Nonstandard étale cohomology

A lot of good properties of étale cohomology only hold for torsion coefficients. We use ultraproducts respectively enlargement construction to define a cohomology theory that inherits the important properties of étale cohomology while allowing greater flexibility with the coefficients. In particular, choosing coefficients ∗Z/P∗Z (for P an infinite prime and ∗Z the enlargement of Z) gives a Weil cohomology, and choosing ∗Z/lh∗Z (for l a finite prime and h an infinite number) allows comparison with ordinary l-adic cohomology. More generally, for every N∈∗Z, we get a category of ∗Z/N∗Z-constructible sheaves with good properties.