(Constructible $\ell$-adic sheaf)
Let \(X\) be a scheme of finite type over a field and \(\ell\) a prime invertible on \(X\). A constructible \(\ell\)-adic sheaf is a projective system \(\mcf = (\mcf_n)_{n \geq 1}\) of constructible étale sheaves of \(\ZZ/\ell^n\)-modules with \(\mcf_{n+1} \otimes_{\ZZ/\ell^{n+1}} \ZZ/\ell^n \cong \mcf_n\).
(Finiteness of proper pushforward)
Let \(f \colon X \to Y\) be a morphism of schemes of finite type over a field, with \(\ell\) invertible, and \(\mcf\) a constructible \(\ell\)-adic sheaf on \(X\). Then \(R^i f_! \mcf = (R^i f_! \mcf_n)_n\) is, up to Artin–Rees equivalence, a constructible \(\ell\)-adic sheaf on \(Y\).
For \(Y = \Spec \overline{\FF}_q\), this is the statement that \(H^i_c(X_{\overline{\FF}_q}, \QQ_\ell) = \big(\varprojlim_n H^i_c(X, \ZZ/\ell^n)\big) \otimes \QQ_\ell\) is finite-dimensional, which is what the Grothendieck–Lefschetz trace formula needs.
Dependencies
D-SHFCONSTRConstructible sheavesusesT-BASECHANGEFlat, proper and smooth base changeuses