4
$\begingroup$

Let $G$ be a finite abelian group and let $S$ be a variety over $\mathbb{F}_p$. It is natural (I think) to expect that the cohomology group $H^i(S,G)$ is finite. But with respect to which cohomology?

For instance, if $n:=\# G$ is invertible on $S$, then $H^i_{et}(S,G)$ is finite.

But I now wonder:

What if $n$ is not invertible on $S$? Could the latter set be infinite?

What if we use fppf cohomology? Can $H^i_{fppf}(S,G)$ be infinite?

Improve this question
$\endgroup$
2
  • 7
    $\begingroup$ Consider Artin-Schreier extensions of $\mathbb{A}^1_k$, i.e., $\text{Spec}\ k[t][x]/\langle x^p-x-f(t) \rangle$ for $f(t)\in k[t]$. By the Artin-Schreier sequence, this $\mathbb{Z}/p\mathbb{Z}$-torsor over $\mathbb{A}^1_k$ is classified by the image $[f]$ in the cokernel of $A:k[t]\to k[t]$, $A(g) = g^p-g$. A $k$-basis for this quotient consists of the images $[t^d]$ of the infinitely many monomials $t^d$ with $d$ relatively prime to $p$. $\endgroup$
    – Jason Starr
    Commented Oct 22, 2016 at 13:18
  • 7
    $\begingroup$ To supplement Jason's comment $H_{et}^1(\mathbb{A}_k^1,\mathbb{Z}/p)$ is infinite over a field of char $p$. $\endgroup$
    – Donu Arapura
    Commented Oct 22, 2016 at 15:25

1 Answer 1

Reset to default
4
$\begingroup$

I am just posting my comment above as an answer. Donu's comment is exactly correct: I failed to point out that the argument proves that $H^1_{et}(\mathbb{A}^1_k,\mathbb{Z}/p\mathbb{Z})$ is infinite.

Consider Artin-Schreier extensions of $\mathbb{A}^1_k$, i.e., $\text{Spec}\ k[t][x]/\langle x^p-x-f(t)\rangle$ for $f(t)\in k[t]$. By the Artin-Schreier sequence, this $\mathbb{Z}/p\mathbb{Z}$-torsor over $\mathbb{A}^1_k$ is classified by the image $[f]$ in the cokerenel of $A:k[t]\to k[t], \ A(g) = g^p-g$. A $k$-basis for this quotient consists of the image $[t^d]$ of the infinitely many monomials $t^d$ with $d$ relatively prime to $p$.

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ Maybe it is worth mentioning that this also proves that $H^1_{fppf}(\mathbb{A}^1_k,\mathbb{Z}/p)$ is infinite, since étale torsors are also fppf torsors (and in fact, since $\mathbb{Z}/p$ is an étale group scheme, the vice versa is also true). $\endgroup$
    – Denis Nardin
    Commented Oct 24, 2016 at 12:17

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .