# Cohomology classes functorial under etale morphisms

Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms.

I am intersted in morphisms between constant functor $\mathcal{C}\to Vect_k$ which maps all objects to $k$ and all morphisms to identity and the functor of algebraic de Rham cohomology $X\mapsto H^i_{dR}(X)$.

Explicitly, this means fixing a class $c_X\in H^i_{dR}(X)$ for every $X$ such that $f^*c_Y=c_X$ for any etale $f:X\to Y$. For even $i$ Chern classes of tangent bundle give an example(since tangent bundle is preserved under etale morphisms).

Are there any nonzero functorial classes for odd $i$? Is it true that any functorial class in even cohomology is a polynomial in Chern classes?

• A useless remark: you can take different polynomials in the Chern classes for different dimensions of the varieties... – Dan Petersen Oct 11 '16 at 6:17
• As noticed by Dan Petersen, one should consider one dimension at a time (assuming that $c_{X \sqcup Y} = c_X \oplus c_Y$). The case of dimension 1 is easy : - for $i=0$, any functorial class is proportional to $c_0$, - for $i=1$ any functorial class is zero (the H^1 of a curve injects into the H^1 of any nonempty open subset, which can be chosen etale over $\mathbb{P}^1$), - for $i=2$, any functorial class is a multiple of $c_1$. – js21 Oct 11 '16 at 15:53