Remark 59.83.3 (0A47): Invariance under extension of algebraically closed ground field

Remark 59.83.3 (Invariance under extension of algebraically closed ground field). Let $k$ be an algebraically closed field of characteristic $p > 0$. In Section 59.63 we have seen that there is an exact sequence

\[ k[x] \to k[x] \to H^1_{\acute{e}tale}(\mathbf{A}^1_ k, \mathbf{Z}/p\mathbf{Z}) \to 0 \]

where the first arrow maps $f(x)$ to $f^ p - f$. A set of representatives for the cokernel is formed by the polynomials

\[ \sum \nolimits _{p \not| n} \lambda _ n x^ n \]

with $\lambda _ n \in k$. (If $k$ is not algebraically closed you have to add some constants to this as well.) In particular when $k'/k$ is an algebraically closed extension, then the map

\[ H^1_{\acute{e}tale}(\mathbf{A}^1_ k, \mathbf{Z}/p\mathbf{Z}) \to H^1_{\acute{e}tale}(\mathbf{A}^1_{k'}, \mathbf{Z}/p\mathbf{Z}) \]

is not an isomorphism in general. In particular, the map $\pi _1(\mathbf{A}^1_{k'}) \to \pi _1(\mathbf{A}^1_ k)$ between étale fundamental groups (insert future reference here) is not an isomorphism either. Thus the étale homotopy type of the affine line depends on the algebraically closed ground field. From Lemma 59.83.2 above we see that this is a phenomenon which only happens in characteristic $p$ with $p$-power torsion coefficients.

The Stacks project

Comments (0)

Post a comment