by Abe, Hiroki; Hoshino, Mitsuo

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 ∈ I

e

_i
with the e

_i
orthogonal idempotents; (b) e

_i

x = xe

_i
for all i ∈ I and x ∈ R; (c) e

_i

A
e

_j
 ≠ 0 for all i, j ∈ I; (d) e

_i

A

_A
 ≇ 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
 ≅ Hom
_R
(Ae

_π(i),R

_R
) and
_A

Ae

_π(i) ≅ 
_A
Hom
_R
(e

_i

A,
_R

R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e

_i
) = e

_π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map $\bigoplus_{i \in J} Ae_{i} \otimes_{R} e_{i}A_{A} \to A_{A}$ is a tilting complex.

DOI: 10.1007/s10468-007-9065-2
Online Date: 6/26/2007
Print publication date: 6/1/2008
View article on SpringerLink