4
$\begingroup$

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\rightarrow k_E^{sep}$ on the seperable closure of $k_E$ denoted by $k_E^{sep}=(k_E)^{sep}$ (eg. if $k=\mathbb{F}^p$, we can choose $\varphi:=(\cdot)^p$).

Now let $\mathbb{G}$ be a linear algebraic group over $k$ and let $\mathbb{N}$ act on $\mathbb{G}(k_E^{sep})$ via $1_{\mathbb{N}}\mapsto \mathbb{G}(\varphi)$. I want to know, if there are nice classes of such $\mathbb{G}$ (eg. reductive $\mathbb{G}$ would be great :P), such that $$H^1(\mathbb{N},\mathbb{G}(k_E^{sep}))=1,$$ i.e. for every $A\in\mathbb{G}(k_E^{sep})$ there exists a $B\in\mathbb{G}(k_E^{sep})$, such that $B=A\cdot\mathbb{G}(\varphi)(B)$.

For $\mathbb{G}=GL_n$ this seems to be true, since for a finite dimensional $k_E^{sep}$-vector space $V$ equiped with a $\varphi$-semilinear map $\varphi_V$, there always exists a $k_E^{sep}$-basis $(v_i)_i$, such that $\varphi_V(v_i)=v_i$. See [Peter Schneider: Galois representations and $(\varphi,\Gamma)$-modules, Proposition 3.2.4] or if you are capable to read some german [https://ivv5hpp.uni-muenster.de/u/pschnei/publ/lectnotes/Theorie-des-Anstiegs.pdf, Satz 2.1].

Improve this question
$\endgroup$
7
  • 1
    $\begingroup$ I've never seen monoid (as opposed to group) cohomology before. Is your definition of the vanishing of $\mathrm H^1$ all that you are asking? (That is, would a positive answer to the reformulated question—which I can't provide—suffice?) $\endgroup$
    – LSpice
    Commented Jul 17, 2018 at 11:52
  • $\begingroup$ Also, does $k_E^{sep}$ mean $(k^{\mathrm{sep}})_E = k^{\mathrm{sep}}((X))$, the maximal unramified extension of $k_E$, or $(k_E)^{\mathrm{sep}}$, the separable closure of $k_E$? $\endgroup$
    – LSpice
    Commented Jul 17, 2018 at 11:53
  • 1
    $\begingroup$ I meant the seperable closure of $k_E$. For $H^1$ you can define a 1-cocycle to be a map $\alpha:\mathbb{N}\rightarrow\mathbb{G}(k_E^{sep})$, which satisfies $$\alpha(n+m)=\alpha(n)\cdot\mathbb{G}(\varphi^n)(\alpha(m))$$ and a subset of 1-coborders to be the maps $\alpha_A(n)=A^{-1}\cdot\mathbb{G}(\varphi^n)(A)$ for some $A\in\mathbb{G}(k_E^{sep})$. The vanishing of $H^1$ means, that the 1-coborders are exactly the 1-cocycles. $\endgroup$
    – Estus
    Commented Jul 17, 2018 at 11:57
  • $\begingroup$ (I'm sorry, I see now that you already explicitly said in your post that $k_E^{sep}$ was the separable closure.) $\endgroup$
    – LSpice
    Commented Jul 17, 2018 at 12:21
  • 4
    $\begingroup$ This feels (which I guess is your motivation) like Lang–Steinberg + Hensel's lemma. Does any approach like that seem plausible? $\endgroup$
    – LSpice
    Commented Jul 17, 2018 at 12:28

0

Reset to default

You must log in to answer this question.