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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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  • $\begingroup$ I thought that was the definition of "pure". What is the definition of pure in Huybrechts and Lehn? Also, please be advised that there is at least one other definition of pure in local commutative algebra: the definition that is used in SGA 2. $\endgroup$
    – Jason Starr
    Commented Jan 20, 2017 at 8:52
  • $\begingroup$ @JasonStarr The definition for pure is as follows: $E $ is pure of dimension d if $dim(F) = d$ for all non-trivial coherent subsheaves $F \subset E$ $\endgroup$
    – Anoop singh
    Commented Jan 20, 2017 at 10:39
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    $\begingroup$ For (1), assume that all associated primes of $E$ have the same dimension $d$, then $\dim E=d$. If $0\neq F\subset E$, then on the one hand, we have $\dim F\leq d$. But the associated primes of $F$ are also associated primes of $E$ and thus we have $\dim F\geq d$. $\endgroup$
    – Mohan
    Commented Jan 20, 2017 at 11:17
  • $\begingroup$ @JasonStarr ,@Mohan How do we relate torsion free sheaf and pure sheaf? Pure sheaf implies torsion free sheaf. What is the proof of this statement? $\endgroup$
    – Anoop singh
    Commented Jan 23, 2017 at 5:19
  • $\begingroup$ This is a cross-post of math.stackexchange.com/q/2099967/33855 $\endgroup$
    – Earthliŋ
    Commented Feb 15, 2019 at 12:02

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