1
$\begingroup$

Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\operatorname{GL}_2(R)/B(R)$, where $B(R)$ is the standard Borel subgroup (upper triangular matrices).

Is the functor $F$ representable by a sheaf?

I think that the answer should be no, because if $R_1 = \mathbb{Q}_p[[t]]$, then the orbits of the following pair of $\operatorname{SL}_2(R_1)$ matrices under right multiplication by $B(R_1)$ are disjoint $$ \begin{pmatrix} (1-t)^{-1}& 0\\ 1& 1-t \end{pmatrix},\quad \begin{pmatrix} 1& 0\\ 1& 1 \end{pmatrix} $$ however, if $R_2 = \mathbb{Q}_p$ and $R_1 \longrightarrow R_2$ is the $\mathbb{Q}_p$ ring homomorphism sending $t$ to $0$, then the two cosets become identical.

Is this proof correct?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Do you intend to switch from $\operatorname{GL}_2$ to $\operatorname{SL}_2$? Also, I don't know what "$\mathbb Q_p$ ring homomorphism" means. Should it be "$\mathbb Q_p$-algebra homomorphism"? $\endgroup$
    – LSpice
    Commented Jan 30, 2023 at 21:08
  • $\begingroup$ Actually the answer is yes to both questions $\endgroup$
    – kindasorta
    Commented Jan 30, 2023 at 21:40

0

Reset to default

You must log in to answer this question.