“An attempt to receoncile the scheme-theoretic point of view with the birational one”.
Colliot-Thelene: Birational invariants, Purity, and the Gersten conjecture. In Proc. Symp. Pure Math. vol 58.1 (1995).
Smooth integral varieties over a field .
There is a general formalism, starting from a functor from rings, or k-algs, (possibly only with the flat morphisms) to Ab. The output is a constravariant functor on (smooth integral???) k-varieties, or something like that. For this to work, one needs some of the following properties of the original functor:
The unramified cohomology groups is defined as a subgroup of etale cohomology of the function field, with torsion (twisted roots of unity) coefficients.
Artin and Mumford: Examples of unirational but nonrational varieties.
arXiv: Experimental full text search
AG (Algebraic geometry)
Cat
Unramified cohomology of quadrics: http://www.math.uiuc.edu/K-theory/0221 and http://www.math.uiuc.edu/K-theory/0338 and http://www.math.uiuc.edu/K-theory/0359
Monnier: Unramfied cohomology and quadratic forms (2000)
arXiv:1001.4574 Birational invariants and A^1-connectedness from arXiv Front: math.KT by Aravind Asok We study some aspects of the relationship between A^1-homotopy theory and birational geometry. We study the so-called A^1-singular chain complex and zeroth A^1-homology sheaf of smooth algebraic varieties over a field k. We exhibit some ways in which these objects are similar to their counterparts in classical topology and similar to their motivic counterparts (the (Voevodsky) motive and zeroth Suslin homology sheaf). We show that if k is infinite the zeroth A^1-homology sheaf is a birational invariant of smooth proper varieties, and we explain how these sheaves control various cohomological invariants, e.g., unramified étale cohomology. In particular, we deduce a number of vanishing results for cohomology of A^1-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of A^1-connectedness by means of vanishing of unramified invariants.
arXiv:1102.0375 Classes de cycles motiviques étales from arXiv Front: math.AG by Bruno Kahn Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (étale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Thélène and Voisin. If k is separably closed, finite or p-adic, this describes it as an extension of a finite group F by a divisible group D, where F is the torsion subgroup of the cokernel of the l-adic cycle map. If k is finite and X is projective and of abelian type, verifying the Tate conjecture, D=0. If k is separably closed, we relate D to an l-adic Griffiths group. If k is the separable closure of a finite field and X comes from a variety over a finite field as described above, then D = 0 as soon as H^3(X,Q_l) is entirely of coniveau > 0, but an example of Schoen shows that this condition is not necessary.
nLab page on Unramified cohomology