Homological Systems in Triangulated Categories

We introduce the notion of homological systems Θ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in triangulated categories. We prove that, attached to the homological system Θ, there are two standardly stratified algebras A and B, which are derived equivalent. Furthermore, it is proved that the category F(Θ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {F}({\Theta }),$\end{document} of the Θ-filtered objects in a triangulated category T,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {T},$\end{document} admits in a very natural way a structure of an exact category, and then there are exact equivalences between the exact category F(Θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {F}({\Theta })$\end{document} and the exact categories of the Δ-good modules associated to the standardly stratified algebras A and B. Some of the obtained results can be seen also under the light of the cotorsion pairs in the sense of Iyama-Nakaoka-Yoshino (see 6.6 and 6.7 ). We recall that cotorsion pairs are studied extensively in relation with cluster tilting categories, t-structures and co-t-structures.