Associated Primes

Support of a module M is     \text{supp}(M) = \{\mathfrak{p} \in Spec R: M_\mathfrak{p} \neq 0\} which is basically the collection of points in the SpecR that is non-trivial under the localization of the module to the point \mathfrak{p}.

Now, we move on to defining the associated primes of the module. There are finitely many associated points in SpecR.

An associated point is called an embedding point if it is in the closure of some other associated point. This means that if R can be reduced, then there are no embedded points.

We also define that \text{Supp}(M)= \overline{Ass_R M} where M is an R-module. This means that the support of M is the closure of the associated points of M.

We call an element of R a zero-divisor if it vanishes at an associated point. Now we define annihilator ideal Ann_R\ m for m \subset R of a module M Ann_R\ m := \{a \in R: am =0\}.

There is the localization at these associated primes. Given that R is a Noetherian ring and a module M over R, then there is an injection M \rightarrow \Pi_{\mathfrak{p}\in Ass M} M_\mathfrak{p} with a similar injection in the localization at the SpecR. (Support of the module and the associated primes commute with localization.) For any reference, see the frequently mentioned notes on this blog.

This entry was posted in . Bookmark the post. Print it.

Leave a Reply