I have started reading the book "GEOMETRY OF MODULI OF SHEAVES" by Daniel Huybrechts and Manfred Lehn. I have come across the following statement (page 3 of the same book)
1.Structure sheaf of a closed subscheme $\mathbb Y$ is pure if it has no components of dimension less than $\mathbf dim (Y)$ and no embedded points.
My attempt: I want to show that the given condition somehow implies all its associated points have same dimension.Because then by characterization of pure sheaf we are done.
Now I have $\mathbb X$ is a Noetherian scheme and Y is closed,so Y have decomposition by finitely many irreducibile components each of which posses a unique generic point.
Since none of the components have dimension less than $\mathbb dim(Y)$ ,so closure of each of those generic points have dimension exactly equal to $\mathbb dim(Y)$.
Now,somehow from the fact that there are no embedded points (i.e no associated point of its structure sheaf is specialization of other associated points) ,I want to conclude that this finitely many generic points are precisely all the associated points of the structure sheaf of the closed subscheme Y,which concludes the proof.
Till now I am only able to find out a strategy but to implement it I am having troubles.For example,I have tried but failed to see how can I think those generic points as associated points keeping the standard definition of associated points in mind.
and Secondly, I am nowhere using the speciality of the appearance of the structure sheaf of the closed subscheme which is by definition inverse image of appropriate quotient sheaf .
Any help from anyone is welcome.