Let be a variety over a separably closed field , and let be some object in . Laumon proved the beautiful result that the usual and compactly supported Euler characteristics and 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 ; after a blowup, we can assume that is an effective Cartier divisor. Write for the inclusion of the boundary. By the usual triangle , we reduce to showing that . Filtering by its perverse cohomology sheaves, we reduce further to the case where also is perverse. Cover by open affines such that is the divisor of a function . By an easy Mayer-Vietoras argument, it’s now enough to show that for every open contained in some , .
But now we win: for any choice of such , there is an exact triangle in where is the unipotent nearby cycles functor associated with , and the first arrow is the logarithm of the unipotent part of the monodromy. Since 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 , the class is identically zero. From here it’s easy to get a relative version of Laumon’s result.
Theorem. Let be any map of -varieties. Then for any , there is an equality in .