In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Ext functors of
homological algebraHomological 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 functorIn 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 functorIn 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 topologyAlgebraic 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...
, but are common in many areas of mathematics.
Let be a
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
and let be the
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
of
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may 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 functorIn 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 . Define
i.e., take an injective resolution
compute
and take the
cohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
of this complex. Note that is
excluded from the complex.
Similarly, the functor , for a fixed module B, is a contravariant left exact functor, and thus we also have right
derived functorIn 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 , but instead of the injective resolution used above, choose a projective resolution
and proceed dually by computing
and taking the cohomology.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Ext functors of
homological algebraHomological 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 functorIn 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 functorIn 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 topologyAlgebraic 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...
, but are common in many areas of mathematics.
Definition and computation
Let be a
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
and let be the
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
of
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may 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 functorIn 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 . Define
i.e., take an injective resolution
compute
and take the
cohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
of this complex. Note that is
excluded from the complex.
Similarly, the functor , for a fixed module B, is a contravariant left exact functor, and thus we also have right
derived functorIn 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 , but instead of the injective resolution used above, choose a projective resolution
and proceed dually by computing
and taking the cohomology. Again note that is excluded.
These two constructions turn out to yield isomorphic results, and so both may be used for calculation of Ext.
Properties of Ext
The Ext functor exhibits some convenient properties, useful in computations.
- for if either is injective
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...
or is projective.
- The converse also holds: if for all , then for all , and is injective; if for all , then for all , and is projective.
Ext and 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-modules
Two extensions
are said to be equivalent
(as extensions of A by B) if there is a commutative diagramIn mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition...
.
An extension of A by B is called split
if it is equivalent to the extension
There is a bijective correspondence between equivalence classIn mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
es of extensions
of by and elements of
Given two extensions and
we can construct the Baer sum
, by forming the pullbackSuppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
of E → A and E ' → A. We form the quotient , with . The extension
thus formed is called the Baer sum of the extensions E and E.
The Baer sum ends up being an
abelian groupAn 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...
operation on the set of equivalence classes, with the extension
acting as the identity.
Ext in abelian categories
This identification enables us to define even for abelian categories without reference to
projectivesIn 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
injectivesIn 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-extensions
under the
equivalence relationIn mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets...
generated by the relation that identifies two extensions
and
if there are maps for all in so that every resulting
square commutesIn mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition...
.
The Baer sum of the two
n-extensions above is formed by letting be the
pullbackSuppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
of and over , and be the
pushoutIn 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 .The pushout is the categorical dual of the...
of and under . Then we define the Baer sum of the extensions to be
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
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
is a
group ringIn 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...
, or a k-
algebraIn 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...
, for a field k or even a
noetherian ringIn abstract algebra, a Noetherian ring, named after Emmy Noether, is a ring that satisfies the ascending chain condition on ideals. Explicitly this means: given an increasing sequence of idealsthere exists an for which...
k, 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 precisely 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 homology precisely .
Another interpretation, not in fact relying on the existence of projective or injective modules is that of
Yoneda splices. Then we take the viewpoint above that an element of is an exact sequence starting in and ending in . This is then spliced with an element in , by replacing and
with
where the middle arrow is the composition of the functions and .
These viewpoints turn out to be equivalent whenever both make sense.
Using similar interpretations, we find that is a
module-Engineering:* Modular design, design of a system in parts* Modular Function Deployment, a method for product modularization* Ontology modularization, a methodological principle in ontology engineering* Modular programming, a software design technique...
over , again for sufficiently nice situations.
Interesting examples
If is the
integral group ringIn 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
groupIn 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 cohomologyIn 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...
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 -
algebraIn 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 algebraIn 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.To understand the basic idea of this construction, first note that...
for a
Lie algebraIn 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 cohomologyIn 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.