In mathematics, specifically in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, an inverse image functor is a
contravariant construction of
sheaves; here “contravariant” in the sense given a map
, the inverse image
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
is a functor from the
category of sheaves on ''Y'' to the category of sheaves on ''X''. The
direct image functor In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Definition
Suppose we are given a sheaf
on
and that we want to transport
to
using a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
.
We will call the result the ''inverse image'' or
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* '' The Sheaf'', a student-run newspaper s ...
. If we try to imitate the
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topol ...
by setting
:
for each open set
of
, we immediately run into a problem:
is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a
presheaf
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
and not a sheaf. Consequently, we define
to be the
sheaf associated to the presheaf:
:
(Here
is an open subset of
and the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions suc ...
runs over all open subsets
of
containing
.)
For example, if
is just the inclusion of a point
of
, then
is just the
stalk of
at this point.
The restriction maps, as well as the
functoriality
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
of the inverse image follows from the
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
of
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
s.
When dealing with
morphisms
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
of
locally ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf o ...
s, for example
schemes in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, one often works with
sheaves of -modules, where
is the structure sheaf of
. Then the functor
is inappropriate, because in general it does not even give sheaves of
-modules. In order to remedy this, one defines in this situation for a sheaf of
-modules
its inverse image by
:
.
Properties
* While
is more complicated to define than
, the
stalks are easier to compute: given a point
, one has
.
*
is an
exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
, as can be seen by the above calculation of the stalks.
*
is (in general) only right exact. If
is exact, ''f'' is called
flat.
*
is the
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
of the
direct image functor In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
. This implies that there are natural unit and counit morphisms
and
. These morphisms yield a natural adjunction correspondence:
:
.
However, the morphisms
and
are ''almost never'' isomorphisms.
For example, if
denotes the inclusion of a closed subset, the stalk of
at a point
is canonically isomorphic to
if
is in
and
otherwise. A similar adjunction holds for the case of sheaves of modules, replacing
by
.
References
* . See section II.4.
{{DEFAULTSORT:Inverse Image Functor
Algebraic geometry
Sheaf theory