Support of a module

In algebra, the support of a module M over a commutative ring A is the set of all prime ideals ${\mathfrak {p}}$ of A such that $M_{\mathfrak {p}}\neq 0$. It is denoted by $\operatorname {Supp} (M)$. In particular, $M=0$ if and only if its support is empty.

• If $0\to M'\to M\to M''\to 0$ be an exact sequence of A-modules. Then

  $\operatorname {Supp} (M)=\operatorname {Supp} (M')\cup \operatorname {Supp} (M'').$

• If $M$ is a sum of submodules $M_{\lambda }$, then $\operatorname {Supp} (M)=\bigcup _{\lambda }\operatorname {Supp} (M_{\lambda }).$
• If $M$ is a finitely generated A-module, then $\operatorname {Supp} (M)$ is the set of all prime ideals containing the annihilator of M. In particular, it is closed.
• If $M,N$ are finitely generated A-modules, then

  $\operatorname {Supp} (M\otimes _{A}N)=\operatorname {Supp} (M)\cap \operatorname {Supp} (N).$

• If $M$ is a finitely generated A-module and I is an ideal of A, then $\operatorname {Supp} (M/IM)$ is the set of all prime ideals containing $I+\operatorname {Ann} (M).$ This is $V(I)\cap \operatorname {Supp} (M)$.

See also

• Associated prime

References

• Template:EGA