Ext functor
Encyclopedia
In mathematics
, the Ext functors of homological algebra
are derived functor
s of Hom functor
s. They were first used in algebraic topology
, but are common in many areas of mathematics.
be a ring
and let
be the category
of modules
over R. Let
be in 
and set 
, for fixed 
in 
. This is a left exact functor and thus has right derived functor
s
. The Ext functor is defined by
This can be calculated by taking any injective resolution
and computing
Then
is the homology
of this complex. Note that
is excluded from the complex.
An alternative definition is given using the functor
. For a fixed module B, this is a contravariant left exact functor, and thus we also have right derived functor
s
, and can define
This can be calculated by choosing any projective resolution
and proceeding dually by computing
Then
is the homology of this complex. Again note that 
is excluded.
These two constructions turn out to yield isomorphic results, and so both may be used to calculate the Ext functor.
Two extensions
are said to be equivalent (as extensions of A by B) if there is a commutative diagram
.
An extension of A by B is called split if it is equivalent to the trivial extension
There is a bijective correspondence between equivalence classes of extensions
of
by 
and elements of
and 
we can construct the Baer sum, by forming the pullback
of 
and 
. We form the quotient 
, that is we mod out by the relation 
~ 
. The extension
where the first arrow is
and the second 
thus formed is called the Baer sum of the extensions E and E.
Up to
equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The extension
has for opposite the same extension with exactly one of the central arrows turned to their opposite eg the morphism g is replaced by -g.
The set of extensions up to equivalence is an abelian group
that is a realization of the functor
even for abelian categories 
without reference to projectives
and injectives
. We simply take
to be the set of equivalence classes of extensions of 
by 
, forming an abelian group under the Baer sum. Similarly, we can define higher Ext groups 
as equivalence classes of n-extensions
under the equivalence relation
generated by the relation that identifies two extensions
and
if there are maps
for all 
in 
so that every resulting square commutes
.
The Baer sum of the two n-extensions above is formed by letting
be the pullback
of
and 
over 
, and 
be the pushout
of
and 
under 
. Then we define the Baer sum of the extensions to be
is considered as an equivalence class of maps 
for a projective resolution 
of 
; so, then we can pick a long exact sequence 
ending with 
and lift the map 
using the projectivity of the modules 
to a chain map 
of degree -n. It turns out that homotopy classes of such chain maps correspond precisely to the equivalence classes in the definition of Ext above.
Under sufficiently nice circumstances, such as when the ring
is a group ring
over a field
, or an augmented 
-algebra
, we can impose a ring structure on
. The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of 
.
One interpretation is in terms of these homotopy classes of chain maps. Then the product of two elements is represented by the composition of the corresponding representatives. We can choose a single resolution of
, and do all the calculations inside 
, which is a differential graded algebra, with cohomology precisely 
.
The Ext groups can also be interpreted in terms of exact sequences; this has the advantage that it does not rely on the existence of projective or injective modules. Then we take the viewpoint above that an element of
is a class, under a certain equivalence relation, of exact sequences of length 
starting with 
and ending with 
. This can then be spliced with an element in 
, by replacing
and 
with
where the middle arrow is the composition of the functions
and 
. This product is called the Yoneda splice.
These viewpoints turn out to be equivalent whenever both make sense.
Using similar interpretations, we find that
is a module
over
, again for sufficiently nice situations.
is the integral group ring
for a group
, then 
is the group cohomology
with coefficients in 
.
For
the finite field on 
elements, we also have that 
, and it turns out that the group cohomology doesn't depend on the base ring chosen.
If
is a 
-algebra
, then
is the Hochschild cohomology 
with coefficients in the module M.
If
is chosen to be the universal enveloping algebra
for a Lie algebra
, then 
is the Lie algebra cohomology
with coefficients in the module M.
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...
, the Ext functors of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
are 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 :...
s of Hom functor
Hom functor
In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
s. They were first used in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, but are common in many areas of mathematics.
