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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. The name "Ext" comes from the connection between the [[functor]]s and extensions in abelian categories.
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Let ''R'' be a [[ring (mathematics)|ring]] and let Mod<sub>''R''</sub> be the [[Category (mathematics)|category]] of [[module (mathematics)|modules]] over ''R''. Let ''B'' be in Mod<sub>''R''</sub> and set ''T''(''B'') = Hom<sub>''R''</sub>(''A,B''), for fixed ''A'' in Mod<sub>''R''</sub>. This is a [[left exact functor]] and thus has right [[derived functor]]s ''R<sup>n</sup>T''. The Ext functor is defined by
 
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</ul>


:<math>\operatorname{Ext}_R^n(A,B)=(R^nT)(B).</math>
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:<math>0 \rightarrow B \rightarrow I^0 \rightarrow I^1 \rightarrow \dots, </math>
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:<math>0 \rightarrow \operatorname{Hom}_R(A,I^0) \rightarrow \operatorname{Hom}_R(A,I^1) \rightarrow \dots.</math>
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Then (''R<sup>n</sup>T'')(''B'') is the [[homology (mathematics)|homology]] of this complex. Note that Hom<sub>''R''</sub>(''A,B'') is excluded from the complex.  
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An alternative definition is given using the functor ''G''(''A'')=Hom<sub>''R''</sub>(''A,B''). For a fixed module ''B'', this is a [[Covariance and contravariance of functors|contravariant]] [[left exact functor]], and thus we also have right [[derived functor]]s ''R<sup>n</sup>G'', and can define
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:<math>\operatorname{Ext}_R^n(A,B)=(R^nG)(A).</math>
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This can be calculated by choosing any [[projective resolution]]
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:<math>\dots \rightarrow P^1 \rightarrow P^0 \rightarrow A \rightarrow 0, </math>
</ul>
 
and proceeding dually by computing
 
:<math>0\rightarrow\operatorname{Hom}_R(P^0,B)\rightarrow  \operatorname{Hom}_R(P^1,B) \rightarrow \dots.</math>
 
Then (''R<sup>n</sup>G'')(''A'') is the homology of this complex. Again note that Hom<sub>''R''</sub>(''A,B'') is excluded.
 
These two constructions turn out to yield [[isomorphic]] results, and so both may be used to calculate the Ext functor.
== Ext and extensions == <!-- "Extension of modules" redirects here -->
 
===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''-modules
 
:<math>0\rightarrow B\rightarrow E\rightarrow A\rightarrow0.</math>
 
Two extensions
 
:<math>0\rightarrow B\rightarrow E\rightarrow A\rightarrow0</math>
:<math>0\rightarrow B\rightarrow E^\prime\rightarrow A\rightarrow0</math>
 
are said to be '''equivalent''' (as extensions of ''A'' by ''B'') if there is a [[commutative diagram]]
 
[[Image:EquivalenceOfExtensions.png]].
 
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'''
 
:<math>0\rightarrow B\rightarrow A\oplus B\rightarrow A\rightarrow0.</math>
 
There is a bijective correspondence between [[equivalence class]]es of extensions
 
:<math>0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0</math>
 
of ''A'' by ''B'' and elements of
 
:<math>\operatorname{Ext}_R^1(A,B).</math>
 
===The Baer sum of extensions===
Given two extensions
 
:<math>0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0</math>
:<math>0\rightarrow B\rightarrow E^\prime\rightarrow A\rightarrow 0</math>
 
we can construct the '''Baer sum''', by forming the [[Pullback (category theory)|pullback]] over <math>A</math>,
 
<math>\Gamma = \left\{ (e, e') \in E \oplus E' \; | \; g(e) = g'(e')\right\}.</math>
 
We form the quotient
 
<math>Y = \Gamma / \{(f(b), 0) - (0, f'(b))\;|\;b \in B\}</math>,
 
that is, we [[mod out]] by the relation <math>(f(b)+e, e') \sim (e, f'(b)+e')</math>. The extension
 
