P-adic order

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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• Template:EGA