p adic – Universal field extension in which $nu(a) = 1$ for each valuation $nu$

Let $K$ be a field, and let $a in K$ be an element.

I seek the universal ring $A$ with a map $K rightarrow A$, where each valuation $nu : A rightarrow Lambda cup { infty }$, with $Lambda = (Lambda, +, 0)$ a totally ordered abelian group sends $a$ to $0$.

This is analogous to localization, except with valuations instead of prime ideals.

ag.algebraic geometry – Open immersion of affinoid adic spaces

If $R$ and $S$ are complete Huber rings with $varphi: R to S$ a continuous map, then is it true in general that if $mathrm{Spa}(S, S^circ) to mathrm{Spa}(R, R^circ)$ is an open immersion of adic spaces (here $S^circ$ and $R^circ$ are the power-bounded subrings) then $mathrm{Spec}(S) to mathrm{Spec}(R)$ is injective?

For example, this is true if $R$ and $S$ both have the discrete topology, because if $frak p$ and $frak q$ are two prime ideals in $S$ which are equal after restricting to $R$ then $(frak p, |cdot|_{rm triv})$ and $(frak q, |cdot|_{rm triv})$ (trivial valuations), which are both points in $mathrm{Spa}(S,S)$, restrict to the trivial valuation on $R/varphi^{-1}(frak p)$.

But I’m not sure how generally to expect that this is true.

p adic numbers – Continous morphisms of a local field with conditions in positive characteristic

Let $P$ be a an irreducible polynomial of $k:=mathbb F_q(T)$, $Omega_P$ be the completion of an algebraic closure $overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-adique valuation. Consider an algebraic closure $overline k$ of $k$ in $Omega_P$ and $alphainoverline k$. Does it exist a continuous $mathbb F_q$-morphism $sigma$ of $Omega_P$ such that $sigma(T)=T+xi$ with $xiinmathbb F_q$ and $sigma(alpha)=alpha$. If not in a whole generality, can one determine the $alpha’s$ such that the problem admits an answer?

ra.rings and algebras – When adic completion preserves projectives?

Lets take a ring $R$ with a finitely generated ideal $mathfrak p$ in it. What properties of the ring (…and maybe ideal, but I’d really like all f.g. ideals to be like that) imply that $mathfrak p$-adic completion of any projective module is again projective? I’m more interested in nontrivial necessary conditions than sufficient ones. (do not have much hope for full characterisation)

P. S. Ring being right perfect + left coherent + gd $leq 2$ is equivalent to “projectives closed under arbitrary limits”. I tried to cook up an example which does not look like that to no avail, and would be happy to see one.

p adic numbers – Is this topology on $mathbb{Q}$ well studied?

Let $|cdot|_p$ denote the $p$-adic norm on $mathbb{Q}$. For the whole set of primes $P$ consider the topology which is generated with prebase of open sets $V_{p,varepsilon}(x) = {yinmathbb{Q} : |x-y|_p<varepsilon}$, $pin P$. Is there any reference for this topology?

p adic number theory – Dirac Measure and Iwasawa Algebras

I am reading through some lecture notes on p-adic L-functions, and one of the exercises asks for $a in mathbb{Z}_p$ that we define the Dirac measure $delta_a$ by

$int_{mathbb{Z}_p} phi cdot delta_a = phi(a)$

and show that the $O_L$-module generated by the $delta_a$ for $a in mathbb{N}$ is dense in the Iwasawa algebra $Lambda (mathbb{Z}_p)$. I am not sure about this, is it related to the argument(s) by which one shows that the Dirac measures are dense in the space of measures? The lecture notes can be found here.

ag.algebraic geometry – Intuition for the adic unit disc $ operatorname {Spa} (C langle T rangle, C ^ { circ} langle T rangle) $

Let $ C $ be an algebraically closed field, complete compared to a non-Archimedean assessment (e.g. $ C = mathbb {C} _p $). The points of the adic spectrum $ operatorname {Spa} (C langle T rangle, C ^ { circ} langle T rangle) $ have been completely classified, consisting of five families of points. The following is from Scholze Perfectoid spaces.

  1. Classic points: Take $ x in C ^ { circ} $and consider the assessment $ f mapsto | f (x) | $.

Points of type 2 and 3 are the "spokes" of the tree, and they follow the following recipe. Let $ 0 leq r <1 $and correct $ x in C ^ circ $. Develop a given power series $ f $ as $ sum_n a_n (T-x) ^ n $and send it to $ sup | a_n | r ^ n $.

