Definition 64.10.1. We denote by K_{perf}(\Lambda ) the category whose objects are bounded complexes of finite projective \Lambda -modules, and whose morphisms are morphisms of complexes up to homotopy. The functor K_{perf}(\Lambda )\to D(\Lambda ) is fully faithful (Derived Categories, Lemma 13.19.8). Denote D_{perf}(\Lambda ) its essential image. An object of D(\Lambda ) is called perfect if it is in D_{perf}(\Lambda ).
64.10 Perfectness
Let \Lambda be a (possibly noncommutative) ring, \text{Mod}_{\Lambda } the category of left \Lambda -modules, K(\Lambda ) = K(\text{Mod}_\Lambda ) its homotopy category, and D(\Lambda )= D(\text{Mod}_\Lambda ) the derived category.
Proposition 64.10.2. Let K\in D_{perf}(\Lambda ) and f\in \text{End}_{D(\Lambda )}(K). Then the trace \text{Tr}(f)\in \Lambda ^\natural is well defined.
Proof. We will use Derived Categories, Lemma 13.19.8 without further mention in this proof. Let P^\bullet be a bounded complex of finite projective \Lambda -modules and let \alpha : P^\bullet \to K be an isomorphism in D(\Lambda ). Then \alpha ^{-1}\circ f\circ \alpha corresponds to a morphism of complexes f^\bullet : P^\bullet \to P^\bullet well defined up to homotopy. Set
\text{Tr}(f) = \sum _ i (-1)^ i \text{Tr}(f^ i : P^ i \to P^ i) \in \Lambda ^\natural .
Given P^\bullet and \alpha , this is independent of the choice of f^\bullet . Namely, any other choice is of the form \tilde{f}^\bullet = f^\bullet + dh +hd for some h^ i : P^ i \to P^{i-1}(i\in \mathbf{Z}). But
\begin{eqnarray*} \text{Tr}(dh) & = & \sum _ i (-1)^ i \text{Tr}(P^ i\xrightarrow {dh} P^ i) \\ & = & \sum _ i (-1)^ i \text{Tr}(P^{i-1}\xrightarrow {hd} P^{i-1}) \\ & = & -\sum _ i (-1)^{i-1}\text{Tr}(P^{i-1}\xrightarrow {hd} P^{i-1}) \\ & = & - \text{Tr}(hd) \end{eqnarray*}
and so \sum _ i (-1)^ i \text{Tr} ((dh+hd)|_{P^ i})=0. Furthermore, this is independent of the choice of (P^\bullet , \alpha ): suppose (Q^\bullet , \beta ) is another choice. The compositions
Q^\bullet \xrightarrow {\beta } K \xrightarrow {\alpha ^{-1}} P^\bullet \quad \text{and}\quad P^\bullet \xrightarrow {\alpha } K \xrightarrow {\beta ^{-1}} Q^\bullet
are representable by morphisms of complexes \gamma _1^\bullet and \gamma _2^\bullet respectively, such that \gamma _1^\bullet \circ \gamma _2^\bullet is homotopic to the identity. Thus, the morphism of complexes \gamma _2^\bullet \circ f^\bullet \circ \gamma _1^\bullet : Q^\bullet \to Q^\bullet represents the morphism \beta ^{-1}\circ f\circ \beta in D(\Lambda ). Now
\begin{eqnarray*} \text{Tr}(\gamma _2^\bullet \circ f^\bullet \circ \gamma _1^\bullet |_{Q^\bullet }) & = & \text{Tr}(\gamma _1^\bullet \circ \gamma _2^\bullet \circ f^\bullet |_{P^\bullet })\\ & = & \text{Tr}(f^\bullet |_{P^\bullet }) \end{eqnarray*}
by the fact that \gamma _1^\bullet \circ \gamma _2^\bullet is homotopic to the identity and the independence of the choice of f^\bullet we saw above. \square
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