Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Abstract

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: =_ΓΓ_Λ ⊗_Λ − and F: =_ΛΛ_Γ ⊗_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).

Key Words:

• Split algebras
• Standardly stratified algebras
• Stratifying systems

2010 Mathematics Subject Classification:

• Primary: 16G10
• Secondary: 18G99

ACKNOWLEDGMENTS

We would like to thank an anonymous referee for his/her comments and suggestions.

Notes

Communicated by D. Zacharia.