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 ${\mathfrak {p}}$ of A such that $M_{{\mathfrak {p}}}\neq 0$ .^[1] It is denoted by $\operatorname {Supp}(M)$ . The support is, by definition, a subset of the spectrum of A.

Properties

$M=0$ if and only if its support is empty.
Let $0\to M'\to M\to M''\to 0$ be an exact sequence of A-modules. Then
$\operatorname {Supp}(M)=\operatorname {Supp}(M')\cup \operatorname {Supp}(M'').$ Note that this union may not be a disjoint union.
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 in the Zariski topology on Spec(A).
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 $V(I)\cap \operatorname {Supp}(M)$ .

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.^[2]

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 ${\tilde {M}}$ on the affine scheme Spec(R). Moreover, if $\{U_{\alpha }=\operatorname {Spec} (A_{\alpha })\}$ 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.^[3]

Examples

There is a useful technical lemma which can be used to compute the support of modules: given a finitely generated $R$ -module $M$ , a prime ideal ${\mathfrak {p}}$ is in the support if and only if it contains the annihilator of $M$ ^[4]. For example, the annihilator of

{\frac {\mathbb {C} [x,y,z,w]}{(x^{4}+y^{4}+z^{4}+w^{4})}}\in {\text{Mod}}(\mathbb {C} [x,y,z,w])

is the ideal $(x^{4}+y^{4}+z^{4}+w^{4})$ . This implies that

{\text{Supp}}(M)\cong {\text{Spec}}(\mathbb {C} [x,y,z,w]/(x^{4}+y^{4}+z^{4}+w^{4}))

the vanishing locus of the polynomial. Looking at the short exact sequence

0\to I\to R\to R/I\to 0

we might think that the support of $(x^{4}+y^{4}+z^{4}+w^{4})$ is isomorphic to

{\text{Spec}}(\mathbb {C} [x,y,z,w]_{(x^{4}+y^{4}+z^{4}+w^{4})})

which is the complement of the vanishing locus of the polynomial. However, since $\mathbb {C} [x,y,z,w]$ is an integral domain, the ideal $(x^{4}+y^{4}+z^{4}+w^{4})$ is isomorphic to $\mathbb {C} [x,y,z,w]$ as a module, so its support is the entire space. Its an easy commutative algebra exercise to see that the support of a module is always closed under specialization.

Now, if we take two polynomials $f_{1},f_{2}\in R$ in an integral domain which form a complete intersection ideal $(f_{1},f_{2})$ , the tensor property shows us that

{\text{Supp}}\left({\frac {R}{(f_{1})}}\otimes _{R}{\frac {R}{(f_{2})}}\right)={\text{Supp}}\left({\frac {R}{(f_{1})}}\right)\cap {\text{Supp}}\left({\frac {R}{(f_{2})}}\right)\cong {\text{Spec}}(R/(f_{1},f_{2}))

There are similar properties for the support of sheaves of modules. Using the exact sequence

0\to {\mathcal {O}}_{X}(-D)\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{D}\to 0

for a divisor D in a smooth projective variety $X$ , we get the exact same support computations. This is because if we look at the open subset $U=X-D$ we have

{\mathcal {O}}_{X}(-D)(U)\cong {\mathcal {O}}_{X}(U)

from the definition of the associated line bundle (this is because $U\cap D=\varnothing$ ).

References

↑ EGA 0_I, 1.7.1.
↑ The Stacks Project authors (2017). Stacks Project, Tag 01B4.
↑ The Stacks Project authors (2017). Stacks Project, Tag 01AS.
↑ Eisenbud, David. Commutative Algebra with a View Towards Algebraic Geometry. corollary 2.7. p. 67.

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.
Atiyah, M. F., and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9 MR 242802

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[1] EGA 0_I, 1.7.1.

[2] The Stacks Project authors (2017). Stacks Project, Tag 01B4.

[3] The Stacks Project authors (2017). Stacks Project, Tag 01AS.

[4] Eisenbud, David. Commutative Algebra with a View Towards Algebraic Geometry. corollary 2.7. p. 67.

Support of a module

Properties

Support of a quasicoherent sheaf

Examples

See also

References