τ-Rigid Modules for Algebras with Radical Square Zero

For a radical square zero algebra [Formula: see text] and an indecomposable right [Formula: see text]-module [Formula: see text], when [Formula: see text] is Gorenstein of finite representation type or [Formula: see text] is [Formula: see text]-rigid, [Formula: see text] is [Formula: see text]-rigid if and only if the first two projective terms of a minimal projective resolution of [Formula: see text] have no non-zero direct summands in common. In particular, we determine all [Formula: see text]-tilting modules for Nakayama algebras with radical square zero.

The Projective Leavitt Complex

For a finite quiverQwithout sources, we consider the corresponding radical square zero algebraA. We construct an explicit compact generator for the homotopy category of acyclic complexes of projectiveA-modules. We call such a generator the projective Leavitt complex ofQ. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex ofQis quasi-isomorphic to the Leavitt path algebra ofQop. Here,Qopis the opposite quiver ofQ, and the Leavitt path algebra ofQopis naturally${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.