Support of a module

In commutative 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 $M_\mathfrak{p} \ne 0$.^[1] It is denoted by $\operatorname{Supp}(M)$. In particular, $M = 0$ if and only if its support is empty.

• Let $0 \to M' \to M \to M'' \to 0$ 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 $\operatorname{Supp}(M) = \cup_\lambda \operatorname{supp}(M_\lambda).$
• 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

1. EGA 0_I, 1.7.1.

• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas" Check |url= value (help). Publications Mathématiques de l'IHÉS 4. doi:10.1007/bf02684778. MR 0217083.