In mathematics

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...

, more specifically sheaf theory, a branch of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 and 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...

, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves

Image functors for sheaves

In mathematics, especially in sheaf theory, a domain applied in areas such as topology, logic and algebraic geometry, there are four image functors for sheaves which belong together in various senses....

. It is needed to express Verdier duality

Verdier duality

In mathematics, Verdier duality is a generalization of the Poincaré duality of manifolds to locally compact spaces with singularities. Verdier duality was introduced by , as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck...

 in its most general form.

Definition

Let f: X → Y be a continuous map of topological spaces or a morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 of 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...

. Then the exceptional inverse image is a functor

Rf^!: D(Y) → D(X)

where D(–) denotes the derived category

Derived category

In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...

 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...

 of abelian groups or modules over a fixed ring.

It is defined to be the right adjoint of the total derived functor

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 :...

 Rf_! of the direct image with compact support

Direct image with compact support

In mathematics, in the theory of sheaves the direct image with compact support is an image functor for sheaves.-Definition:Let f: X → Y be a continuous mapping of topological spaces, and Sh the category of sheaves of abelian groups on a topological space...

. Its existence follows from certain properties of Rf_! and general theorems about existence of adjoint functors, as does the unicity.

The notation Rf^! is an abuse of notation insofar as there is in general no functor f^! whose derived functor would be Rf^!.

Examples and properties

• If f is an immersion of a locally closed subspace, then it is possible to define

f^!(F) := f^∗ G,

where the sections of G on some open subset U of Y are the sections s ∈ F(U) whose support

Support (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...

 is contained in X. The functor f^! is left exact, and the above Rf^!, whose existence is guaranteed by general structural arguments, is indeed the derived functor of this f^!. Moreover f^! is right adjoint to f_!

Direct image with compact support

In mathematics, in the theory of sheaves the direct image with compact support is an image functor for sheaves.-Definition:Let f: X → Y be a continuous mapping of topological spaces, and Sh the category of sheaves of abelian groups on a topological space...

, too.

• Slightly more generally, a similar statement holds for any quasi-finite morphism
  Quasi-finite morphism
  In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is finite type and satisfies any of the following equivalent conditions:...
  
   such as an étale morphism
  Étale morphism
  In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not...
  
  .

• If f is an open immersion, the exceptional inverse image equals the usual inverse image
  Inverse image functor
  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.-Definition:...
  
  .