User pz - MathOverflow

Galois descent for semilinear endomorphisms

2013-03-26T15:46:54Z

Let $K \subset L$ be a finite Galois extension, $\sigma$ an automorphism of $L$ (not necessarily fixing $K$) and let $E$ be a finite-dimensional vector space over $L$ together with an $\sigma$-linear endomorphism $\phi : E \rightarrow E$. I would like to do Galois descent. So my question is "Under what conditions does $\phi$ ascent from a $\sigma$-linear endomorphism defined over $K$?".

Thanks.

EDIT: I guess I should describe my problem more precisely.

$k$ a perfect field of char. $p>0$, $K$ the fraction field of the Witt ring $W(k)$, $F$ finite extension of $\mathbb{Q}_p$, $L$ the compositum of $K$ and $F$ in $\bar{K}$, $\tau$ the Frobenius automorphism of $L$ over $F$ and $\sigma$ the Frobenius automorphism of $K$.

A $\tau$-$L$-space is a finite dimensional vector space $V$ over $L$ together with a $\tau$-semilinear bijection $\Phi: V\rightarrow V$.

If I'm given an $\sigma^f$-isocrystal $V$ over $k$, then I can associate to it an $\tau$-$L$-space as follows. Let $F$ be the unique unramified extension of $\mathbb{Q}_p$ of degree $f$, $L=KF$ then $\tau|_K = \sigma^f$ and tensoring with $L$ gives a $\tau$-$L$-space.

I'm interested in the other direction. Let $F$ be an unramified extension of $\mathbb{Q}_p$ of degree $f$ and $E$ an $\tau$-$L$-space. Does $E$ come from a $\sigma^f$-isocrystal over $k$?

Two different definitions of $\sigma$-L-spaces in Kottwitz I and II

2013-03-22T20:06:02Z

In his papers "Isocrystals with additional structure" I and II, Kottwitz defines the notion of $\sigma$-$L$-spaces. In the first one the situation is the following

$k$ an algebraically closed field of char. $p>0$, $K$ the fraction field of the Witt ring $W(k)$, $F$ finite extension of $\mathbb{Q}_p$, $L$ the compositum of $K$ and $F$ in $\bar{K}$, $\sigma$ the Frobenius automorphism of $L$ over $F$.

He later defines a $\sigma$-$L$-space to be a finite dimensional vector space $V$ over $L$ together with a $\sigma$-semilinear bijection $\Phi: V\rightarrow V$.

In "Isocrystals with additional structure II" he considers a different situation.

Again $F$ is a finite extension of $\mathbb{Q}_p$, $F^{nr}$ the maximal unramified extension of $F$ in some algebraic closure $\bar{F}$, but now $L$ is the completion of $F^{nr}$, $\sigma$ is the continuous extension of the Frobenius automorphism of $F^{nr}$ over $F$.

He then defines $\sigma$-$L$-spaces exactly as above.

I don't see how these two definitions are the same, since the second definition makes no reference to the Witt ring. What is the relation between these two definitions?

EDIT: Kottwitz assumes in his paper that $k$ is algebraically closed. But actually I'm interested in the more general situation of $k$ just a perfect field of char. $p$. Do the definitions agree in this more general situation?

Thanks.

Answer by pz for Constructible topology on schemes

2013-01-25T15:17:10Z

In Hochster's paper 'Prime Ideal Structure in Commutative Rings' the author uses it to characterize spectral spaces. This in turn is used in Huber's work on Adic spaces, cf. 'Huber - Étale cohomology of Rigid Analytic Varieties and Adic Spaces' and 'Scholze - Perfectoid spaces'.

Structure of f.g. modules over a non-commutative ring

2013-01-14T13:47:15Z

To what extent is the structure theorem for finitely generated modules over principal ideal domains true over non-commutative domains? I'm in particular interested in non-commutative euclidean domains especially the twisted polynomial ring $K\langle X \rangle$ over a field $K$ (i.e. such that $Xa = \sigma(a)X$ for some automorphism $\sigma$ of K).

$\sigma$-conjugate iff $p$-adically close

2012-12-05T14:08:37Z

First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}_p$-algebra, $R_W = R \otimes_Z W $, $\phi$ the $R$-linear extension of $\sigma$ to $R_W$. Also denote by $\phi$ the endomorphism of $R_W((u))$ given by $\phi(\sum_i a_i u^i)=\sum_i \phi(a_i) u^{pi}$.

$LG(R) = GL_d(R_W((u)))$
$LG^+(R) = GL_d(R_W[[u]])$
$LG^{\leq m}(R) = { A \in GL_d(R_W((u))) \mid A, A^{-1} \in u^{-m}M_d(R_W[[u]]) }$

Proposition 2.2 in Pappas, Rapoport - $\phi$-modules and coefficient spaces, available here states the following

Suppose $n > \frac{2m}{p-1}$

For each $g\in U_n(R)$, $A\in LG^{\leq m}(R)$ there is a unique $H\in U_n(R)$ such that $g^{-1} A \phi(g) = H^{-1}A$.
Conversely for each $A\in LG^{\leq m}(R)$, $H\in U_n(R)$ there is a unique $g\in U_n(R)$ such that $g^{-1} A \phi(g) = H^{-1}A$.

In the footnote the authors state that the analogous fact in classical Dieudonné theory is also true. However it seems that the proof given there does not work in that case, the problem being that the Frobenius on the Witt vectors leaves the uniformizer $p$ fixed whereas in the situation of Proposition 2.2 above the uniformizer $u$ is raised to the $p$-th power by $\phi$. In the first statement this forces me to assume $m$ to be $0$, but the comparison of the coefficients in uniqueness part of 2. doesn't seem to be working at all.

How can one prove this result in the classical case?

EDIT: Scholze uses a similar thing in 'The Langlands-Kottwitz Method and Deformation Spaces of p-Divisible Groups'. This is Lemma 4.4 there. But he does not get explicit bounds.

Comment by pz

2013-03-27T11:52:47Z

I edited my Question to give mor information.

Comment by pz

2013-03-27T11:45:02Z

@Keerthi Madapusi Pera: Yes, see my EDIT.

Comment by pz

2013-03-23T21:18:10Z

Thanks for the comments. I was not aware that $F^{nr}$ is the compositum of $F$ and $K$ in this case. Can you perhaps give a reference? Kottwitz assumes in his paper that $k$ is algebraically closed. But actually I'm interested in the more general situation of $k$ just a perfect field of char. $p$. Do the definitions agree in this more general situation?

Comment by pz

2013-01-25T18:21:00Z

@Ricky I think you are right. Huber uses spectral spaces in some proofs, but I think he doesn't do anything with them that could not be done without them. They are afaik not crucial to the theory of adic spaces.

Comment by pz

2013-01-16T21:22:05Z

Thanks. That's what I'm looking for.