Ext functor

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another.
In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book "Homological Algebra".
Definition.
Let "R" be a ring and let "R"-Mod be the category of modules over "R". (One can take this to mean either left "R"-modules or right "R"-modules.) For a fixed "R"-module "A", let "T"("B") = Hom"R"("A", "B") for "B" in "R"-Mod. (Here Hom"R"("A", "B") is the abelian group of "R"-linear maps from "A" to "B"; this is an "R"-module if "R" is commutative.) This is a left exact functor from "R"-Mod to the category of abelian groups Ab, and so it has right derived functors "RiT". The Ext groups are the abelian groups defined by
for an integer "i". By definition, this means: take any injective resolution
remove the term "B", and form the cochain complex:
For each integer "i", Ext("A", "B") is the cohomology of this complex at position "i". It is zero for "i" negative. For example, Ext("A", "B") is the kernel of the map Hom"R"("A", "I"0) → Hom"R"("A", "I"1), which is isomorphic to Hom"R"("A", "B").
An alternative definition uses the functor "G"("A")=Hom"R"("A", "B"), for a fixed "R"-module "B". This is a contravariant functor, which can be viewed as a left exact functor from the opposite category ("R"-Mod)op to Ab. The Ext groups are defined as the right derived functors "RiG":
That is, choose any projective resolution
remove the term "A", and form the cochain complex:
Then Ext("A", "B") is the cohomology of this complex at position "i".
Cartan and Eilenberg showed that these constructions are independent of the choice of projective or injective resolution, and that both constructions yield the same Ext groups. Moreover, for a fixed ring "R", Ext is a functor in each variable (contravariant in "A", covariant in "B").
For a commutative ring "R" and "R"-modules "A" and "B", Ext("A", "B") is an "R"-module (using that Hom"R"("A", "B") is an "R"-module in this case). For a non-commutative ring "R", Ext("A", "B") is only an abelian group, in general. If "R" is an algebra over a ring "S" (which means in particular that "S" is commutative), then Ext("A", "B") is at least an "S"-module.
Properties of Ext.
Here are some of the basic properties and computations of Ext groups.
Ext and extensions.
Equivalence of extensions.
The Ext groups derive their name from their relation 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 diagram:
Note that the Five lemma implies that the middle arrow is an isomorphism. An extension of "A" by "B" is called split if it is equivalent to the trivial extension
There is a one-to-one correspondence between equivalence classes of extensions of "A" by "B" and elements of Ext("A", "B"). The trivial extension corresponds to the zero element of Ext("A", "B").
The Baer sum of extensions.
The Baer sum is an explicit description of the abelian group structure on Ext("A", "B"), viewed as the set of equivalence classes of extensions of "A" by "B". Namely, given two extensions
and
first form the pullback over formula_21,
Then form the quotient module
The Baer sum of "E" and "E′" is the extension
where the first map is formula_25 and the second is formula_26.
Up to equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The negative of an extension 0 → "B" → "E" → "A" → 0 is the extension involving the same module "E", but with the homomorphism "E" → "A" replaced by its negative.
Construction of Ext in abelian categories.
Nobuo Yoneda defined the abelian groups Ext("A", "B") for objects "A" and "B" in any abelian category C; this agrees with the definition in terms of resolutions if C has enough projectives or enough injectives. First, Ext("A","B") = HomC("A", "B"). Next, Ext("A", "B") is the set of equivalence classes of extensions of "A" by "B", forming an abelian group under the Baer sum. Finally, the higher Ext groups Ext("A", "B") are defined as equivalence classes of "n-extensions", which are exact sequences
under the equivalence relation generated by the relation that identifies two extensions
if there are maps formula_29 for all "m" in {1, 2, ..., "n"} so that every resulting square commutes, that is, if there is a chain map ξ → ξ' which is the identity on "A" and "B".
The Baer sum of two "n"-extensions as above is formed by letting formula_30 be the pullback of formula_31 and formula_32 over "A", and formula_33 be the pushout of formula_34 and formula_35 under "B". Then the Baer sum of the extensions is
The derived category and the Yoneda product.
An important point is that Ext groups in an abelian category C can be viewed as sets of morphisms in a category associated to C, the derived category "D"(C). The objects of the derived category are complexes of objects in C. Specifically, one has
where an object of C is viewed as a complex concentrated in degree zero, and ["i"] means shifting a complex "i" steps to the left. From this interpretation, there is a bilinear map, sometimes called the Yoneda product:
which is simply the composition of morphisms in the derived category.
The Yoneda product can also be described in more elementary terms. For "i" = "j" = 0, the product is the composition of maps in the category C. In general, the product can be defined by splicing together two Yoneda extensions.
Alternatively, the Yoneda product can be defined in terms of resolutions. (This is close to the definition of the derived category.) For example, let "R" be a ring, with "R"-modules "A", "B", "C", and let "P", "Q", and "T" be projective resolutions of "A", "B", "C". Then Ext("A","B") can be identified with the group of chain homotopy classes of chain maps "P" → "Q"["i"]. The Yoneda product is given by composing chain maps:
By any of these interpretations, the Yoneda product is associative. As a result, formula_40 is a graded ring, for any "R"-module "A". For example, this gives the ring structure on group cohomology formula_41 since this can be viewed as formula_42. Also by associativity of the Yoneda product: for any "R"-modules "A" and "B", formula_43 is a module over formula_40.