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 $M_{{\mathfrak {p}}}\neq 0$ .^[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 $V(I)\cap \operatorname {Supp}(M)$ .

References

↑ EGA 0_I, 1.7.1.

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

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Support of a module

See also

References