Euler class of a bounded projective complex is derived invariant and triangle additive

${\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A)$ is an essentially small additive category, so $\chi$ takes values in the group of [F3]. It is additive because the direct sum of two finitely generated projective modules is finitely generated projective and the zero module is finitely generated projective. It is essentially small: every finitely generated projective $P$ is a direct summand of a finite free module $A^n$ [F8], so $P\cong \mathrm{im}(e)$ for an idempotent matrix $e\in M_n(A)$, and the idempotents in ${\bigcup }_{n\ge 0}{M}_n(A)$ form a set. For a bounded complex $P$, the sum $\chi (P)={\sum }_n(-1{)}^n[P^n]$ is finite by boundedness; it uses the classes of the terms in $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{Proj}}_{\mathrm{f}\mathrm{g}}(A))$. The graded category of finite graded projectives is additive and essentially small by [F10], with the same finiteness of the alternating sum.

Every acyclic bounded complex $P$ of finitely generated projectives has $\chi (P)=0$. Induct on the finite number of degrees in which $P$ is nonzero. Let $b$ be the largest such degree, so $P^b\mathrm{\ne }0$ and $P^{b+1}=0$; acyclicity gives $P^b=\mathrm{im}(d^{b-1})$, and the sequence $0\to Z^{b-1}\to P^{b-1}\stackrel{d^{b-1}}{\to }{P}^b\to 0$, with $Z^{b-1}=\ker d^{b-1}$, splits by projectivity of $P^b$ [F8]; hence $[P^{b-1}]=[Z^{b-1}]+[P^b]$ in the split group and $Z^{b-1}$ is finitely generated projective. Let $T$ be the complex with $T^n=P^n$ for $n\le b-2$, $T^{b-1}=Z^{b-1}$ and $T^n=0$ for $n\ge b$, with the restricted differentials. Then $T$ is a bounded complex of finitely generated projectives with fewer nonzero terms, and it is acyclic: in degrees $n\le b-2$ its cohomology is that of $P$, and $H^{b-1}(T)=Z^{b-1}\mathrm{/}\mathrm{im}(d^{b-2})=H^{b-1}(P)=0$. By induction $\chi (T)=0$, while the split relation gives $\chi (P)-\chi (T)=(-1{)}^{b-1}([P^{b-1}]-[Z^{b-1}])+(-1{)}^b[P^b]=0$. Hence $\chi (P)=0$. The graded case repeats the argument with the degreewise kernel $Z^{b-1}$ and the graded splitting of [F10].

For a chain map $f:P\to Q$ of bounded finite-projective complexes, $\chi (\mathrm{Cone}(f))=\chi (Q)-\chi (P)$. By [F7] the cone has terms $\mathrm{Cone}(f{)}^n=Q^n\oplus P^{n+1}$; the split relations of [F3] give $\chi (\mathrm{Cone}(f))={\sum }_n(-1{)}^n[Q^n]+{\sum }_n(-1{)}^n[P^{n+1}]=\chi (Q)-\chi (P)$. The cone is bounded with finitely generated projective terms by [F2], so the left side is defined.

$\chi$ is unchanged by quasi-isomorphism of bounded finite-projective complexes. If $f:P\to Q$ is a quasi-isomorphism, then $\mathrm{Cone}(f)$ is acyclic [F5], so $\chi (\mathrm{Cone}(f))=0$ by step 1.2 and step 1.3 gives $\chi (Q)=\chi (P)$.

$\chi$ takes the same value on any two bounded finite-projective representatives of one object of $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(A)$. Let $X$ be perfect and let $P,P^{\mathrm{\prime }}$ be bounded finite-projective complexes with isomorphisms $X\cong P$ and $X\cong P^{\mathrm{\prime }}$ in $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$; composing gives an isomorphism $P\to P^{\mathrm{\prime }}$ in the derived category. By [F2] this derived morphism is represented by a chain map $f:P\to P^{\mathrm{\prime }}$, and $f$ is a quasi-isomorphism because its image in $D$ is an isomorphism. Step 2.1 gives $\chi (P)=\chi (P^{\mathrm{\prime }})$, so $\chi (X):=\chi (P)$ is well defined on perfect objects, independently of the chosen representatives.

$\chi$ is unchanged by homotopy equivalence: a homotopy equivalence of bounded finite-projective complexes is a quasi-isomorphism by [F6], so step 2.1 applies. Together with step 3.1 this is the invariance asserted for bounded finite-projective complexes and for the represented perfect object.

$\chi$ is additive on distinguished triangles of perfect objects. Let $X\to Y\to Z\to X[1]$ be distinguished, choose bounded finite-projective representatives $P,Q$ with isomorphisms $u:P\to X$, $v:Q\to Y$, and let $g:=v^{-1}\circ (X\to Y)\circ u$. By [F2] and [F4], $g$ is represented by a chain map $f:P\to Q$, that is, $vQ(f)=Q(X\to Y)u$; the cone triangle $P\to Q\to \mathrm{Cone}(f)\to P[1]$ is distinguished [F7]. TR3 [F9] supplies $c:\mathrm{Cone}(f)\to Z$ making $(u,v,c)$ a morphism of triangles, and the first two components are isomorphisms, so the triangulated five lemma [F9] makes $c$ an isomorphism. Therefore $Z\cong \mathrm{Cone}(f)$ in $D(A\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$ and, by step 3.1, $\chi (Z)=\chi (\mathrm{Cone}(f))=\chi (Q)-\chi (P)=\chi (Y)-\chi (X)$, which is the asserted additivity.

Steps 1.1–4.2 establish the definition, quasi-isomorphism and homotopy invariance, independence of representatives, and triangle additivity for complexes of finitely generated projective left modules; the graded assertions use the graded splitting and degreewise biproducts of [F10] at every occurrence of a splitting or a direct sum. No global-dimension hypothesis and no choice principle is used, and the argument nowhere asserts an Euler class for an arbitrary bounded complex of modules.