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 of A such that
.[1] It is denoted by
. In particular,
if and only if its support is empty.
- Let
be an exact sequence of A-modules. Then
- If
is a sum of submodules
, then
- If
is a finitely generated A-module, then
is the set of all prime ideals containing the annihilator of M. In particular, it is closed.
- If
are finitely generated A-modules, then
- If
is a finitely generated A-module and I is an ideal of A, then
is the set of all prime ideals containing
This is
.
See also
References
- ↑ EGA 0I, 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.