Direct image with compact support
Encyclopedia
In mathematics
, in the theory of sheaves
the direct image with compact (or proper) support is an image
functor
for sheaves.
s, and Sh(–) the category
of sheaves of abelian group
s on a topological space. The direct image with compact (or proper) support
sends a sheaf F on X to f!(F) defined by
where U is an open subset of Y. The functoriality of this construction follows from the very basic properties of the support and the definition of sheaves.
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...
, in the theory of sheaves
Sheaf (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...
the direct image with compact (or proper) support is an image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
for sheaves.
Definition
Let f: X → Y be a continuous mapping of topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s, and Sh(–) the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of sheaves of abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
s on a topological space. The direct image with compact (or proper) support
- f!: Sh(X) → Sh(Y)
sends a sheaf F on X to f!(F) defined by
- f!(F)(U) := {s ∈ F(f −1(U)) : f|supp(s):suppSupport (mathematics)In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
(s)→U is properProper mapIn mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...
},
where U is an open subset of Y. The functoriality of this construction follows from the very basic properties of the support and the definition of sheaves.