The $Q$-shaped derived category of a ring — compact and perfect objects

Abstract:

A chain complex can be viewed as a representation of a certain self-injective quiver with relations, $Q$. To define $Q$, include a vertex $q_n$ and an arrow $q_n \xrightarrow {\partial } q_{n-1}$ for each integer $n$. The relations are $\partial ^2 = 0$.

Replacing $Q$ by a general self-injective quiver with relations, it turns out that some of the key properties of chain complexes generalise. Indeed, consider the representations of such a $Q$ with values in ${}_{A}\mspace {-1mu}\operatorname {Mod}$ where $A$ is a ring. We showed in earlier work that these representations form the objects of the $Q$-shaped derived category, $\mathcal {D}_{Q}(A)$, which is triangulated and generalises the classic derived category $\mathcal {D}_{}(A)$. This follows ideas of Iyama and Minamoto.

While $\mathcal {D}_{Q}(A)$ has many good properties, it can also diverge dramatically from $\mathcal {D}_{}(A)$. For instance, let $Q$ be the quiver with one vertex $q$, one loop $\partial$, and the relation $\partial ^2 = 0$. By analogy with perfect complexes in the classic derived category, one may expect that a representation with a finitely generated free module placed at $q$ is a compact object of $\mathcal {D}_{Q}(A)$, but we will show that this is, in general, false.

The purpose of this paper, then, is to compare and contrast $\mathcal {D}_{Q}(A)$ and $\mathcal {D}_{}(A)$ by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant.