Let $A$ be a commutative noetherian ring. 


Let $K\_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated projective $A$-modules. The functor $K\_{parf}(A) \longrightarrow D(A)$ is fully faithful and we denote its essential image by $D\_{parf}(A)$. An object of $D\_{parf}(A)$ is called a **perfect complex**.


I have to give a talk about the Trace formula on tuesday and the audience and me are not so comfortable with derived categories (yet). (See below for what we want to do this tuesday.)


**Question 1.** Is a complex of $A$-modules perfect iff it is quasi-isomorphic to a bounded complex of finitely generated projective $A$-modules? Is there some condition involving ``homotopy'' missing?


Let $k$ be an algebraically closed field and let $\mathcal{F}$ be a locally constant finite sheaf of finitely generated projective $A$-modules on a finite type separated $k$-scheme $X$.


This defines an object $$K=\ldots \longrightarrow 0\longrightarrow \mathcal{F} \longrightarrow 0 \longrightarrow \ldots $$ of the derived category $D(X,A)$ of sheaves of $A$-modules. 


**Question 2.** Can one explicitly write down a complex representing $R\Gamma\_c(X,\mathcal{F})$? (One may assume $X$ is proper over $k$ for simplicity.) 


One can show that $R\Gamma\_c(X,\mathcal{F})$ is a perfect complex after identifying $A$-modules with sheaves of $A$-modules on $\textrm{Spec} \ k$. If the answer to Question 2 is positive, I could avoid introducing derived categories on tuesday. 


[ADDED]


To prove Grothendieck's generalized Lefschetz trace formula, it suffices to prove the following statement.


**Theorem.** Let $X\_0$ be a smooth affine geometrically integral curve over $\mathbf{F}\_q$ and let $\Lambda$ be a finite ring which gets killed by a power of a prime number $\ell$ invertible in $\mathbf{F}\_q$. Suppose that $X\_0(k) = \emptyset$. Then, for any locally constant finite sheaf $\mathcal{F}$ of finite projective $\Lambda$-modules, we have that $$ 0= \textrm{Tr}(\textrm{Frob}^\ast, R\Gamma\_c(X,K)).$$


So one has to do three things:


1. Understand the statement above.
2. Show that this statement implies the Trace formula
3. Prove the statement above.


1 and 3 don't really need the language of derived categories, I believe. That's why I asked this question. Next week we will take care of 2 after introducing derived categories.



Perfect complexes and RGamma(X,F) without mentioning derived categories
For question 1 the answer is yes, in fact the question is tautological. As for question 2 (I assume that $R\Gamma\_c$ stands for the sections with compact support which in the case of proper $X$ coincide with $R\Gamma$), of course you can use any standard complex, for example you can use Cech complex.


