In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the direct image with compact (or proper) support is an
image functor for
sheaves that extends the
compactly supported
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set ...
global sections functor
In mathematics, a sheaf (: sheaves) 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 da ...
to the relative setting. It is one of
Grothendieck's six operations
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphi ...
.
Definition
Let
be a
continuous mapping
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
of
locally compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
Hausdorff topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, and let
denote the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* 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 commu ...
s on a topological space. The direct image with compact (or proper)
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
* Support (art), a solid surface upon which a painting is executed
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Su ...
is the
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 ...
:
that sends a sheaf
on
to the sheaf
given by the formula
:
for every open subset
of
. Here, the notion of a
proper map
In mathematics, a function (mathematics), function between topological spaces is called proper if inverse images of compact space, compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
...
of spaces is unambiguous since the spaces in question are locally compact Hausdorff. This defines
as a subsheaf of 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 ...
sheaf
and the functoriality of this construction then follows from basic properties of the support and the definition of sheaves.
The assumption that the spaces be locally compact Hausdorff is imposed in most sources (e.g., Iversen or Kashiwara–Schapira). In slightly greater generality, Olaf Schnürer and
Wolfgang Soergel have introduced the notion of a "locally proper" map of spaces and shown that the functor of direct image with compact support remains well-behaved when defined for separated and locally proper continuous maps between arbitrary spaces.
Properties
* If
is proper, then
equals
.
* If
is an open
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
, then
identifies with the extension by zero functor.
References
* {{Citation , last1=Iversen , first1=Birger , title=Cohomology of sheaves , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, location=Berlin, New York , series=Universitext , isbn=978-3-540-16389-3 , mr=842190 , year=1986, esp. section VII.1
Sheaf theory
Theory of continuous functions
Functors