While I feel I understand the definition and main properties of associated primes, as well as examples, the technical role played by them is not clear to me. Could someone elaborate their role by pointing out the most important properties and highlighting it by some instructive example of how it is used? I am particularly interested in connections with homological techniques in commutative algebra, as well algebraic geometry (= theory of schemes).
($A$ a Noetherian ring, $M$ a finite type $A$-module.) I know (for instance) that if $M_p=0$ for all $p\in Ass(M)$, then $M=0$; so in some sense $Ass(M)$ is a finite version of $Supp(M)$ ($Ass(M)$ is a finite set). Also that if $p\in Ass(M)$, then $\kappa(p)\subset M_p$, where $\kappa(p)$ is the residue field of $A_p$. Furthermore $p\in Ass(M)$ if and only if $M_p$ has depth $0$. Also the associated primes occur in a certain filtration of $M$, allowing for devissage arguments.