Frobenius extensions and tilting complexes

Let n (>=) 1 be an integer and p 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 = Sigma(i is an element of I)e(i) with the e(i) orthogonal idempotents; ( b) ei(x) = xe(i) for all i is an element of I and x is an element of R; (c) e(i) Ae(i) not equal 0 for all i, j is an element of I; (d) e(i)A(A) not congruent to e(j)A(A) unless i = j; ( e) every e(i)Ae(i) is a local ring whenever R is; (f) e(i)A(A) congruent to HomR( Ae p( i), RR) and AAe p( i) not congruent to AHomR( eiA, RR) for all i. I; and ( g) there exists a ring automorphism.. Aut( A) such that.( ei) = e p( i) for all i. I. Furthermore, for any nonempty pi-stable subset J of I, the mapping cone of the multiplication map circle plus i=J Ae(i) circle times R e(i)A(A) -> A(A) is a tilting complex.