The adic tame site

For a scheme of characteristic $p > 0$ (or mixed characteristic) étale cohomology with $p$-torsion coefficients does not behave very well: Smooth base change, cohomological purity, Poincaré duality, just to name a few, only hold for coefficients prime to the characteristic. The reason for this failure is the existence of wild ramification. This talk presents a modification of the étale topology that does not admit for wild ramification, called the tame site. For coefficients away from the characteristic the étale and tame cohomology groups are isomorphic and for $p$-torsion coefficients they are better behaved than the étale cohomology groups.