Perfect complexes over a ring and its graded version

Let $A$ be a unital associative ring and let $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ be the derived category of left $A$-modules in the cochain convention of Derived category of an abelian category. An object $X$ of $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ is perfect when it is isomorphic there to a bounded cochain complex of finitely generated projective left $A$-modules (Bounded, bounded below, and bounded above complexes, Projective modules and the lifting property, Generated submodule, cyclic and finitely generated modules, module basis and free module). The isomorphism is taken in the derived category, so a perfect object is presented by a zigzag of quasi-isomorphisms to its bounded finite-projective representative; no single representative is singled out as canonical. Write $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ for the strictly full subcategory of $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ whose objects are the perfect ones: it contains every morphism of $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ between perfect objects, and it is closed under isomorphism in $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$.

For a unital graded $k$-algebra $A$, the graded version uses the derived category $D({\mathrm{GrMod}}_0(A))$ of the abelian category of graded left $A$-modules with degree-zero maps (Associative graded algebras, bimodules, and internal shifts, Finite graded projective modules). A graded left $A$-module is finite graded projective when it is a finitely generated projective object of ${\mathrm{GrMod}}_0(A)$; a graded complex has degree-zero differentials when every differential is a degree-zero map of graded modules. An object of $D({\mathrm{GrMod}}_0(A))$ is graded perfect when it is isomorphic there to a bounded cochain complex of finite graded projective left $A$-modules with degree-zero differentials; the strictly full subcategory of these objects is written $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}^{\mathrm{g}\mathrm{r}}(A)$.

Two operations on complexes are kept separate throughout. The cochain shift $[1]$ is the translation of the derived category, $X[1{]}^n=X^{n+1}$ with the sign convention of Derived category of an abelian category; the internal shift $\{1\}$ reindexes the internal grading of a graded module or graded complex, $(M\{1\}{)}_d=M_{d-1}$ (Associative graded algebras, bimodules, and internal shifts). They act on different structures and commute with one another. They need not produce distinct isomorphism classes: for the zero complex, $0[1]\cong 0\{1\}\cong 0$.

A bounded cochain complex of arbitrary left $A$-modules need not be perfect: boundedness alone neither supplies finitely generated projective terms nor permits their recovery, and the definition above asks for such a representative up to isomorphism in the derived category. The definition itself does not identify $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ with the bounded derived category of finitely generated left $A$-modules; that comparison needs additional hypotheses.