Ext functor on arvau

Wikipedia / 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 algebrass, and associative algebrass 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 in 1934. It was named by Samuel Eilenberg and Saunders MacLane in 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 1956.

Definition

be the category is commutative.) This is a left exact functor to the category of abelian groups, and so it has right derived functors. 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:

is the cohomology is the kernel, which is isomorphic.

. This is a contravariant functor, which can be viewed as a left exact functor from the opposite category:

That is, choose any projective resolution

, and form the cochain complex:

One may wonder why the choice of resolution has been left vague so far. In fact, 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 HomR(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

groups.

.
is projective (for example, free is injective.

The converses also hold:

If Ext(A, B) = 0 for all B, then A is projective (and hence Ext(A, B) = 0 for all i > 0).
If Ext(A, B) = 0 for all A, then B is injective (and hence Ext(A, B) = 0 for all i > 0).
.
is a principal ideal domain.
is not a zero divisor, then
-torsion subgroup for any finitely generated abelian group.
groups when the first module is the quotient of a commutative ring by any regular sequence, using the Koszul complex. is the polynomial ring is the exterior algebra; this is an example of Koszul duality.
By the general properties of derived functors, there are two basic exact sequences. First, a short exact sequence-modules induces a long exact sequence of the form
induces a long exact sequence of the form
.
Ext takes direct sums (possibly infinite) in the first variable and productss in the second variable to products. That is:
Let A be a finitely generated module over a commutative Noetherian ring R. Then Ext commutes with localization, in the sense that for every multiplicatively closed set S in R, every R-module B, and every integer i,

==Ext and extensions==

Equivalence of extensions

, an extension of A by B-modules

Two extensions

are said to be equivalent) if there is a commutative diagram:

Note that the Five lemma is called split if it is equivalent to the trivial extension

There is a one-to-one correspondence between equivalence classes. This can be made precise as follows.

The Baer sum of extensions

The Baer sum. Namely, given two extensions

and

first form the pullback,

Then form the quotient module

is the extension

Up to 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

for all m in {1, 2, ..., n} so that every resulting square commutes that is, if there is a chain map which is the identity on A and B.

The Baer sum of two n be the pullback over A be the pushout 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:

is a graded ring, for any R-module A. For example, this gives the ring structure on group cohomology. Also by associativity of the Yoneda product: for any R-modules A and B.

Important special cases

Group cohomology is defined by
1. ,
is a representation is the group ring.
For an algebra-bimodule, Hochschild cohomology is defined by
Lie algebra cohomology is a Lie algebra is the universal enveloping algebra.
For a topological space, sheaf cohomology Here Ext is taken in the abelian category of sheaves is the sheaf of locally constant, one can consider any sheaf of rings and take Ext in the category of \mathcal O_X-modules.
, taking the right derived functors of the sheaf Hom, the internal Hom. They are related to the global Ext groups via the local-to-global Ext spectral sequence.
For a commutative Noetherian local ring with residue field is the universal enveloping algebra of a graded Lie algebra, known as the homotopy Lie algebra has characteristic has to be viewed as an "adjusted Lie algebra".) There is a natural homomorphism of graded Lie algebras from the André–Quillen cohomology has characteristic zero.