Definition and computation
Let be a ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
and let
be the categoryCategory (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 modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over R. Let
be in and set , for fixed in . This is a left exact functor and thus has right derived functorDerived 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 :...
s
. The Ext functor is defined byThis can be calculated by taking any injective resolution
and computing
Then
is the homologyHomology (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 this complex. Note that
is excluded from the complex.An alternative definition is given using the functor
. For a fixed module B, this is a contravariant left exact functor, and thus we also have right derived functorDerived 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 :...
s
, and can defineThis can be calculated by choosing any projective resolution
and proceeding dually by computing
Then
is the homology of this complex. Again note that is excluded.These two constructions turn out to yield isomorphic results, and so both may be used to calculate the Ext functor.
Equivalence of extensions
Ext functors derive their name from the relationship to extensions of modules. Given R-modules A and B, an extension of A by B is a short exact sequence of R-modulesTwo extensions
are said to be equivalent (as extensions of A by B) if there is a commutative diagram
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
.
An extension of A by B is called split if it is equivalent to the trivial extension
There is a bijective correspondence between equivalence classes of extensions
of
by and elements ofThe Baer sum of extensions
Given two extensions and we can construct the Baer sum, by forming the pullback
Pullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...
of and . We form the quotient , that is we mod out by the relation ~ . The extensionwhere the first arrow is
and the second thus formed is called the Baer sum of the extensions E and E.Up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The extension
has for opposite the same extension with exactly one of the central arrows turned to their opposite eg the morphism g is replaced by -g.The set of extensions up to equivalence is an 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...
that is a realization of the functor
Construction of Ext in abelian categories
This identification enables us to define even for abelian categories without reference to projectivesProjective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
and injectives
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
. We simply take
to be the set of equivalence classes of extensions of by , forming an abelian group under the Baer sum. Similarly, we can define higher Ext groups as equivalence classes of n-extensionsunder the equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
generated by the relation that identifies two extensions
andif there are maps
for all in so that every resulting square commutesCommutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
.
The Baer sum of the two n-extensions above is formed by letting
be the pullbackPullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...
of
and over , and be the pushoutPushout (category theory)
In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...
of
and under . Then we define the Baer sum of the extensions to beFurther properties of Ext
The Ext functor exhibits some convenient properties, useful in computations.-
for 
if either 
is injectiveInjective moduleIn mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
or
is projective.
- A converse also holds: if
for all 
, then 
for all 
, and 
is injective; if 
for all 
, then 
for all 
, and 
is projective.
Ring structure and module structure on specific Exts
One more very useful way to view the Ext functor is this: when an element of is considered as an equivalence class of maps for a projective resolution of ; so, then we can pick a long exact sequence ending with and lift the map using the projectivity of the modules to a chain map of degree -n. It turns out that homotopy classes of such chain maps correspond precisely to the equivalence classes in the definition of Ext above.Under sufficiently nice circumstances, such as when the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
is a group ringGroup ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
over a field
, or an augmented -algebraAlgebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
, we can impose a ring structure on
. The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of .One interpretation is in terms of these homotopy classes of chain maps. Then the product of two elements is represented by the composition of the corresponding representatives. We can choose a single resolution of
, and do all the calculations inside , which is a differential graded algebra, with cohomology precisely .The Ext groups can also be interpreted in terms of exact sequences; this has the advantage that it does not rely on the existence of projective or injective modules. Then we take the viewpoint above that an element of
is a class, under a certain equivalence relation, of exact sequences of length starting with and ending with . This can then be spliced with an element in , by replacing and with
where the middle arrow is the composition of the functions
and . This product is called the Yoneda splice.These viewpoints turn out to be equivalent whenever both make sense.
Using similar interpretations, we find that
is a moduleModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over
, again for sufficiently nice situations.Interesting examples
If is the integral group ringGroup ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
for a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, then is the group cohomologyGroup cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
with coefficients in .For
the finite field on elements, we also have that , and it turns out that the group cohomology doesn't depend on the base ring chosen.If
is a -algebraAlgebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
, then
is the Hochschild cohomology with coefficients in the module M.If
is chosen to be the universal enveloping algebraUniversal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
for a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
, then is the Lie algebra cohomologyLie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups...
with coefficients in the module M.See also
- Tor functorTor functorIn homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....
- The Grothendieck group is a construction centered on extensions
- The universal coefficient theorem for cohomology is one notable use of the Ext functor