In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.

de Rham cohomology with compact support for smooth manifolds

Given a manifold X, let be the real vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 of k-forms on X with compact support, and d be the standard exterior derivative

Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

. Then the de Rham cohomology groups with compact support are the homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

 of the chain complex

Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

 :


i.e., is the vector space of closed

Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero , and an exact form is a differential form that is the exterior derivative of another differential form β...

 q-forms modulo

Modulo (jargon)

The word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...

 that of exact q-forms.

Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate covariant behavior; for example, given the inclusion mapping j for an open set U of X, extension of forms on U to X (by defining them to be 0 on X–U) is a map inducing a map
.

They also demonstrate contravariant behavior with respect to proper map

Proper map

In 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 :...

s - that is, maps such that the inverse image of every compact set is compact. Let f: Y → X be such a map; then the pullback


induces a map
.

If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence


called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem

Jordan curve theorem

In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...

, which is obtained for X = R² and Z a simple closed curve in X.

De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then


where all maps are induced by extension by zero is also exact.