Frobenius Extensions and Tilting Complexes

概要

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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.
Springer Netherlandsの論文
2008-06-00