Dtjytyk

Is it possible to create a QR code using text?

Finitely generated matrix groups whose eigenvalues are all algebraic

Bullying boss launched a smear campaign and made me unemployable

How could indestructible materials be used in power generation?

Did 'Cinema Songs' exist during Hiranyakshipu's time?

Was the Stack Exchange "Happy April Fools" page fitting with the '90's code?

My ex-girlfriend uses my Apple ID to log in to her iPad. Do I have to give her my Apple ID password to reset it?

Are British MPs missing the point, with these 'Indicative Votes'?

How to show a landlord what we have in savings?

Placement of More Information/Help Icon button for Radio Buttons

Machine learning testing data

How does a dynamic QR code work?

Processor speed limited at 0.4 Ghz

How to travel to Japan while expressing milk?

how do we prove that a sum of two periods is still a period?

Can compressed videos be decoded back to their uncompresed original format?

Why was Sir Cadogan fired?

Send out email when Apex Queueable fails and test it

Unlock My Phone! February 2018

What is the fastest integer factorization to break RSA?

OP Amp not amplifying audio signal

Can a virus destroy the BIOS of a modern computer?

Convert seconds to minutes

Is there a hemisphere-neutral way of specifying a season?

Completion of the local ring at a point on arithmetic surfaces.

Completion of regular local ringsRegular schemes and base changeThe local ring of the generic point of a prime divisorlocal equation of a divisor, localization and local ring at a pointDivisor on an arithmetic surface and “base change”completion morphism for $mathscr{H}_x$, the local ring of germs of holomorphic functions in a neighborhood of xDiscrete valuation on the local ring of a nonsingular curvecomparison between valuation ring and proper morphismLocal ring of a surface: polynomial expressionComputation of completion of a local ring

2

$begingroup$

Let $K$ be a number field and consider a arithmetic surface $Xto B=operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$.

Now pick a closed point $xin X$ such that $xmapsto bin B$ and consider $widehat{mathcal O_{X,x}}$. In other words the completion of the local ring $mathcal O_{X,x}$ with respect to its maximal ideal.

Can we express $widehat{mathcal O_{X,x}}= A[[t]]$? What is $A$ in
this case? Do we have $A=O_L$ where $L$ is a complete discrete
valuation field and a finite extension of $K_b$ (here $K_b$ is the completion of $K$ at $b$)?

For sure we have an embedding $mathcal O_{B,b}[t]hookrightarrowmathcal
O_{X,x}$

Thanks in advance

algebraic-geometry power-series surfaces local-rings

share|cite|improve this question

edited Feb 18 at 15:08

notsure

asked Feb 17 at 15:52

notsurenotsure

547

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I asked a similar question in Mathoverflow some time ago but I didn't get an answer: Completion of a local ring of an arithmetic surface
  $endgroup$
  – yamete kudasai
  Feb 18 at 15:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Take a look at Liu's book exercise 4, 3.35 (d) at page 149. But it assumes the existence of a section.
  $endgroup$
  – notsure
  Feb 18 at 15:03
  

  
  
  
  

add a comment | 

2

$begingroup$

Let $K$ be a number field and consider a arithmetic surface $Xto B=operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$.

Now pick a closed point $xin X$ such that $xmapsto bin B$ and consider $widehat{mathcal O_{X,x}}$. In other words the completion of the local ring $mathcal O_{X,x}$ with respect to its maximal ideal.

Can we express $widehat{mathcal O_{X,x}}= A[[t]]$? What is $A$ in
this case? Do we have $A=O_L$ where $L$ is a complete discrete
valuation field and a finite extension of $K_b$ (here $K_b$ is the completion of $K$ at $b$)?

For sure we have an embedding $mathcal O_{B,b}[t]hookrightarrowmathcal
O_{X,x}$

Thanks in advance

algebraic-geometry power-series surfaces local-rings

share|cite|improve this question

edited Feb 18 at 15:08

notsure

asked Feb 17 at 15:52

notsurenotsure

547

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I asked a similar question in Mathoverflow some time ago but I didn't get an answer: Completion of a local ring of an arithmetic surface
  $endgroup$
  – yamete kudasai
  Feb 18 at 15:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Take a look at Liu's book exercise 4, 3.35 (d) at page 149. But it assumes the existence of a section.
  $endgroup$
  – notsure
  Feb 18 at 15:03
  

  
  
  
  

add a comment | 

$begingroup$

Let $K$ be a number field and consider a arithmetic surface $Xto B=operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$.

Now pick a closed point $xin X$ such that $xmapsto bin B$ and consider $widehat{mathcal O_{X,x}}$. In other words the completion of the local ring $mathcal O_{X,x}$ with respect to its maximal ideal.

Can we express $widehat{mathcal O_{X,x}}= A[[t]]$? What is $A$ in
this case? Do we have $A=O_L$ where $L$ is a complete discrete
valuation field and a finite extension of $K_b$ (here $K_b$ is the completion of $K$ at $b$)?

For sure we have an embedding $mathcal O_{B,b}[t]hookrightarrowmathcal
O_{X,x}$

Thanks in advance

algebraic-geometry power-series surfaces local-rings

share|cite|improve this question

edited Feb 18 at 15:08

notsure

asked Feb 17 at 15:52

notsurenotsure

547

$endgroup$

share|cite|improve this question

edited Feb 18 at 15:08

notsure

asked Feb 17 at 15:52

notsurenotsure

547

share|cite|improve this question

edited Feb 18 at 15:08

notsure

asked Feb 17 at 15:52

notsurenotsure

547

asked Feb 17 at 15:52

notsurenotsure

547

asked Feb 17 at 15:52

notsurenotsure

547