Frobenius Extensions and Tilting Complexes

Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑ i ∈ Iei with the ei orthogonal idempotents; (b) eix = xei for all i ∈ I and x ∈ R; (c) eiAej ≠ 0 for all i, j ∈ I; (d) eiAA ≇ ejAA unless i = j; (e) every eiAei is a local ring whenever R is; (f) eiAA ≅ HomR(Aeπ(i),RR) and AAeπ(i) ≅ AHomR(eiA,RR) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(ei) = eπ(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bigoplus_{i \in J} Ae_{i} \otimes_{R} e_{i}A_{A} \to A_{A}$\end{document} is a tilting complex.