The bounded projective comparison for the derived category

The bounded localization has a small family of roofs. For each finite list of integers $\mathbf{r}=(r_1,\dots ,r_s)$ put $F_{\mathbf{r}}:={⨁}_{j=1}^s{A}_m\{r_j\}$. By [L11], every object of $A_m\text{-mod}$ is a quotient $F_{\mathbf{r}}\mathrm{/}N$ for some graded submodule $N\subseteq F_{\mathbf{r}}$. The finite lists $\mathbf{r}$ form a set and, for each one, the graded submodules of $F_{\mathbf{r}}$ form a set. Thus the pairs $(\mathbf{r},N)$ are a set of presentation codes. Let ${\mathcal{M}}_m^{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ have these codes as objects and all degree-zero $A_m$-module maps between their quotient modules as morphisms. Its object collection is a set, and its morphism collection is a union of sets of maps between fixed quotient modules, hence a set by [L10]. Every finitely generated graded module is isomorphic to the quotient of one of these codes. A bounded complex has only finitely many nonzero terms, so finite choice of a code and an isomorphism for each such term, followed by transport of its differentials, gives an isomorphic bounded complex over ${\mathcal{M}}_m^{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$. The category of these coded bounded complexes is small: its objects are finite-support sequences of codes with differentials from the set of code morphisms, and its morphisms and homotopy classes are sets. Every bounded complex is isomorphic to one of them, and a quasi-isomorphism remains one after transport. For fixed bounded endpoints $X,Y$, the maps from each coded middle complex to $X$ and $Y$ form sets, since they are families of functions between fixed underlying sets; taking their union over the set of coded middle complexes still gives a set. Replacing the middle complex of any bounded roof or comparison by an isomorphic code complex therefore gives small cofinal denominator families as required by [F3]. The bounded roof calculus and the bounded instance of [L6] apply, and $D^b(A_m\text{-mod})$ has Hom sets. This construction uses only finitely many choices for each bounded complex; it selects no skeleton of the large category.

Bounded projective complexes are K-projective, without dependent choice. Let $P$ be a complex with every $P^n$ projective and $P^n=0$ for $n\mathrm{\notin }[a,b]$, and let $f:P\to A$ be a chain map into an acyclic complex $A$. We construct maps $h^n:P^n\to A^{n-1}$ with $d_A{h}^n+h^{n+1}{d}_P=f^n$ by descending induction on $n\le b$: put $h^{b+1}=0$, and given $h^{n+1}$ the map $g:=f^n-h^{n+1}{d}_P:P^n\to A^n$ satisfies $d_{A}g=d_A{f}^n-d_A{h}^{n+1}{d}_P=f^{n+1}{d}_P-(f^{n+1}-h^{n+2}{d}_P)d_P=0$, so $g$ lands in $\ker d_A^n=\mathrm{im}(d_A^{n-1})$ by acyclicity of $A$ at $A^n$, the map $d_A^{n-1}:A^{n-1}\to \ker d_A^n$ is an epimorphism, and the projective lifting property of [L11] applied to the projective $P^n$ supplies $h^n$ with $d_A{h}^n=g$. The induction has finitely many stages and each stage chooses one lift, so $f$ is null-homotopic and ${\mathrm{Hom}}_K(P,A[r])=0$ for all $r$ after shifting; hence $P$ is K-projective by [F5], with no countable or dependent choice.

