In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals p of A such that M p ≠ 0 . It is denoted by Supp ( M ) . The support is, by definition, a subset of the spectrum of A.
M = 0 if and only if its support is empty.Let 0 → M ′ → M → M ″ → 0 be an exact sequence of A-modules. Then Supp ( M ) = Supp ( M ′ ) ∪ Supp ( M ″ ) . If M is a sum of submodules M λ , then Supp ( M ) = ∪ λ supp ( M λ ) . If M is a finitely generated A-module, then Supp ( M ) is the set of all prime ideals containing the annihilator of M. In particular, it is closed in the Zariski topology on Spec(A).If M , N are finitely generated A-modules, then Supp ( M ⊗ A N ) = Supp ( M ) ∩ Supp ( N ) . If M is a finitely generated A-module and I is an ideal of A, then Supp ( M / I M ) is the set of all prime ideals containing I + Ann ( M ) . This is V ( I ) ∩ Supp ( M ) .If F is a quasicoherent sheaf on a scheme X, the support of F is the set of all points x∈X such that the stalk Fx is nonzero. This definition is similar to the definition of the support of a function on a space X, and this is the motivation for using the word "support". Most properties of the support generalize from modules to quasicoherent sheaves word by word. For example, the support of a coherent sheaf (or more generally, a finite type sheaf) is a closed subspace of X.
If M is a module over a ring A, then the support of M as a module coincides with the support of the associated quasicoherent sheaf M ~ on the affine scheme Spec(R). Moreover, if { U α = Spec ( A α ) } is an affine cover of a scheme X, then the support of a quasicoherent sheaf F is equal to the union of supports of the associated modules Mα over each Aα.
