Corina Sáenz

On Standardly Stratified Algebras

Abstract Let A be a finite dimensional algebra over an algebraically closed field k. For any fixed partial ordering of an index set,Λ say,labelling the simple A-modules L(i),there are standard modules,denoted by Δ(i),i ∈ Λ. By definition,Δ(i) is the largest quotient of the projective cover of L(i) having composition factors L(j) with j ≤ i. Denote by ℱ(Δ) the category of A-modules which have a filtration whose quotients are isomorphic to standard modules. The algebra A is said to be standardly stratified if all projective A-modules belong to ℱ(Δ). In this paper we define a “stratifying system” and we show that this produces a module Y,whose endomorphism ring A is standardly stratified. In particular,we construct stratifying systems for special biserial self-injective algebras.

Stratifying systems via relative projective modules

ABSTRACT In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992), and also the “Comparison method” given in Platzeck and Reiten (2001) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992) and Proposition 2.1 in Ringel (1991) follows. #Communicated by A. Happel. ‡Dedicated to Raymundo Bautista on his 60th birthday.

On the Relative Socle for Stratifying Systems

Let A be a finite dimensional k-algebra, (Θ, ≤) be a stratifying system in mod(A) and ℱ(Θ) be the class of Θ-filtered A-modules. In this article, we give the definition and also study some of the properties of the relative socle in ℱ(Θ). We approach the relative socle in three ways. Namely, we view it as (1) a Θ-semisimple subobject of M having the largest Θ-length, (2) a maximal Θ-semisimple subobject of M, and (3) a minimal Θ-essential subobject of M.

Wide subcategories of finitely generated Λ-modules

We explore some properties of wide subcategories of the category modu(Λ) of finitely generated left Λ-modules, for some artin algebra Λ. In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Furthermore, for a wide class 𝒳 in modu(Λ), we give necessary and sufficient conditions to see that 𝒳 = pres(P), for some projective Λ-module P; and finally, a connection with ring epimorphisms is given.

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =ΓΓΛ ⊗Λ − and F: =ΛΛΓ ⊗Γ −. In this article, by assuming that I Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).

Quasi-hereditary algebras related to local algebras

For ⋀ a finite dimensional local algebra with radical N where Nn = 0 ≠ N n-1 define (as a right A-module), then A(⋀) is quasi-hereditary and it has a unique heredity ideal J 1(A).Assume ⋀ satisfies the right socle condition (the socle series and the radical series of ⋀⋀ coincide). We show that then the algebra Ai/J1 (Ai-1 ) is isomorphic to A i-1 where Ai = A (⋀/Ni ). for 1 ≤ i ≤ n. Moreover we determine the canonical module for A(⋀) and we show that the Ringel dual of A(⋀) is isomorphic to the endomorphism ring of as a left module, provided that ⋀ also satisfies the left socle condition.