Degreewise split extensions give distinguished triangles. Let $0\to X^{\mathrm{\prime }}\stackrel{i}{\to }X\stackrel{q}{\to }{X}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a degreewise split short exact sequence of bounded cochain complexes. Choose degreewise maps $k:X\to X^{\mathrm{\prime }}$ and $j:X^{\mathrm{\prime }\mathrm{\prime }}\to X$ with $ki=1$, $qj=1$, $kj=0$ and $1_X=ik+jq$; these need not be chain maps. Set $\eta :=k(d_{X}j-j{d}_{X^{\mathrm{\prime }\mathrm{\prime }}}):X^{\mathrm{\prime }\mathrm{\prime }}\to X^{\mathrm{\prime }}[1]$. Since $q(d_{X}j-j{d}_{X^{\mathrm{\prime }\mathrm{\prime }}})=0$, one has $d_{X}j-j{d}_{X^{\mathrm{\prime }\mathrm{\prime }}}=i\eta$, and the cochain identity $d_X^{2}j=j{d}_{X^{\mathrm{\prime }\mathrm{\prime }}}^2=0$ gives $d_{X^{\mathrm{\prime }}}\eta +\eta d_{X^{\mathrm{\prime }\mathrm{\prime }}}=0$. Thus $\psi :X^{\mathrm{\prime }\mathrm{\prime }}\to \mathrm{Cone}(i)$, $\psi (z)=(jz,-\eta z)$, is a chain map. The projection $\phi :\mathrm{Cone}(i)\to X^{\mathrm{\prime }\mathrm{\prime }}$, $\phi (x,x^{\mathrm{\prime }})=q(x)$, is also a chain map, and $\phi \psi =1_{X^{\mathrm{\prime }\mathrm{\prime }}}$. For $H(x,x^{\mathrm{\prime }}):=(0,kx)$ one computes $dH+Hd=1_{\mathrm{Cone}(i)}-\psi \phi$, so $\phi$ and $\psi$ are inverse isomorphisms in $K(A_m\text{-mod})$. The canonical cone triangle ends with the projection $\mathrm{Cone}(i)\to X^{\mathrm{\prime }}[1]$; composing that projection with $\psi$ gives the connecting map $\delta =-\eta :X^{\mathrm{\prime }\mathrm{\prime }}\to X^{\mathrm{\prime }}[1]$. Consequently $X^{\mathrm{\prime }}\stackrel{i}{\to }X\stackrel{q}{\to }{X}^{\mathrm{\prime }\mathrm{\prime }}\stackrel{\delta }{\to }{X}^{\mathrm{\prime }}[1]$ is distinguished by the isomorphism-closure clause of [L8].

The base of the replacement induction. Let $X=M[-a]$ be a complex concentrated in degree $a$. By [L1] choose a finite graded projective resolution $0\to Q_L\to \cdots \to Q_0\stackrel{\epsilon }{\to }M\to 0$ with $Q_k=0$ for $k>L$. Put $P^{a-k}=Q_k$ for $0\le k\le L$ and $P^n=0$ otherwise, with $d_P^{a-k}:Q_k\to Q_{k-1}$ the resolution differential for $k\ge 1$; its differential raises cohomological degree by one. The augmentation $P^a=Q_0\stackrel{\epsilon }{\to }{X}^a=M$ and zero maps in other degrees give a quasi-isomorphism $P\to X$, since the resolution is exact below degree $a$ and has cohomology $M$ in degree $a$. The complex $P$ is bounded and has finite graded projective terms, so $X\cong \mathrm{\Theta }(P)$ in $D^b$ by [F2] and [F3]. If $X=0$, use the zero complex.

$\mathrm{\Theta }$ is full and faithful. Let $P,Q$ be bounded complexes of finite graded projectives. The complex $P$ is K-projective by step 1.2 and [F5]. Apply the bounded roof statement [L6] using the small coded denominator families of step 1.1: localization sends ${\mathrm{Hom}}_{K^b}(P,Q)$ bijectively to ${\mathrm{Hom}}_{D^b}(P,Q)$. The morphisms of $K^b$ are the same homotopy classes as in $K(A_m\text{-mod})$, so this is exactly the map induced by $\mathrm{\Theta }$.

