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:

- Understand the statement above.
- Show that this statement implies the Trace formula
- 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.