Tilting in $Q$-shaped derived categories
Sira Gratz, Henrik Holm, Peter Jorgensen, Greg Stevenson
TL;DR
This work develops a tilting-theoretic bridge between Q-shaped derived categories $\mathsf{D}_Q(A)$ and classical derived categories $\mathsf{D}(B)$. By selecting $Q$ as shifts of indecomposable projectives over a self-injective $\mathbb{Z}$-graded algebra $\Lambda$ and applying Yamaura’s tilting framework, the authors prove $\mathsf{D}_Q(A) \cong \mathsf{D}(\Gamma\otimes_k A)$ for a finite-dimensional algebra $\Gamma$, with $A$ any $k$-algebra; this encompasses and generalizes the Iyama–Kato–Miyachi result $\mathsf{D}_N(A) \cong \mathsf{D}(T_{N-1}(A))$ when $\Lambda = k[X]/(X^N)$. The paper also shows how $i^*$ interacts with compact generation and base change, and provides explicit examples via mesh categories of type $A$ and exterior algebras, including Beilinson-type tilting giving connections to $\mathbb{P}^{n-1}_A$ for commutative $A$. Overall, the results offer a broad, tilting-based route to convert Q-shaped diagrammatic derived categories into familiar derived categories, enabling concrete computability and decompositions.
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 $Λ$. 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.