The inductive step. Let $X$ be bounded with $X^n=0$ for $n\mathrm{\notin }[a,b]$ and $X^a\mathrm{\ne }0$, and suppose $b>a$. Let $X^{\mathrm{\prime }}$ be the upper brutal subcomplex, with $X^{\mathrm{\prime }n}=0$ for $n\le a$ and $X^{\mathrm{\prime }n}=X^n$ for $n>a$, and let $X^{\mathrm{\prime }\mathrm{\prime }}:=X^a[-a]$ have its only nonzero term in degree $a$. Since the differential raises degree, $X^{\mathrm{\prime }}$ really is a subcomplex, $X^{\mathrm{\prime }\mathrm{\prime }}$ is the quotient complex, and $0\to X^{\mathrm{\prime }}\stackrel{i}{\to }X\stackrel{q}{\to }{X}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ is degreewise split. Both pieces have fewer nonzero terms than $X$, and step 1.3 gives a connecting map $\delta :X^{\mathrm{\prime }\mathrm{\prime }}\to X^{\mathrm{\prime }}[1]$ making the triangle distinguished. By induction and step 1.4 choose quasi-isomorphisms $p^{\mathrm{\prime }}:P^{\mathrm{\prime }}\to X^{\mathrm{\prime }}$ and $p^{\mathrm{\prime }\mathrm{\prime }}:P^{\mathrm{\prime }\mathrm{\prime }}\to X^{\mathrm{\prime }\mathrm{\prime }}$ from bounded complexes of finite graded projectives. The complex $P^{\mathrm{\prime }\mathrm{\prime }}$ is K-projective by step 1.2, so the bounded form of [L6] represents $Q(p^{\mathrm{\prime }}[1]{)}^{-1}Q(\delta )Q(p^{\mathrm{\prime }\mathrm{\prime }})$ by a chain map $w:P^{\mathrm{\prime }\mathrm{\prime }}\to P^{\mathrm{\prime }}[1]$; injectivity in [L6] gives $p^{\mathrm{\prime }}[1]w=\delta p^{\mathrm{\prime }\mathrm{\prime }}$ in $K^b(A_m\text{-mod})$. Put $P:=\mathrm{Cone}(-w[-1])$. This is bounded, and each term is a finite direct sum of finite graded projectives, hence finite graded projective by [L12]. The minus sign ensures that the rotated cone triangle has connecting map $w$.

The comparison of triangles and the induction closes. The cone triangle of $-w[-1]:P^{\mathrm{\prime }\mathrm{\prime }}[-1]\to P^{\mathrm{\prime }}$ is distinguished and, after one rotation by TR2 of [L8], reads $P^{\mathrm{\prime }}\stackrel{f}{\to }P\stackrel{g}{\to }{P}^{\mathrm{\prime }\mathrm{\prime }}\stackrel{w}{\to }{P}^{\mathrm{\prime }}[1]$: rotation changes the sign of the shifted first arrow, so $-(-w[-1])[1]=w$. Rotating twice more gives the distinguished triangle $P^{\mathrm{\prime }\mathrm{\prime }}\stackrel{w}{\to }{P}^{\mathrm{\prime }}[1]\stackrel{-f[1]}{\to }P[1]\stackrel{-g[1]}{\to }{P}^{\mathrm{\prime }\mathrm{\prime }}[1]$. Rotating the triangle of step 1.3 in the same way gives $X^{\mathrm{\prime }\mathrm{\prime }}\stackrel{\delta }{\to }{X}^{\mathrm{\prime }}[1]\to X[1]\to X^{\mathrm{\prime }\mathrm{\prime }}[1]$. The first arrows commute with $p^{\mathrm{\prime }\mathrm{\prime }}$ and $p^{\mathrm{\prime }}[1]$ in $K$ by step 2.2, so TR3 of [L8] completes them to a morphism of triangles with third component represented by a chain map $u[1]:P[1]\to X[1]$. After applying the bounded localization $Q$, the components $p^{\mathrm{\prime }\mathrm{\prime }}$ and $p^{\mathrm{\prime }}[1]$ are isomorphisms, so [L9] makes $Q(u[1])$, and therefore $Q(u)$, an isomorphism in $D^b(A_m\text{-mod})$. Thus $X\cong \mathrm{\Theta }(P)$, closing the finite induction.

Conclusion. Claims 1 and 2 are steps 2.1 and 3.1. The functor $\mathrm{\Theta }$ is exact for the triangulations because $K^b({\mathrm{proj}}^{gr}{A}_m)$ is closed under shifts and under cones of its maps $w[-1]$, which are again bounded complexes of finite graded projectives by step 2.2, and the distinguished triangles of $K^b(A_m\text{-mod})$ formed by bounded projective complexes are carried to distinguished triangles by the triangulated localization of [L8]. Thus $\mathrm{\Theta }$ is exact and fully faithful, and step 3.1 gives an objectwise replacement for each bounded complex. The bounded-above result [L13] includes supplied replacements and lifts as hypotheses; here finite homological dimension supplies a replacement for each bounded complex and the finite induction supplies the lifts needed for that object. No choice principle, and in particular no dependent choice, was used: step 1.2 is a finite induction with one lift per stage, step 1.4 uses the explicit resolution of [L1], and step 2.2 makes one lift and one cone per stage of a finite induction on the span.