Support of a module

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.[1] 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) = \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

References

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