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Cycle associated to a closed subscheme

1

$begingroup$

Let $X$ be an algebraic variety (i.e. an integral $k$-scheme, such that $X to mathrmSpec ;k$ is separated and of finite type). Let $Y$ be a closed subscheme and $Y_1, dots Y_n$ the irreducible components of $Y_mathrmred$. Let $xi_i$ be the generic point of $Y_i$. Since $Y$ is noetherian, then $mathcalO_Y,xi_i$ is a noetherian ring. We define the effective cycle associated to $Y$ as $<Y>:=sum_i=1^nl_iY_i$, where $l_i$ is the length of the local ring $mathcalO_Y,xi_i$. In the book "3264 and all that" it is said that the ring $mathcalO_Y,xi_i$ has a finite composition series. I know that the length of $mathcalO_Y,xi_i$ is finite if and only if it is an artinian ring. But in this case i just see that it is noetherian. Why is it artinian too?

EDIT: I try to answer my own. Let $xi$ be one of the $xi_i$ and $Z$ the irreducible component associated to $xi$. Choose $U$ to be an affine open subset of $Y$ such that $xi in U$. Then $Z cap U$ is an irreducible component of $U$, with generic point $xi$. If $U= mathrmSpec ;A$, then $xi=mathfrakp$ is a minimal prime ideal and hence $mathcalO_Y,xi=mathcalO_U,mathfrakp$ is noetherian of dimension $0$, i.e. an artinian ring. Do you think it works?

algebraic-geometry schemes intersection-theory

share|cite|improve this question

edited Mar 26 at 19:47

ciccio

asked Mar 26 at 17:04

cicciociccio

8917

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, it works $ $.
  $endgroup$
  – Georges Elencwajg
  Mar 28 at 9:23
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $X$ be an algebraic variety (i.e. an integral $k$-scheme, such that $X to mathrmSpec ;k$ is separated and of finite type). Let $Y$ be a closed subscheme and $Y_1, dots Y_n$ the irreducible components of $Y_mathrmred$. Let $xi_i$ be the generic point of $Y_i$. Since $Y$ is noetherian, then $mathcalO_Y,xi_i$ is a noetherian ring. We define the effective cycle associated to $Y$ as $<Y>:=sum_i=1^nl_iY_i$, where $l_i$ is the length of the local ring $mathcalO_Y,xi_i$. In the book "3264 and all that" it is said that the ring $mathcalO_Y,xi_i$ has a finite composition series. I know that the length of $mathcalO_Y,xi_i$ is finite if and only if it is an artinian ring. But in this case i just see that it is noetherian. Why is it artinian too?

EDIT: I try to answer my own. Let $xi$ be one of the $xi_i$ and $Z$ the irreducible component associated to $xi$. Choose $U$ to be an affine open subset of $Y$ such that $xi in U$. Then $Z cap U$ is an irreducible component of $U$, with generic point $xi$. If $U= mathrmSpec ;A$, then $xi=mathfrakp$ is a minimal prime ideal and hence $mathcalO_Y,xi=mathcalO_U,mathfrakp$ is noetherian of dimension $0$, i.e. an artinian ring. Do you think it works?

algebraic-geometry schemes intersection-theory

share|cite|improve this question

edited Mar 26 at 19:47

ciccio

asked Mar 26 at 17:04

cicciociccio

8917

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, it works $ $.
  $endgroup$
  – Georges Elencwajg
  Mar 28 at 9:23
  
  

  
  
  
  

add a comment | 

$begingroup$

Let $X$ be an algebraic variety (i.e. an integral $k$-scheme, such that $X to mathrmSpec ;k$ is separated and of finite type). Let $Y$ be a closed subscheme and $Y_1, dots Y_n$ the irreducible components of $Y_mathrmred$. Let $xi_i$ be the generic point of $Y_i$. Since $Y$ is noetherian, then $mathcalO_Y,xi_i$ is a noetherian ring. We define the effective cycle associated to $Y$ as $<Y>:=sum_i=1^nl_iY_i$, where $l_i$ is the length of the local ring $mathcalO_Y,xi_i$. In the book "3264 and all that" it is said that the ring $mathcalO_Y,xi_i$ has a finite composition series. I know that the length of $mathcalO_Y,xi_i$ is finite if and only if it is an artinian ring. But in this case i just see that it is noetherian. Why is it artinian too?

EDIT: I try to answer my own. Let $xi$ be one of the $xi_i$ and $Z$ the irreducible component associated to $xi$. Choose $U$ to be an affine open subset of $Y$ such that $xi in U$. Then $Z cap U$ is an irreducible component of $U$, with generic point $xi$. If $U= mathrmSpec ;A$, then $xi=mathfrakp$ is a minimal prime ideal and hence $mathcalO_Y,xi=mathcalO_U,mathfrakp$ is noetherian of dimension $0$, i.e. an artinian ring. Do you think it works?

algebraic-geometry schemes intersection-theory

share|cite|improve this question

edited Mar 26 at 19:47

ciccio

asked Mar 26 at 17:04

cicciociccio

8917

$endgroup$

share|cite|improve this question

edited Mar 26 at 19:47

ciccio

asked Mar 26 at 17:04

cicciociccio

8917

share|cite|improve this question

edited Mar 26 at 19:47

ciccio

asked Mar 26 at 17:04

cicciociccio

8917

asked Mar 26 at 17:04

cicciociccio

8917

asked Mar 26 at 17:04

cicciociccio

8917