Tilting in $Q$-shaped derived categories

Abstract

The main result of this paper is that there is sometimes a triangulated equivalence between $D_Q(A)$, the $Q$-shaped derived category of an algebra $A$, and $D(B)$, the classic derived category of a different algebra $B$. By construction, $D_Q(A)$ consists of $Q$-shaped diagrams of $A$-modules for a suitable small category $Q$. Our result concerns the case where $Q$ consists of shifts of indecomposable projective modules over a self-injective $\mathbb{Z}$-graded algebra $\hat{I}›$. A notable special case is the result by Iyama, Kato, and Miyachi that $D_N(A)$, the $N$-derived category of $A$, is triangulated equivalent to $D(T_{N-1}A)$, the classic derived category of $T_{N-1}(A)$, which denotes upper diagonal $(N-1)\times (N-1)$-matrices over $A$. Several other special cases will also be discussed.

Reference added. 10 pages