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edit approved Jan 21, 2017 at 7:26
Martin Sleziak
edit approved Jan 21, 2017 at 7:26
Martin Sleziak
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I have started reading the book "The Geometry of moduliModuli Spaces of sheaves"Sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3page no.3 the last line.

"$E$ is pure if and only if all associated points of $E$ have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

  1. How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?
  2. Can anyone suggest me some reference in this regard?

I have started reading the book "The Geometry of moduli of sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.

"$E$ is pure if and only if all associated points of $E$ have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

  1. How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?
  2. Can anyone suggest me some reference in this regard?

I have started reading the book "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.

"$E$ is pure if and only if all associated points of $E$ have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

  1. How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?
  2. Can anyone suggest me some reference in this regard?
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edit approved Jan 21, 2017 at 4:17
Konstantinos Kanakoglou
edit approved Jan 21, 2017 at 4:17
Konstantinos Kanakoglou
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I have started reading the book "The Geometry of moduli of sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.

"E"$E$ is pure if and only if all associated points of E$E$ have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

(1). How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?

(2). Can anyone suggest me some reference in this regard?

  1. How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?
  2. Can anyone suggest me some reference in this regard?

I have started reading the book "The Geometry of moduli of sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.

"E is pure if and only if all associated points of E have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

(1). How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?

(2). Can anyone suggest me some reference in this regard?

I have started reading the book "The Geometry of moduli of sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.

"$E$ is pure if and only if all associated points of $E$ have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

  1. How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?
  2. Can anyone suggest me some reference in this regard?
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Anoop singh
Anoop singh
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A criterion for purity

I have started reading the book "The Geometry of moduli of sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.

"E is pure if and only if all associated points of E have the same dimension." Definition for associated points of a sheaf is as follows:

$Ass(E)=\{x∈X|m_x∈AssE_x\}$.

(1). How to prove this statement specially the converse part that if all associated points have the same dimension, then $E$ is pure?

(2). Can anyone suggest me some reference in this regard?