:<math>0\rightarrow B\rightarrow Y\rightarrow A\rightarrow 0</math>
 
where the first arrow is <math>b \mapsto [(f(b), 0)] = [(0, f'(b))]</math> and the second <math>(e, e') \mapsto g(e) = g'(e')</math> 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 0 → ''B'' → ''E'' → ''A'' → 0 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 Ext{{su|b=''R''|p=1}}(''A'', ''B'')
 
== Construction of Ext in abelian categories ==
This identification enables us to define Ext{{su|b='''Ab'''|p=1}}(''A'', ''B'') even for [[abelian categories]] '''Ab''' without reference to [[Projective module|projectives]] and [[Injective module|injectives]]. We simply take Ext{{su|b='''Ab'''|p=1}}(''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 Ext{{su|b='''Ab'''|p=''n''}}(''A'', ''B'') as equivalence classes of ''n-extensions''
 
:<math>0\rightarrow B\rightarrow X_n\rightarrow\cdots\rightarrow X_1\rightarrow A\rightarrow0</math>
 
under the [[equivalence relation]] generated by the relation that identifies two extensions
 
:<math>0\rightarrow B\rightarrow X_n\rightarrow\cdots\rightarrow X_1\rightarrow A\rightarrow0</math>
:<math>0\rightarrow B\rightarrow X'_n\rightarrow\cdots\rightarrow X'_1\rightarrow A\rightarrow0</math>
 
if there are maps ''X<sub>m</sub>'' → ''X′<sub>m</sub>'' for all ''m'' in {1, 2, ..., ''n''} so that every resulting [[Commutative diagram|square commutes]].
 
The Baer sum of the two ''n''-extensions above is formed by letting ''X{{su|b=1|p=′′}}'' be the [[Pullback (category theory)|pullback]] of ''X<sub>1</sub>'' and ''X{{su|b=1|p=′}}'' over ''A'', and ''X{{su|b=n|p=′′}}'' be the [[Pushout (category theory)|pushout]] of ''X<sub>n</sub>'' and ''X{{su|b=n|p=′}}'' under ''B'' quotiented by the skew diagonal copy of B. Then we define the Baer sum of the extensions to be
 
:<math>0\rightarrow B\rightarrow X''_n\rightarrow X_{n-1}\oplus X'_{n-1}\rightarrow\cdots\rightarrow X_2\oplus X'_2\rightarrow X''_1\rightarrow A\rightarrow0.</math>
 
== Further properties of Ext ==
The Ext functor exhibits some convenient properties, useful in computations.
 
* Ext{{su|b=''R''|p=''i''}}(''A'', ''B'') = 0 for ''i'' > 0 if either ''B'' is [[injective module|injective]] or ''A'' [[projective module|projective]].
 
* A converse also holds: if Ext{{su|b=''R''|p=1}}(''A'', ''B'') = 0 for all ''A'', then Ext{{su|b=''R''|p=''i''}}(''A'', ''B'') = 0 for all ''A'', and ''B'' is injective; if Ext{{su|b=''R''|p=1}}(''A'', ''B'') = 0 for all ''B'', then Ext{{su|b=''R''|p=''i''}}(''A'', ''B'') = 0 for all ''B'', and ''A'' is projective.
 
* <math>\operatorname{Ext}^n_R \left (\bigoplus_\alpha A_\alpha,B \right )\cong\prod_\alpha\operatorname{Ext}^n_R(A_\alpha,B)</math>
 
* <math>\operatorname{Ext}^n_R \left (A,\prod_\beta B_\beta \right )\cong\prod_\beta\operatorname{Ext}^n_R(A,B_\beta)</math>
 
