In
mathematics, the direct image with compact (or proper) support is an
image functor for
sheaves that extends the
compactly supported 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 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 morp ...
.
Definition
Let ''f'': ''X'' → ''Y'' be a
continuous mapping
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 ...
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 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 po ...
s, and let Sh(–) denote 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 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 com ...
s on a topological space. The direct image with compact (or proper) support is the
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, an ...
:''f''
!: Sh(''X'') → Sh(''Y'')
that sends a sheaf ''F'' on ''X'' to the sheaf ''f''
!(''F'') given by the formula
:''f''
!(''F'')(''U'') :=
for every open subset ''U'' of ''Y.'' Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff. This defines ''f''
!(''F'') as a subsheaf of the
direct image sheaf ''f''
∗(''F''), 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 ''f'' is proper, then ''f''
! equals ''f''
∗.
* If ''f'' 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 that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
, then ''f''
! 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 ...
, 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