Support of a module

Definition

Let $A$ be a commutative unital ring and $M$ be a module over $A$. The support of $M$ is the subset of $Spec(A)$ (the spectrum of $A$) comprising those prime ideals $P$ such that $M_{P}\neq 0$

Here, $M_{P}$ denotes the localization of $M$ at the prime ideal $P$.

Facts

• If a prime ideal $P$ is contained in the support of $M$, then any prime ideal containing $P$ is in the support of $M$.
• The support of a module is a union of closed subsets. (This follows from the preceding). Conversely any union of closed subsets, arises as the support of a module.
• For a finitely generated module, the support of the module equals the Galois correspondent closed set to the annihilator of the module (the ideal that annihilates all elements).