In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the direct image functor is a construction in
sheaf theory
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 ...
that generalizes the
global sections functor
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 ...
to the relative case. It is of fundamental importance in
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 geometrical ...
. Given a sheaf ''F'' defined on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' and 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 valu ...
''f'': ''X'' → ''Y'', we can define a new sheaf ''f''
∗''F'' on ''Y'', called the direct image sheaf or the pushforward sheaf of ''F'' along ''f'', such that the global sections of ''f''
∗''F'' is given by the global sections of ''F''. This assignment gives rise to a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
''f''
∗ from the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of sheaves on ''X'' to the category of sheaves on ''Y'', which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of
quasi-coherent sheaves and
étale sheaves 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 ...
.
Definition
Let ''f'': ''X'' → ''Y'' be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s on a topological space. The direct image functor
:
sends a sheaf ''F'' on ''X'' to its direct image presheaf ''f''
∗''F'' on ''Y'', defined on open subsets ''U'' of ''Y'' by
:
This turns out to be a sheaf on ''Y'', and is called the direct image sheaf or pushforward sheaf of ''F'' along ''f''.
Since a
morphism of sheaves φ: ''F'' → ''G'' on ''X'' gives rise to a morphism of sheaves ''f''
∗(φ): ''f''
∗(''F'') → ''f''
∗(''G'') on ''Y'' in an obvious way, we indeed have that ''f''
∗ is a functor.
Example
If ''Y'' is a point, and ''f'': ''X'' → ''Y'' the unique continuous map, then Sh(''Y'') is the category Ab of abelian groups, and the direct image functor ''f''
∗: Sh(''X'') → Ab equals the
global sections functor
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 ...
.
Variants
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if ''f'': (''X'', ''O
X'') → (''Y'', ''O
Y'') is a morphism of
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 of ...
s, we obtain a direct image functor ''f''
∗: Sh(''X'',''O
X'') → Sh(''Y'',''O
Y'') from the category of sheaves of ''O
X''-modules to the category of sheaves of ''O
Y''-modules. Moreover, if ''f'' is now a morphism of
quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
and
quasi-separated In algebraic geometry, a morphism of schemes from to is called quasi-separated if the diagonal map from to is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme is called quasi-separated if ...
schemes, then ''f''
∗ preserves the property of being quasi-coherent, so we obtain the direct image functor between categories of quasi-coherent sheaves.
A similar definition applies to sheaves on
topoi
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a noti ...
, such as
étale sheaves. There, instead of the above preimage ''f''
−1(''U''), one uses the
fiber product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...
of ''U'' and ''X'' over ''Y''.
Properties
* Forming sheaf categories and direct image functors itself defines a functor from the category of topological spaces to the category of categories: given continuous maps ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'', we have (''gf'')
∗=''g''
∗''f''
∗.
* The direct image functor is
right 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 ...
to the
inverse image functor In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor ...
, which means that for any continuous
and sheaves
respectively on ''X'', ''Y'', there is a natural isomorphism:
:
.
* If ''f'' is the inclusion of a closed subspace ''X'' ⊆ ''Y'' then ''f''
∗ is
exact
Exact may refer to:
* Exaction, a concept in real property law
* ''Ex'Act'', 2016 studio album by Exo
* Schooner Exact, the ship which carried the founders of Seattle
Companies
* Exact (company), a Dutch software company
* Exact Change, an Ameri ...
. Actually, in this case ''f''
∗ is an
equivalence between the category of sheaves on ''X'' and the category of sheaves on ''Y'' supported on ''X''. This follows from the fact that the stalk of
is
if
and zero otherwise (here the closedness of ''X'' in ''Y'' is used).
* If ''f'' is the morphism of affine schemes
determined by a ring homomorphism
, then the direct image functor ''f''
∗ on quasi-coherent sheaves identifies with the
restriction of scalars
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'',
*f_! M = S\otimes_R M, the induced module.
*f_* M = \operator ...
functor along φ.
Higher direct images
The direct image functor is
left exact, but usually not right exact. Hence one can consider the right
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s of the direct image. They are called higher direct images and denoted ''R
q f''
∗.
One can show that there is a similar expression as above for higher direct images: for a sheaf ''F'' on ''X'', the sheaf ''R
q f''
∗(''F'') is the sheaf associated to the presheaf
:
,
where ''H
q'' denotes
sheaf cohomology.
In the context of algebraic geometry and a morphism
of quasi-compact and quasi-separated schemes, one likewise has the right derived functor
:
as a functor between the (unbounded)
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
of quasi-coherent sheaves. In this situation,
always admits a right adjoint
.
This is closely related, but not generally equivalent to, the
exceptional inverse image functor In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duali ...
, unless
is also
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
.
See also
*
Proper base change theorem
References
* , esp. section II.4
{{DEFAULTSORT:Direct Image Functor
Sheaf theory
Theory of continuous functions