Tilting in $Q$-shaped derived categories

TL;DR

This paper explores conditions under which $Q$-shaped derived categories of an algebra are triangulated equivalent to classic derived categories of different algebras, focusing on categories formed from shifts of projective modules.

Contribution

It establishes new triangulated equivalences between $Q$-shaped and classic derived categories, especially for categories built from shifts of projective modules over self-injective graded algebras.

Findings

Triangulated equivalence between $D_Q(A)$ and $D(B)$ in certain cases.

Special case: $D_N(A)$ is equivalent to $D(T_{N-1}A)$ for upper diagonal matrices.

Discussion of additional special cases and their implications.

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 $\Lambda$. 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.