Projective Localization, The Support of a Module

The Support of a Module

The support of an R module M is the set of prime ideals P in R such that the localization M_P is nonzero.

Since 0 is a local property, sup(M) is a nonempty set of spec R whenever M is nonzero.

Tensor the exact sequence 0 → A → B → C → 0 with the flat module R_P. The middle module B_P is nonzero iff A_P or C_P is nonzero. Therefore sup(B) is the union of sup(A) and sup(C).

Tensor a (possibly infinite) direct sum of modules with R_P, giving the direct sum of the individual localizations. Therefore the support of a direct sum is the union of the component supports.

Let L and M be finitely generated R modules, and consider the support of L tensor M. Write L×M×R_P = L×R_P×M×R_P = L_P×M_P. If either localization is zero the result is zero. Conversely, assume L_PM_P = 0. The tensor product of two finitely generated modules over a local ring is 0, hence one of the two modules is 0. Therefore the support of L×M is the intersection of sup(L) and sup(M).