Inverse image functor
Encyclopedia
In mathematics
, the inverse image functor is a contravariant construction of sheaves. The direct image functor
is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
on Y and that we want to transport 
to X using a continuous map f : X → Y. We will call the result the inverse image or pullback sheaf
. If we try to imitate the direct image by setting 
for each open set U of X, we immediately run into a problem: f(U) is not necessarily open. The best we can do is to approximate it by open sets, and even then we will get a presheaf, not a sheaf. Consequently we define 
to be the sheaf associated to the presheaf:
(U is an open subset of X and the colimit runs over all open subsets V of Y containing f(U)).
For example, if f is just the inclusion of a point y of Y, then
is just the stalk
of
at this point.
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limit
s.
When dealing with morphisms f : X → Y of locally ringed spaces, for example schemes
in algebraic geometry
, one often works with sheaves of
-modules, where 
is the structure sheaf of Y. Then the functor f−1 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
.
However, these morphisms are almost never isomorphisms.
For example, if
denotes the inclusion of a closed subset, the stalks 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 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the inverse image functor is a contravariant construction of sheaves. The direct image functor
Direct image functor
In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.-Definition:...
is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Definition
Suppose given a sheaf on Y and that we want to transport to X using a continuous map f : X → Y. We will call the result the inverse image or pullback sheafSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
. If we try to imitate the direct image by setting for each open set U of X, we immediately run into a problem: f(U) is not necessarily open. The best we can do is to approximate it by open sets, and even then we will get a presheaf, not a sheaf. Consequently we define to be the sheaf associated to the presheaf:(U is an open subset of X and the colimit runs over all open subsets V of Y containing f(U)).
For example, if f is just the inclusion of a point y of Y, then
is just the stalkSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of
at this point.The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
s.
When dealing with morphisms f : X → Y of locally ringed spaces, for example schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, one often works with sheaves of
-modules, where is the structure sheaf of Y. Then the functor f−1 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 f∗, the stalksStalk (sheaf)The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.-Motivation and definition:Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a...
are easier to compute: given a point
, one has 
. -
is an exact functor, as can be seen by the above calculation of the stalks. -
is (in general) only right exact. If 
is exact, f is called flatFlat morphismIn mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,is a flat map for all P in X...
. -
is the left adjoint of the direct image functorDirect image functorIn mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.-Definition:...
f∗. This implies that there are natural unit and counit morphisms
and 
. These morphisms yield a natural adjunction correspondence:
.
However, these morphisms are almost never isomorphisms.
For example, if
denotes the inclusion of a closed subset, the stalks 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 .