ScalingStacks

Definition 2.13. A (non-unital) algebra $A$ with a distinguished set of idempotents $\{e_{i}|i\in I\}$ is called locally unital if the canonical map

$$\bigoplus_{i,j\in I}e_{i}Ae_{j}\to A$$

is an isomorphism. A right module over a locally unital algebra $A$ is called finitely generated (projective) if it is isomorphic to a quotient (direct summand) of a finite direct sum of the modules $e_{i}A$ for $i\in I.$ The perfect derived category of $A$ is the bounded homotopy category of the finitely generated projective right modules

$$\operatorname{D_{perf}}(A)=\operatorname{K}^{b}(\operatorname{mod_{fgp}-}A).$$

If $A$ is moreover $\mathbb{Z}$-graded we consider the category $\operatorname{mod}^{\mathbb{Z}}\operatorname{-}A$ of graded right modules over $A$ with morphisms of degree $0$ and the graded perfect derived category

$$\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(A)=\operatorname{K}^{b}(\operatorname{mod}^{\mathbb{Z}}_{\operatorname{fgp}}\operatorname{-}A).$$