- Ext functor
In
mathematics , the Ext functors ofhomological algebra arederived functor s ofHom functor s. They were first used inalgebraic topology , but are common in many areas of mathematics.Definition and computation
Let R be a ring and let mathrm{Mod}_R be the category of modules over "R". Let B be in mathrm{Mod}_R and set T(B) = operatorname{Hom}_{mathrm{Mod}_R}(A,B), for fixed A in mathrm{Mod}_R. (This is a
left exact functor and thus has rightderived functor s R^nT). To this end, define:operatorname{Ext}_R^n(A,B)=(R^nT)(B),
i.e., take an injective resolution
:J(B)leftarrow Bleftarrow 0,
compute
:operatorname{Hom}_{mathrm{Mod}_R}(A,J(B))leftarrowoperatorname{Hom}_{mathrm{Mod}_R}(A,B)leftarrow0,
and take the
cohomology of this complex.Similarly, we can view the functor G(A)=operatorname{Hom}_{mathrm{Mod}_R}(A,B) for a fixed module B as a
contravariant left exact functor , and thus we also have rightderived functor s R^nG, but instead of the injective resolution used above, choose a projective resolution P(A), and proceed dually by calculating from:P(A) ightarrow A ightarrow 0,compute
:operatorname{Hom}_{mathrm{Mod}_R}(P(A),B)leftarrowoperatorname{Hom}_{mathrm{Mod}_R}(A,B)leftarrow0,
and then take the cohomology.
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.
* operatorname{Ext}^i_{mathrm{Mod}_R}(A,B)=0 for i>0 if either B is injective or A is projective.
* The converse also holds: if operatorname{Ext}^1_{mathrm{Mod}_R}(A,B)=0 for all A, then operatorname{Ext}^i_{mathrm{Mod}_R}(A,B)=0 for all A, and B is injective; if operatorname{Ext}^1_{mathrm{Mod}_R}(A,B)=0 for all B, then operatorname{Ext}^i_{mathrm{Mod}_R}(A,B)=0 for all B, and A is projective.
* operatorname{Ext}^n_{mathrm{Mod}_R}(igoplus_alpha A_alpha,B)congprod_alphaoperatorname{Ext}^n_{mathrm{Mod}_R}(A_alpha,B)
* operatorname{Ext}^n_{mathrm{Mod}_R}(A,prod_eta B_eta)congprod_etaoperatorname{Ext}^n_{mathrm{Mod}_R}(A,B_eta)
Ext and extensions
Ext functors derive their name from the relationship to extensions. Given R-modules A and B, there is a bijective correspondence between
equivalence class es of extensions :0 ightarrow B ightarrow C ightarrow A ightarrow 0of A by B and elements of:operatorname{Ext}_R^1(A,B).Given two extensions:0 ightarrow B ightarrow C ightarrow A ightarrow 0 and:0 ightarrow B ightarrow C' ightarrow A ightarrow 0we can construct the Baer sum, by forming the
pullback Gamma of C ightarrow A and C' ightarrow A. We form the quotient Y=Gamma/Delta, with Delta={(-b,b):bin B}. The extension:0 ightarrow B ightarrow Y ightarrow A ightarrow 0 thus formed is called the Baer sum of the extensions C and C'.The Baer sum ends up being an
abelian group operation on the set of equivalence classes, with the extension:0 ightarrow B ightarrow Aoplus B ightarrow A ightarrow 0 acting as the identity.Ext in abelian categories
This identification enables us to define operatorname{Ext}^1_{mathcal{C(A,B) even for
abelian categories mathcal{C} without reference to projectives and injectives. We simply take operatorname{Ext}^1_{mathcal{C(A,B) to be the set of equivalence classes of extensions of A by B, forming an abelian group under the Baer sum. Similarly, we can define higher Ext groups operatorname{Ext}^n_{mathcal{C(A,B) as equivalence classes of "n-extensions":0 ightarrow B ightarrow X_n ightarrowcdots ightarrow X_1 ightarrow A ightarrow0
under the
equivalence relation generated by the relation that identifies two extensions:0 ightarrow B ightarrow X_n ightarrowcdots ightarrow X_1 ightarrow A ightarrow0 and
:0 ightarrow B ightarrow X'_n ightarrowcdots ightarrow X'_1 ightarrow A ightarrow0
if there are maps X_m ightarrow X'_m for all m in 1,2,..,n so that every resulting square commutes.
The Baer sum of the two "n"-extensions above is formed by letting X"_1 be the
pullback of X_1 and X'_1 over A, and X"_n be the pushout of X_n and X'_n under B. Then we define the Baer sum of the extensions to be:0 ightarrow B ightarrow X"_n ightarrow X_{n-1}oplus X'_{n-1} ightarrowcdots ightarrow X_2oplus X'_2 ightarrow X"_1 ightarrow A ightarrow0.Ring structure and module structure on specific Exts
One more very useful way to view the Ext functor is this: when an element of operatorname{Ext}^n_{mathrm{Mod}_R}(A,B) is considered as an equivalence class of maps f: P_n ightarrow B for a projective resolution P_* of A ; so, then we can pick a long exact sequence Q_* ending with B and lift the map f using the projectivity of the modules P_m to a chain map f_*: P_* ightarrow Q_* 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 R is a
group ring , or a k-algebra, for a field k or even anoetherian ring k, we can impose a ring structure on operatorname{Ext}^*_{mathrm{Mod}_R}(k,k). The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of operatorname{Ext}^*_{mathrm{Mod}_R}(k,k).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 k, and do all the calculations inside operatorname{Hom}_{mathrm{Mod}_R}(P_*,P_*), which is a differential graded algebra, with homology precisely operatorname{Ext}_{mathrm{Mod}_R}(k,k).
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 operatorname{Ext}^n_{mathrm{Mod}_R}(A,B) is an exact sequence starting in A and ending in B. This is then spliced with an element in operatorname{Ext}^m_{mathrm{Mod}_R}(B,C), by replacing :ightarrow X_1 ightarrow B ightarrow 0 and 0 ightarrow B ightarrow Y_n ightarrowwith:ightarrow X_1 ightarrow Y_n ightarrowwhere the middle arrow is the composition of the functions X_1 ightarrow B and B ightarrow Y_n.
These viewpoints turn out to be equivalent whenever both make sense.
Using similar interpretations, we find that operatorname{Ext}_{mathrm{Mod}_R}^*(k,M) is a
module over operatorname{Ext}^*_{mathrm{Mod}_R}(k,k), again for sufficiently nice situations.Interesting examples
If mathbb ZG is the integral group ring for a group G, then operatorname{Ext}^*_{mathrm{Mod}_{mathbb ZG(mathbb Z,M) is the
group cohomology H^*(G,M) with coefficients in M.For mathbb F_p the finite field on p elements, we also have that H^*(G,M)=operatorname{Ext}^*_{mathrm{Mod}_{mathbb F_pG(mathbb F_p,M), and it turns out that the group cohomology doesn't depend on the base ring chosen.
If A is a k-algebra, then operatorname{Ext}^*_{mathrm{Mod}_{Aotimes_k A^{op}(A,M) is the
Hochschild cohomology operatorname{HH}^*(A,M) with coefficients in the module "M".If R is chosen to be the
universal enveloping algebra for aLie algebra mathfrak g, then operatorname{Ext}^*_{mathrm{Mod}_R}(R,M) is theLie algebra cohomology operatorname{H}^*(mathfrak g,M) with coefficients in the module "M".References
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* | year=1994
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