Perfect complexes form an essentially small triangulated subcategory

$D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ is essentially small. Every finitely generated projective left $A$-module is a direct summand of a finite free module [F5]; choosing a finite generating family of $P$ gives a surjection $A^n\twoheadrightarrow P$, which splits by [F5], so $P\cong \mathrm{im}(e)$ for an idempotent $e\in M_n(A)$. The idempotent matrices in ${\bigcup }_{n\ge 0}{M}_n(A)$ form a set, so the isomorphism classes of finitely generated projective left $A$-modules form a set; bounded cochain complexes of these modules are finite-support sequences of such modules with differentials, and they therefore also form a set of objects. By [F1] every object of $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ is isomorphic to one of these complexes, so the set of isomorphism classes $\mathrm{Iso}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ is a set. In the graded case [F6] exhibits each finite graded projective as a degree-zero summand of a finite sum $A\{r_1\}\oplus \cdots \oplus A\{r_n\}$; the finite tuples of shifts and the degree-zero idempotent endomorphisms of their sums form a set, and the same finite-support complex argument applies with [F7].

Shifts preserve bounded finite-projective complexes. If $P$ is a bounded complex of finitely generated projective left $A$-modules, then $(P[1]{)}^n=P^{n+1}$ with differential $-d_P^{n+1}$ [F4], so $P[1]$ is again bounded with finitely generated projective terms, and likewise $P[-1{]}^n=P^{n-1}$ is such a complex. Graded complexes with degree-zero differentials behave identically, since cochain shift changes only cochain degrees and moves the sign of the differential.

The cone of a chain map of bounded finite-projective complexes is again one. For a chain map $f:P\to Q$ of such complexes, [F4] gives $\mathrm{Cone}(f{)}^n=Q^n\oplus P^{n+1}$, which is a finite direct sum of finitely generated projective modules, hence finitely generated projective by [F5]; the support of the cone is contained in the sum of the supports of $P$ and $Q$, hence finite. In the graded case the biproduct is computed degreewise [F7] and a finite direct sum of finite graded projectives is again finite graded projective by [F6].

Every bounded complex $P$ of projective objects is K-projective, with no choice principle needed. Let $E$ be acyclic, $r$ an integer, $f:P\to E[r]$ a chain map, and suppose $P^n=0$ for $n>b$; put $F:=E[r]$, which is acyclic, and set $h^n=0$ for $n>b$. Inductively assume $h^{n+1}:P^{n+1}\to F^n$ satisfies $f^{n+1}=d_F^n{h}^{n+1}+h^{n+2}{d}_P^{n+1}$, and put $u^n:=f^n-h^{n+1}{d}_P^n$. Then $d_F^n{u}^n=d_F^n{f}^n-d_F^n{h}^{n+1}{d}_P^n=f^{n+1}{d}_P^n-(f^{n+1}-h^{n+2}{d}_P^{n+1})d_P^n=0$, so $u^n$ factors through the cycles $Z^n(F)=\ker d_F^n$. Since $H^n(F)=0$, the map $F^{n-1}\to Z^n(F)$ is an epimorphism; by projectivity of $P^n$ the composite $P^n\to Z^n(F)$ lifts to $h^n:P^n\to F^{n-1}$ with $d_F^{n-1}{h}^n=u^n$, which is the homotopy equation in degree $n$. Below the support of $P$ we take $h^n=0$, where both sides vanish. Only finitely many lifts are chosen, one for each degree in the finite support of $P$, so no dependent choice is used and the induction terminates. Hence ${\mathrm{Hom}}_K(P,F)=0$ for every acyclic $F$ and every shift, which is [F2].

Every derived morphism between bounded finite-projective complexes is represented by a chain map, uniquely up to homotopy. A bounded complex of finitely generated projectives has projective terms, so step 1.4 makes it K-projective; the published no-roof proposition [F3] then makes $Q:{\mathrm{Hom}}_K(P,X)\to {\mathrm{Hom}}_D(P,X)$ bijective for every complex $X$, in particular for a bounded finite-projective complex $X$. Surjectivity represents every derived morphism $P\to X$ by a chain map, and injectivity says two chain maps represent the same derived morphism exactly when they are chain homotopic.

$D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ is closed under shifts. Let $X$ be perfect with bounded finite-projective representative $P$, so that $X\cong P$ in $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$. Then $X[1]\cong P[1]$ and $X[-1]\cong P[-1]$; by step 1.2 both $P[1]$ and $P[-1]$ are bounded complexes of finitely generated projectives, so [F1] makes $X[1]$ and $X[-1]$ perfect. In the graded case the same argument uses the graded shift of a bounded complex with degree-zero differentials and finite graded projective terms.

$D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ is closed under cones. Let $X\to Y\to Z\to X[1]$ be a distinguished triangle of $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ with $X,Y$ perfect. Fix bounded finite-projective representatives $P,Q$ and isomorphisms $u:P\to X$, $v:Q\to Y$ in $D$. The composite $g:=v^{-1}\circ (X\to Y)\circ u:P\to Q$ is a derived morphism between bounded finite-projective complexes, so by step 2.1 it is represented by a chain map $f:P\to Q$ with $Q(f)=g$, that is, $vQ(f)=Q(X\to Y)u$. The cone triangle $P\to Q\to \mathrm{Cone}(f)\to P[1]$ is distinguished by [F4], and its cone is a bounded finite-projective complex by step 1.3. Since $v\circ Q(f)=Q(X\to Y)\circ u$, TR3 [F8] supplies a third component $c:\mathrm{Cone}(f)\to Z$ making $(u,v,c)$ a morphism of triangles; the first two components are isomorphisms, so the triangulated five lemma [F9] makes $c$ an isomorphism. Hence $Z\cong \mathrm{Cone}(f)$, and [F1] makes $Z$ perfect. The graded case is identical, with the graded biproduct and graded projectivity supplied by [F6, F7].

Collecting the results: $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ is a strictly full subcategory by [F1], closed under isomorphism by construction, and closed under shifts and cones by steps 2.2 and 3.1. The distinguished triangles with objects in $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ are those of $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ among these objects; the axioms TR1–TR4 hold in $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ [F10] and their completions, being built by shifts and cones from perfect objects, again lie in $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ by steps 2.2 and 3.1, while all morphisms between perfect objects are available because the subcategory is strictly full. Hence $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$, with the inherited translation and triangles, is triangulated, and step 1.1 shows it is essentially small. The graded assertions are proved by the same steps with the graded data.