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 formula_1 of "A" such that formula_2 (i.e. the localization of M at formula_1 is not equal to zero). It is denoted by formula_4. The support is, by definition, a subset of the spectrum of "A".
Support of a quasicoherent sheaf.
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 "F""x" 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 formula_19 on the affine scheme Spec("A"). Moreover, if formula_20 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 "M"α over each "A"α.
Examples.
As noted above, a prime ideal formula_1 is in the support if and only if it contains the annihilator of formula_8. For example, the annihilator of
is the ideal formula_24. This implies that
the vanishing locus of the polynomial. Looking at the short exact sequence
we might think that the support of formula_24 is isomorphic to
which is the complement of the vanishing locus of the polynomial. However, since formula_29 is an integral domain, the ideal formula_24 is isomorphic to formula_29 as a module, so its support is the entire space.
The support of a finite module over a noetherian ring is always closed under specialization.
Now, if we take two polynomials formula_32 in an integral domain which form a complete intersection ideal formula_33, the tensor property shows us that