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:
$Ass(E) = \{x \in X \vert m_x \in AssE_x \}$.
Point has always dimension 0 ,So my question is, what does this mean by saying that associated points have same dimension?
Does this mean that local rings $O_x$ has the same dimension for all $x$ associated point of $E$ ?
Can anyone suggest me a good reference which supports me while studying this book?
I found a P.hd. thesis by Alain Leytem entitled with "Torsion and purity on non-integral schemes and singular sheaves in the fine Simpson moduli spaces of one-dimensional sheaves on the projective plane ".The above asked question is answered there and give a good exposition to the relation between torsion sheave and purity of coherent sheaves over Noetherian scheme. Link is given below: http://orbilu.uni.lu/handle/10993/23380