  1. Assessments as above, but with the constraint that & # 39;$ r in | C ^ { circ} | $". Branching occurs precisely at points like this. I don't know what that notation means and I would appreciate someone telling me.
  2. The remaining evaluations, i.e. the shelves with $ r notin | C ^ { circ} | $.
  3. The "dead ends" of the tree. Let $ D_1 supseteq D_2 supseteq cdots , $ be a sequence of closed discs with $ bigcap_i D_i = varnothing $ (the existence (not) of such sequences is called spherical completeness), and consider the evaluation $ f mapsto inf_i sup_ {x in D_i} | f (x) | $.
  4. Some rankings$ 2 $ evaluations. To fix $ x in C ^ { circ} $ and $ 0 <r <$ 1. Choose a sign $ { lessgtr} in { < , >} $. Let $ Gamma _ { lessgtr} $ to be the abelian group ordered $ mathbb {R} _ {> 0} times gamma ^ { mathbb {Z}} $ such as $ r & # 39; lessgtr gamma lessgtr r $ for everyone $ r & # 39; lessgtr r $. We are now defining our valuation. Take any power series $ f $, develop it as $ sum_n a_n (T-x) ^ n $and send it to $ sup | a_n | gamma ^ n $.

The following illustration is with the list.

Test

I'm having a hard time with the intuition behind this photo. Here is what I can collect. The arc at the bottom represents the classic unitary disc, and one can think that the vertical direction is the & # 39;$ r $-ax & # 39;. Certainly, then, the fact that there are rays makes sense. The type 5 points close to type 2 reflect them in the closure of the neighboring type 2 point.

Question. An open question might be: how should I think about all this? In terms of image in particular, a more specific question is: What does branching mean? Additional questions that I would like to understand: Can I see the open disc (for example) on this image? What is the relationship between my classical intuition for diagrams? What changes if we take, say, $ C = mathbb {Q} _p $ (that is, something that is not algebraically closed)? Do the allowable openings match something in this image?

All ideas are appreciated.

can the compact Lie group $ p $ -adic be integrated into $ O_K ^ n $ or $ text {GL} _n (K) $?

Let $ K $ be a local charecteristic area $ 0 $ and $ G $ be compact $ p $-adi lie group of dimension $ n $, so $ G $ be integrated into $ O_K ^ n $ or $ text {GL} _n (K) $ as a closed subgroup? It is a double question for the real case.

Recall that a $ p $-the adic Lie group is a variety with a locally analytical group structure and a variety is a topological space which is locally isomorphism to an open set in $ K ^ n $. These definitions are in Schneider's book on $ p $– Lie Lie group.

I'm more interested in the case of $ n = $ 1 and $ K = mathbb {Q} _p $, so that I can integrate $ G $ in $ mathbb {Z} _p $ or $ mathbb {Z} _p ^ times $.

Thank you!

ag.algebraic geometry – Dimension of $ ell $ -adic Eilenberg-Maclane space

I am currently studying the $ ell $-adic cohomology function, i.e. the functor $$ F: X rightarrow H ^ i_ {ét} (X, mathbb {Q} _ { ell}). $$
In a certain sense, it is a representable functor, that is to say that there is a $ ell $-adic Eilenberg-Maclane space (see Representability of Weil's cohomology theories in the theory of stable motivational homotopy). I would like to know the size of this space. The normal way to calculate the dimension of a module space is to study the deformations. However, since the spread site of a scheme is determined by the underlying reduced structure, we see that
$$ F (X ( epsilon)) = H ^ i_ {ét} (X ( epsilon), mathbb {Q} _ { ell}) cong H ^ i_ {ét} (X, mathbb {Q } _ { ell}) = F (X). $$
However, confusingly, this implies that this analogue of the Eilenberg-Maclane space is of zero dimension, which seems to me to be very false. Where did I go wrong?

arithmetic geometry – Hodge-Tate weight of $ p $ – Galician adic representations and integration in $ mathbb {C} _p $

Let $ K $ to be a finite extension of $ mathbb {Q} _p $ and $ W $ to be a representation of $ G_K $ dimensional $ 3 $.

assume $ W $ has Hodge-Tate weights -2s, 0, 2s $ or $ s $ is a non-zero integer, so I want to know why there is a $ G_K $-climbing $ W $ in $ mathbb {C} _p $. I also want to know why this integration is unique up to an element of $ K ^ * $ by the uniqueness of the disappearance of the Hodge-Tate weight.

These two facts are used in the paragraph after Proposition 5.3 and in the proof of Proposition 5.4. Please see SEN THEORY AND LOCALLY ANALYTICAL VECTORS.

Thank you!