Euler characteristics and perverse sheaves

Let X be a variety over a separably closed field k, and let A be some object in D^b_c(X,\mathbf{Q}_{\ell}). Laumon proved the beautiful result that the usual and compactly supported Euler characteristics \chi(X,A) and \chi_c(X,A) are always equal. Recently while trying to do something else, I found a quick proof of Laumon’s result, as well as a relative version, and I want to sketch this here.

Pick an open immersion into a compactification j:X \to X'; after a blowup, we can assume that Z=X' - X is an effective Cartier divisor. Write i:Z \to X' for the inclusion of the boundary. By the usual triangle j_!A \to Rj_*A \to i_*i^* Rj_*A \to , we reduce to showing that \chi(X',i_*i^* Rj_*A)=0. Filtering A by its perverse cohomology sheaves, we reduce further to the case where also A is perverse. Cover X' by open affines X_n' such that Z_n= Z \cap X_n' is the divisor of a function f_n. By an easy Mayer-Vietoras argument, it’s now enough to show that for every open U contained in some X_n', \chi(U,(i_* i^{\ast}Rj_{\ast}A)|U) = 0.

But now we win: for any choice of such U \subset X_n', there is an exact triangle R\psi_{f_n}(A|U \cap X) \to R\psi_{f_n}(A|U \cap X) \to (i_* i^{\ast}Rj_{\ast}A)|U \to in D^b_c(U,\mathbf{Q}_{\ell}) where R\psi_{f_n}:\mathrm{Perv}(U \cap X) \to \mathrm{Perv}( U \cap Z) is the unipotent nearby cycles functor associated with f_n, and the first arrow is the logarithm of the unipotent part of the monodromy. Since \chi(U, -) is additive in exact triangles and the first two terms agree, we’re done.

A closer reading of this argument shows that you actually get the following stronger statement: for any A, the class [i_*i^* Rj_*A] \in K_0\mathrm{Perv}(X') is identically zero. From here it’s easy to get a relative version of Laumon’s result.

Theorem. Let f:X \to Y be any map of k-varieties. Then for any A\in D^b_c(X,\mathbf{Q}_\ell), there is an equality [Rf_! A]=[Rf_\ast A] in K_0\mathrm{Perv}(Y).