== Ring structure and module structure on specific Exts ==
One more very useful way to view the Ext functor is this: when an element of Ext{{su|b=''R''|p=''n''}}(''A'', ''B'') = 0 is considered as an equivalence class of maps ''f'': ''P<sub>n</sub>'' → ''B'' for a [[Projective module#Facts|projective resolution]] ''P''<sub>*</sub> of ''A'' ; so, then we can pick a long exact sequence ''Q''<sub>*</sub> ending with ''B'' and lift the map ''f'' using the projectivity of the modules ''P<sub>m</sub>'' to a [[Chain complex#Chain maps|chain map]] ''f''<sub>*</sub>: ''P''<sub>*</sub> → ''Q''<sub>*</sub> of degree -n. It turns out that [[Chain complex#Chain homotopy|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 (mathematics)|ring]] ''R'' is a [[group ring]] over a field ''k'', or an augmented ''k''-[[Algebra over a field|algebra]], we can impose a ring structure on Ext{{su|b=''R''|p=*}}(''k'', ''k''). The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of Ext{{su|b=''R''|p=*}}(''k'', ''k'').
 
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 ''k'', and do all the calculations inside Hom<sub>''R''</sub>(''P''<sub>*</sub>,''P''<sub>*</sub>), which is a differential graded algebra, with cohomology precisely Ext<sub>''R''</sub>(''k,k'').
 
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 Ext{{su|b=''R''|p=''n''}}(''A'', ''B'') is a class, under a certain equivalence relation, of exact sequences of length ''n'' + 2 starting with ''B'' and ending with ''A''. This can then be spliced with an element in Ext{{su|b=''R''|p=''m''}}(''C'', ''A''), by replacing ... → ''X''<sub>1</sub> → ''A'' → 0 and 0 → ''A'' → ''Y<sub>n</sub>'' → ... with:
 
:<math>\cdots \rightarrow X_1\rightarrow Y_n\rightarrow \cdots </math>
 
where the middle arrow is the composition of the functions ''X''<sub>1</sub> → ''A'' and ''A'' → ''Y<sub>n</sub>''. This product is called the ''Yoneda splice''.
 
These viewpoints turn out to be equivalent whenever both make sense.
 
Using similar interpretations, we find that Ext{{su|b=''R''|p=*}}(''k'', ''M'') is a [[Module (mathematics)|module]] over Ext{{su|b=''R''|p=*}}(''k'', ''k''), again for sufficiently nice situations.
 
== Interesting examples ==
If '''Z'''[''G''] is the [[group ring|integral group ring]] for a [[Group (mathematics)|group]] ''G'', then Ext{{su|b='''Z'''[''G'']|p=*}}('''Z''', ''M'') is the [[group cohomology]] H*(''G,M'') with coefficients in ''M''.
 
For '''F'''<sub>''p''</sub> the finite field on ''p'' elements, we also have that H*(''G,M'') = Ext{{su|b='''F'''<sub>''p''</sub>[''G'']|p=*}}('''F'''<sub>''p''</sub>, ''M''), and it turns out that the group cohomology doesn't depend on the base ring chosen.
 
If ''A'' is a ''k''-[[algebra over a field|algebra]], then Ext{{su|b=''A'' ⊗<sub>''k''</sub> ''A''<sup>op</sup>|p=*}}(''A'', ''M'') is the [[Hochschild cohomology]] HH*(''A,M'') with coefficients in the ''A''-bimodule ''M''.
 
If ''R'' is chosen to be the [[universal enveloping algebra]] for a [[Lie algebra]] <math>\mathfrak g</math> over a commutative ring ''k'', then Ext{{su|b=''R''|p=*}}(''k'', ''M'') is the [[Lie algebra cohomology]] <math>\operatorname{H}^*(\mathfrak g,M)</math> with coefficients in the module ''M''.
 
==See also==
* [[Tor functor]]
* The [[Grothendieck group#Grothendieck group and extensions in an abelian category|Grothendieck group]] is a construction centered on extensions
* The [[universal coefficient theorem for cohomology]] is one notable use of the Ext functor
 
==References==
* {{Citation | last1=Gelfand | first1=Sergei I. | last2=Manin | first2=Yuri Ivanovich | author2-link=Yuri Ivanovich Manin | title=Homological algebra | isbn=978-3-540-65378-3 | year=1999 | publisher=Springer | location=Berlin}}
* {{Weibel IHA}}
 
[[Category:Homological algebra]]
[[Category:Binary operations]]

Revision as of 09:30, 1 March 2014

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