$PGL_n$ action on an affine variety

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Let $X$ be an affine variety on which $PGL_n$ acts freely. Then how to see that over the quotient variety $X / PGL_n$, there is a bundle of central simple algebras $M_n(k) times^PGL_n X$. I am not even able to understand the notation $M_n(k) times^PGL_n X$. Can anyone please let me know..

algebraic-geometry group-actions

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asked Aug 3 at 5:04

Amrita

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up vote
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Let $X$ be an affine variety on which $PGL_n$ acts freely. Then how to see that over the quotient variety $X / PGL_n$, there is a bundle of central simple algebras $M_n(k) times^PGL_n X$. I am not even able to understand the notation $M_n(k) times^PGL_n X$. Can anyone please let me know..

algebraic-geometry group-actions

share|cite|improve this question

asked Aug 3 at 5:04

Amrita

61

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  If you don't know what are you asking, how can we help you?
  – Taroccoesbrocco
  Aug 3 at 5:23
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Let $X$ be an affine variety on which $PGL_n$ acts freely. Then how to see that over the quotient variety $X / PGL_n$, there is a bundle of central simple algebras $M_n(k) times^PGL_n X$. I am not even able to understand the notation $M_n(k) times^PGL_n X$. Can anyone please let me know..

algebraic-geometry group-actions

share|cite|improve this question

asked Aug 3 at 5:04

Amrita

61

Let $X$ be an affine variety on which $PGL_n$ acts freely. Then how to see that over the quotient variety $X / PGL_n$, there is a bundle of central simple algebras $M_n(k) times^PGL_n X$. I am not even able to understand the notation $M_n(k) times^PGL_n X$. Can anyone please let me know..

algebraic-geometry group-actions

share|cite|improve this question

asked Aug 3 at 5:04

Amrita

61

share|cite|improve this question

share|cite|improve this question

asked Aug 3 at 5:04

Amrita

61

asked Aug 3 at 5:04

Amrita

61

asked Aug 3 at 5:04

Amrita

61

61

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  If you don't know what are you asking, how can we help you?
  – Taroccoesbrocco
  Aug 3 at 5:23
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  If you don't know what are you asking, how can we help you?
  – Taroccoesbrocco
  Aug 3 at 5:23
  

  
  
  
  

1

1

If you don't know what are you asking, how can we help you?
– Taroccoesbrocco
Aug 3 at 5:23

If you don't know what are you asking, how can we help you?
– Taroccoesbrocco
Aug 3 at 5:23

add a comment | 

1 Answer
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If a group $G$ acts on two varieties $X$ and $Y$ (on the left and right respectively) then one talks of an action on $Xtimes Y$ defined in the following way: $g.(x,y) = (xg^-1, gy)$. The quotient for this action is denoted by $Xtimes ^G Y$.

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answered Aug 3 at 5:14

P Vanchinathan

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1 Answer
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active

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1 Answer
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active

oldest

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up vote
3
down vote

If a group $G$ acts on two varieties $X$ and $Y$ (on the left and right respectively) then one talks of an action on $Xtimes Y$ defined in the following way: $g.(x,y) = (xg^-1, gy)$. The quotient for this action is denoted by $Xtimes ^G Y$.

share|cite|improve this answer

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

add a comment | 

up vote
3
down vote

If a group $G$ acts on two varieties $X$ and $Y$ (on the left and right respectively) then one talks of an action on $Xtimes Y$ defined in the following way: $g.(x,y) = (xg^-1, gy)$. The quotient for this action is denoted by $Xtimes ^G Y$.

share|cite|improve this answer

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

add a comment | 

up vote
3
down vote

up vote
3
down vote

If a group $G$ acts on two varieties $X$ and $Y$ (on the left and right respectively) then one talks of an action on $Xtimes Y$ defined in the following way: $g.(x,y) = (xg^-1, gy)$. The quotient for this action is denoted by $Xtimes ^G Y$.

share|cite|improve this answer

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

If a group $G$ acts on two varieties $X$ and $Y$ (on the left and right respectively) then one talks of an action on $Xtimes Y$ defined in the following way: $g.(x,y) = (xg^-1, gy)$. The quotient for this action is denoted by $Xtimes ^G Y$.

share|cite|improve this answer

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

share|cite|improve this answer

share|cite|improve this answer

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

answered Aug 3 at 5:14

P Vanchinathan

13.9k12035

13.9k12035

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