Class field theory for singular schemes over finite fields

A classical theorem states that the Galois group of the maximal abelian unramified extension of a number field is isomorphic to the class group of the number field. A similar statement holds for global fields of finite characteristic, giving the automorphism group of the maximal abelian etale covering of a smooth and proper curve.This has been generalized by Bloch and Kato-Saito to smooth and proper schemes of any dimension, and further generalized by Spiess-Schmidt to smooth schemes. We discuss a further generalization to arbitrary schemes.