Geometry of chain complexes and outer automorphisms under derived equivalence. (Q2750921)

One of the main results of this paper is that, if \(A\) is a finite dimensional algebra over an algebraically closed field, the identity component \(\text{Out}(A)^0\) of the algebraic group of outer automorphisms of \(A\) is invariant under derived equivalence. (The authors note that this invariance theorem was also proved independently and with different methods by R. Rouquier.) This generalizes a result of Brauer on Morita invariance of \(\text{Out}(A)^0\), and also a result on the tilting-cotilting of \(\text{Out}(A)^0\) due to \textit{F. Guil-Asensio} and \textit{M. Saorín} [Arch. Math. 76, No. 1, 12-19 (2001; Zbl 1036.16017)]. The derived equivalence was introduced by \textit{J. Rickard} [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0642.16034)], and the paper under review is a contribution to the extensive literature on exhibiting invariants under derived equivalence. Note that derived invariance fails for the group \(\Aut(A)\) of all algebra automorphisms of \(A\), as well as for the full Picard group. The authors also obtain in the paper that the derived Picard group \(\text{DPic}(A)\) contains only finitely many two-sided tilting complexes of fixed total dimension which are pairwise non-isomorphic when viewed as one-sided complexes over \(A\).