Under AC, left Noetherian rings of finite left global dimension identify perfect and bounded finite-module derived categories

Under the hypotheses, $\mathcal{A}$ is an essentially small abelian category with enough projectives, and every finitely generated left $A$-module $M$ has a finite resolution $0\to Q_n\to \cdots \to Q_0\to M\to 0$ by finitely generated projective modules, with $n:=\max (1,d)$. Indeed, kernels of maps of finitely generated modules are finitely generated submodules of Noetherian modules [F1], so $\mathcal{A}$ is closed under kernels and cokernels and is abelian; each finitely generated $M$ is a quotient of a finite free module $A^m$ [F1, F2], which is projective, so $\mathcal{A}$ has enough projectives; and isomorphism classes of finitely generated modules form a set because every such module is a quotient of some $A^m$, so $\mathcal{A}$ is essentially small. Iterating finite free covers builds a projective resolution all of whose syzygies are finitely generated [F1]; since $\mathrm{pd}(M)\le d\le n$ and $n\ge 1$, the syzygy theorem [F3] makes ${\mathrm{\Omega }}^n(M)$ projective, and it is finitely generated as a submodule of a finitely generated free module [F1]; truncating the resolution there gives the displayed finite resolution.

Every bounded complex $X$ of finitely generated left $A$-modules, say supported in degrees $[a,b]$, admits a bounded complex $P$ of finitely generated projectives together with a quasi-isomorphism $p:P\to X$. Apply the published replacement construction [F4] inside the abelian category $\mathcal{A}$ of step 1.1, which has enough projectives, using the successive choices licensed by the DC supplied by AC [F8]; this gives a termwise epic quasi-isomorphism $p_{\mathrm{\infty }}:P_{\mathrm{\infty }}\to X$ with $P_{\mathrm{\infty }}^j=0$ for $j>b$ and every $P_{\mathrm{\infty }}^j$ a finitely generated projective. Let $M:=\ker (d_{P_{\mathrm{\infty }}}^{a-1}:P_{\mathrm{\infty }}^{a-1}\to P_{\mathrm{\infty }}^a)$, a finitely generated module [F1]; by step 1.1, $M$ has a finite resolution by finitely generated projectives. Place that resolution in degrees $a-2,a-3,\dots$, the first of its maps being composed onto $M$ followed by the inclusion $M\hookrightarrow P_{\mathrm{\infty }}^{a-1}$, keep $P_{\mathrm{\infty }}$ unchanged in degrees $\ge a-1$, and set the terms below the placed resolution to zero; call the result $P$. Then $P$ is bounded with finitely generated projective terms, and the restricted map $p$ (the map $p_{\mathrm{\infty }}$ above degree $a-1$, and zero in degrees below $a-1$, where $X$ vanishes) is a chain map. Its cohomology matches that of $X$: in degrees $\ge a$ because $P$ agrees with $P_{\mathrm{\infty }}$ there and $p_{\mathrm{\infty }}$ is a quasi-isomorphism [F4]; in degree $a-1$ the kernel of $d_P^{a-1}$ is $M$, which is exactly the image of the placed first differential, so $H^{a-1}(P)=0=H^{a-1}(X)$; and in degrees below $a-1$ the placed resolution is exact and $X$ vanishes. Hence $p$ is a quasi-isomorphism.

The functor $D^b(\mathcal{A})\to D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ induced by the inclusion is fully faithful. Given objects $X,Y$ of $D^b(\mathcal{A})$, step 2.1 supplies bounded complexes $P,Q$ of finitely generated projectives with quasi-isomorphisms onto bounded complexes representing $X$ and $Y$, so in both derived categories $X\cong P$ and $Y\cong Q$. Such $P$ is a bounded-above complex of projective objects of $\mathcal{A}$ and, since each $P^j$ is a direct summand of a finite free module [F2], also a bounded-above complex of projective $A$-modules; hence $P$ is K-projective in both categories by the DC-qualified theorem [F5], with DC supplied by AC [F8]. By the no-roof proposition [F6], ${\mathrm{Hom}}_{D(\mathcal{A})}(P,Q)\cong {\mathrm{Hom}}_{K(\mathcal{A})}(P,Q)$ and ${\mathrm{Hom}}_{D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})}(P,Q)\cong {\mathrm{Hom}}_{K(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})}(P,Q)$, and by [F7] these derived Hom sets are the bounded ones. Chain maps and homotopies between $P$ and $Q$ are the same data in $\mathcal{A}$ and in $A$-Mod because $\mathcal{A}$ is a full subcategory, so the comparison map ${\mathrm{Hom}}_{D^b(\mathcal{A})}(X,Y)\to {\mathrm{Hom}}_{D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})}(X,Y)$ is bijective.

The functor of step 3.1 is essentially surjective onto $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ and has image contained in it. Every perfect object is isomorphic in $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ to a bounded complex $P$ of finitely generated projective left $A$-modules [F9], which is a bounded complex in $\mathcal{A}$ and hence an object of $D^b(\mathcal{A})$ mapping to $P$; conversely every object of $D^b(\mathcal{A})$ is isomorphic by step 2.1 to a bounded complex of finitely generated projectives, whose image is perfect by [F9]. The inclusion of complexes preserves finite biproducts, shifts and cones degreewise, so the induced functor is exact.

Steps 3.1 and 4.1 show that the induced exact functor $D^b(\mathcal{A})\to D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ is fully faithful and essentially surjective, that is, an exact equivalence of triangulated categories; by [F7] it is also compatible with the bounded localizations. By [F11] it induces an isomorphism $E_{\ast }:K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D^b(\mathcal{A}))\to K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ on triangle Grothendieck groups, with inverse induced by any quasi-inverse equivalence.

Write $\phi$ for the isomorphism of [F10] on the projective side, with $\phi ([P])=[P[0]]$, and $\psi$ for the isomorphism $G_0(\mathcal{A})\to K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D^b(\mathcal{A}))$ of [F10], whose inverse sends $[X]$ to ${\sum }_n(-1{)}^n[H^n(X)]$. The composite ${\psi }^{-1}\circ E_{\ast }^{-1}\circ \phi$ is a homomorphism $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))\to G_0(\mathcal{A})$, and on a generator $[P]$ it sends $[P]\mapsto [P[0]]\mapsto [P[0]]\mapsto {\sum }_n(-1{)}^n[H^n(P[0])]=[P]$, because the bounded complex $P[0]$ lies in the equivalence of step 5.1 with $P$ as its image and has cohomology $P$ in degree $0$ and zero elsewhere [F10]. Since the classes $[P]$ generate $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))$ [F10], this composite is exactly the Cartan map $[P]\mapsto [P]$, which is therefore an isomorphism, being a composite of isomorphisms. The ring hypotheses entered only through the equivalence of step 5.1: the two comparison isomorphisms of [F10] hold for every unital ring and every essentially small abelian category respectively, and this does not generalise to a singular or non-left-Noetherian ring.