An elementary construction of tilting complexes

Abstract

Let A be an artin algebra and e∈ A an idempotent with add( eA A )=add( D( A Ae)). Then a projective resolution of Ae eAe gives rise to tilting complexes {P(l) •} l⩾1 for A, where P( l) • is of term length l+1. In particular, if A is self-injective, then End K( Mod-A) (P(l) •) is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes { T(2 l) •} l⩾1 for A, where T(2 l) • is of term length 2 l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes { T( l) •} l⩾1 for A, where T( l) • is of term length l+1.