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 $M_{\mathfrak{p}}\ne 0$. It is denoted by $\mathrm{Supp}(M)$. In particular, $M=0$ if and only if its support is empty.

• If $0\to M^{\prime }\to M\to M^{″}\to 0$ be an exact sequence of A-modules. Then

  $\mathrm{Supp}(M)=\mathrm{Supp}(M^{\prime })\cup \mathrm{Supp}(M^{″}).$

• If $M$ is a sum of submodules $M_{\lambda }$, then $\mathrm{Supp}(M)=\bigcup _{\lambda }\mathrm{Supp}(M_{\lambda }).$
• If $M$ is a finitely generated A-module, then $\mathrm{Supp}(M)$ 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

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

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

See also

• Associated prime

References

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