In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, the support of a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''M'' over a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''A'' is the set of all
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s
of ''A'' such that
(that is, the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
of ''M'' at
is not equal to zero). It is denoted by
. The support is, by definition, a subset of the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of ''A''.
Properties
*
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
its support is
empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
.
* Let
be a
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
of ''A''-modules. Then
*:
:Note that this union may not be a
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
.
* If
is a sum of
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
s
, then
* If
is a
finitely generated ''A''-module, then
is the set of all prime ideals containing the
annihilator of ''M''. In particular, it is closed in the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on Spec ''A''.
*If
are finitely generated ''A''-modules, then
*:
*If
is a finitely generated ''A''-module and ''I'' is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of ''A'', then
is the set of all prime ideals containing
This is
.
Support of a quasicoherent sheaf
If ''F'' is a
quasicoherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
''X'', the support of ''F'' is the set of all points ''x'' in ''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 for word. For example, the support of a
coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
(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 Associated may refer to:
*Associated, former name of Avon, Contra Costa County, California
* Associated Hebrew Schools of Toronto, a school in Canada
*Associated Newspapers, former name of DMG Media, a British publishing company
See also
*Associati ...
quasicoherent sheaf
on the
affine scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
Spec ''A''. Moreover, if
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
is in the support if and only if it contains the annihilator of
.
For example, the annihilator of
:
is the ideal
. This implies that
:
the vanishing locus of the
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
. Looking at the short exact sequence
:
we might think that the support of
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to
:
which is the complement of the vanishing locus of the polynomial. However, since