Triangle K0 of perfect complexes equals split K0 of finite projectives

For finitely generated projective left modules $P,Q$ the degreewise split sequence of complexes $0\to P[0]\to (P\oplus Q)[0]\to Q[0]\to 0$ (the biproduct sequence in degree zero, zero in every other degree) is short exact, and all three complexes are bounded with finitely generated projective terms; by [F2] they are perfect objects of $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$, and [F6] gives a distinguished triangle $P[0]\to (P\oplus Q)[0]\to Q[0]\to P[1]$ of $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$, hence of $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$ by [F2]. Its relation $[(P\oplus Q)[0]]=[P[0]]+[Q[0]]$ holds in $K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ by [F1]. Thus the class function $[P]\mapsto [P[0]]$ on $\mathrm{Iso}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))$ is additive on biproducts, and [F3] gives a unique homomorphism ${\iota }_{\ast }:K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))\to K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ with ${\iota }_{\ast }([P])=[P[0]]$.

By [F4] the assignment $\chi (X):={\sum }_n(-1{)}^n[P^n]$, for any bounded finite-projective complex $P$ representing the perfect object $X$, is a well-defined class function on $\mathrm{Iso}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ with values in $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))$, is additive on distinguished triangles of perfect objects, and is independent of the representative. Extending $\chi$ over the free abelian group on $\mathrm{Iso}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ and applying the quotient universal property of [F8] in the pattern of the functor-induced class functions of [F4] and [F1] produces a unique homomorphism $\bar{\chi }:K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))\to K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))$ with $\bar{\chi }([X])={\sum }_n(-1{)}^n[P^n]$.

The composite $\bar{\chi }\circ {\iota }_{\ast }$ is the identity of $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))$: for a finitely generated projective $P$, $\bar{\chi }({\iota }_{\ast }([P]))=\chi (P[0])=[P]$ because the degree-zero stalk complex has the single term $P$ in degree $0$ by [F7], and the two homomorphisms agree on every generator of $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}$ [F3], with ${\iota }_{\ast }$ and $\bar{\chi }$ as constructed in steps 1.1 and 1.2.

The composite ${\iota }_{\ast }\circ \bar{\chi }$ is the identity of $K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$. Let $P$ be a bounded finite-projective complex with $P^i=0$ for $i<a$ and $i>b$, representing $X$. For every integer $n$ the inclusion ${\sigma }^{\ge n}P\hookrightarrow {\sigma }^{\ge n-1}P$ of [F7] is a degreewise split short exact sequence of complexes with cokernel the degree-$(n-1)$ stalk complex $P^{n-1}[-(n-1)]$, all terms bounded finite projective; by [F6] and [F2] it gives a distinguished triangle of $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$, so [F1] and [F5] give $[{\sigma }^{\ge n-1}P]=[{\sigma }^{\ge n}P]+(-1{)}^{n-1}[P^{n-1}[0]]$. Since ${\sigma }^{\ge b+1}P=0$ and $[{\sigma }^{\ge b+1}P]=0$ by [F5], summing these relations for $n=a+1,\dots ,b+1$ telescopes to $[{\sigma }^{\ge a}P]={\sum }_{j=a}^b(-1{)}^j[P^j[0]]$; and ${\sigma }^{\ge a}P=P$ by [F7]. Hence $[X]=[P]={\sum }_j(-1{)}^j[P^j[0]]={\sum }_j(-1{)}^j{\iota }_{\ast }([P^j])={\iota }_{\ast }(\bar{\chi }([X]))$, using ${\iota }_{\ast }([P^j])=[P^j[0]]$ from step 1.1 and the definition of $\bar{\chi }$ from step 1.2. The classes $[X]$ generate $K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$ [F1], so ${\iota }_{\ast }\circ \bar{\chi }$ is the identity; the zero complex, where the range $a\le j\le b$ is empty, satisfies $[0]=0$ by [F5].

Steps 2.1 and 2.2 exhibit $\bar{\chi }$ as a two-sided inverse of ${\iota }_{\ast }$, so degree-zero inclusion induces the asserted isomorphism $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))\to K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A))$, with inverse sending the class of an object represented by $P$ to ${\sum }_n(-1{)}^n[P^n]$; no finite global-dimension or Noetherian hypothesis was used. The graded comparison is the same argument run in the abelian category ${\mathrm{GrMod}}_0(A)$ of graded modules with degree-zero maps, where finite graded projectives replace finite projectives, the graded Euler lemma and graded closure clause of [F2, F4] replace their ungraded counterparts, internal shifts $\{1\}$ are left untouched, and the biproduct and brutal-truncation sequences are formed degreewise.