theorem for strongly quasi-Frobenius rings. Note that a strongly quasi-Frobenius ring is quasi-Frobenius, and that a ring A is strongly quasi-Frobenius if and only if so is its basic ring. Throughout this note, rings are associative rings with identity and modules are unitary modules. Sometimes, we use the notation J (resp. X,) to signify that the module X considered is a left (resp. right) A-module. Let A be a quasi-Frobenius ring and { e , , ..-, en} a basic set of orthogonal local idempotents in A, i.e., the ei are orthogonal local idempotents in A such that (Zin=, e i ) ~ ( Z t n = , e,) is a basic ring of A. Then the Nakayama permutation of A is

Gorenstein Orders Associated with Modules

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 = EndΛ(M ⊕ Λ), we show that EndΛ(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 EndΛ(Ω n M ⊕ Λ), and that Δ is a Gorenstein R-order if M 𝔭 is projective as a right Λ𝔭-module for every prime ideal 𝔭 of R with ht 𝔭 ≤ 1 and , 1 ≤ i ≤ ht 𝔭 −2, for every prime ideal 𝔭 of R with ht 𝔭 ≥3.

Extension closed reflexive modules

Throughout this note, R stands for a ring with identity, and all modules are unital R-modules. In this note, we ask when the class of all finitely generated reflexive (torsionless) left modules is closed under extensions. This is not always the case, even if R is left and right noetherian (cf. Bass [2, Lemma 3.2] and [3, p. 12]). As will be shown, in case R is a commutat ive noetherian local domain of Krull dimension two, R is a CohenMacaulay ring whenever the class of the finitely generated reflexive modules is closed under extensions. Our main aim is to show that if R is a left and right noetherian ring then both the class of all finitely generated torsionless left modules and the class of all finitely generated reflexive left modules are closed under extensions if and only if for i = 1, 2 the functor Ext , 1 (Ext~ ( , R), R) vanishes on the finitely generated right modules. Note that the latter condition is satisfied if R is a commutat ive Gorenstein ring (see Bass [3]). We will show also that if R is a left and right noetherian ring with a minimal injective resolution 0 ~ RR ~ Eo ~ E1 , . . . such that weak dim E l < i + 1 for i = 0, 1 then both the class of finitely generated torsionless left modules and the class of finitely generated reflexive left modules are closed under extensions. In the proof, it will be shown that if R is a left and right noetherian ring with inj dim RR _--< 2 then the class of the finitely generated reflexive left modules is closed under extensions. On the other hand, even if R is a left and right noetherian ring with inj dim RR = inj dim R R < 2, the class of all finitely generated torsionless left modules is not necessarily closed under extensions. In what follows, we denote by ( )* both the R-dual functors, and for a given module M we denote by eu: M ~ M** the usual evaluation map and by E(M) the injective envelope of M. Recall that a module M is said to be torsionless if eu is a monomorph i sm and to be reflexive if eu is an isomorphism. Note that a module M is torsionless if and only if it is cogenerated by R. We denote by mod R (resp. mod R ~ the category of the finitely presented left (resp. right) modules. Note that if R is left noetherian then every finitely generated left module is finitely presented.

Group-graded and group-bigraded rings

Let I be a nontrivial finite multiplicative group with the unit element e and A = ⨁x∈I Ax an I-graded ring. We construct a Frobenius extension Λ of A and study when the ring extension A of Ae can be a Frobenius extension. Also, formulating the ring structure of Λ, we introduce the notion of I-bigraded rings and show that every I-bigraded ring is isomorphic to the I-bigraded ring Λ constructed above.

Crossed products for matrix rings

ABSTRACT Let R be a ring and n≥2 an integer. We provide a systematic way to define new multiplications on Mn(R), the ring of n×n full matrices with entries in R. The obtained new rings Λ are Auslander–Gorenstein if and only if so is R.

Finiteness of Selfinjective Dimension for Noetherian Algebras

We will study coherent modules of finite weak Gorenstein dimension and characterize noetherian algebras of finite selfinjective dimension in terms of weak Gorenstein dimension.