Editing Ext functor

{{Short description|Construction in homological algebra}}
In [[mathematics]], the '''Ext functors''' are the [[derived functor]]s 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 [[group cohomology|cohomology of groups]], [[Lie algebra cohomology|Lie algebra]]s, and [[Hochschild cohomology|associative algebra]]s can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext<sup>1</sup> classifies [[group extension|extensions]] of one [[module (mathematics)|module]] by another.

In the special case of [[abelian group]]s, 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 (mathematics)|ring]], Ext was defined by [[Henri Cartan]] and Eilenberg in their 1956 book ''Homological Algebra''.<ref>Weibel (1999); Cartan & Eilenberg (1956), section VI.1.</ref>

==Definition==
Let <math>R</math> be a ring and let <math>R\text{-Mod}</math> be the [[category (mathematics)|category]] of modules over <math>R</math>. (One can take this to mean either left <math>R</math>-modules or right <math>R</math>-modules.) For a fixed <math>R</math>-module <math>A</math>, let <math>T(B)=\text{Hom}_R(A,B)</math> for <math>B</math> in <math>R\text{-Mod}</math>. (Here <math>\text{Hom}_R(A,B)</math> is the abelian group of <math>R</math>-linear maps from <math>A</math> to <math>B</math>; this is an <math>R</math>-module if <math>R</math> is [[commutative ring|commutative]].) This is a [[left exact functor]] from <math>R\text{-Mod}</math> to the [[category of abelian groups]] <math>\mathbf{Ab}</math>, and so it has right [[derived functor]]s <math>R^iT</math>. The Ext groups are the abelian groups defined by

:<math>\operatorname{Ext}_R^i(A,B)=(R^iT)(B),</math>

for an [[integer]] ''i''. By definition, this means: take any [[injective resolution]]

:<math>0 \to B \to I^0 \to I^1 \to \cdots,</math>

remove the term ''B'', and form the [[cochain complex]]:

:<math>0 \to \operatorname{Hom}_R(A,I^0) \to \operatorname{Hom}_R(A,I^1) \to \cdots.</math>

For each integer <math>i</math>, <math>\text{Ext}_R^i(A,B)</math> is the [[chain complex|cohomology]] of this complex at position <math>i</math>. It is zero for <math>i</math> negative. For example, <math>\text{Ext}_R^0(A,B)</math> is the [[kernel (linear algebra)|kernel]] of the map <math>\text{Hom}_R(A,I^0)\rightarrow\text{Hom}_R(A,I^1)</math>, which is [[isomorphic]] to <math>\text{Hom}_R(A,B)</math>.

An alternative definition uses the functor <math>G(A)=\operatorname{Hom}_R(A,B)</math>, for a fixed <math>R</math>-module <math>B</math>. This is a [[Covariance and contravariance of functors|contravariant]] functor, which can be viewed as a left exact functor from the [[opposite category]] <math>(R\text{-Mod})^{\text{op}}</math> to <math>\mathbf{Ab}</math>. The Ext groups are defined as the right derived functors <math>R^iG</math>:

:<math>\operatorname{Ext}_R^i(A,B)=(R^iG)(A).</math>

That is, choose any [[projective resolution]]

:<math>\cdots \to P_1 \to P_0 \to A \to 0, </math>

remove the term <math>A</math>, and form the cochain complex:

:<math>0\to \operatorname{Hom}_R(P_0,B)\to \operatorname{Hom}_R(P_1,B) \to \cdots.</math>

Then <math>\operatorname{Ext}_R^i(A,B)</math> is the cohomology of this complex at position <math>i</math>.

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.<ref>Weibel (1994), sections 2.4 and 2.5 and Theorem 2.7.6.</ref> 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{{supsub|''i''|''R''}}(''A'', ''B'') is an ''R''-module (using that Hom<sub>''R''</sub>(''A'', ''B'') is an ''R''-module in this case). For a non-commutative ring ''R'', Ext{{supsub|''i''|''R''}}(''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{{supsub|''i''|''R''}}(''A'', ''B'') is at least an ''S''-module.

==Properties of Ext==
Here are some of the basic properties and computations of Ext groups.<ref>Weibel (1994), Chapters 2 and 3.</ref>

*Ext{{supsub|0|''R''}}(''A'', ''B'') ≅ Hom<sub>''R''</sub>(''A'', ''B'') for any ''R''-modules ''A'' and ''B''.

*Ext{{su|b=''R''|p=''i''}}(''A'', ''B'') = 0 for all ''i'' > 0 if the ''R''-module ''A'' is [[projective module|projective]] (for example, [[free module|free]]) or if ''B'' is [[injective module|injective]].

*The converses also hold:
**If Ext{{su|b=''R''|p=1}}(''A'', ''B'') = 0 for all ''B'', then ''A'' is projective (and hence Ext{{su|b=''R''|p=''i''}}(''A'', ''B'') = 0 for all ''i'' > 0).
**If Ext{{su|b=''R''|p=1}}(''A'', ''B'') = 0 for all ''A'', then ''B'' is injective (and hence Ext{{su|b=''R''|p=''i''}}(''A'', ''B'') = 0 for all ''i'' > 0).

*<math>\operatorname{Ext}^i_{\Z}(A,B) = 0</math> for all <math>i\geq 2</math> and all abelian groups <math>A</math> and <math>B</math>.<ref>Weibeil (1994), Lemma 3.3.1.</ref>

* Generalizing the previous example, <math>\operatorname{Ext}^i_R(A,B)=0</math> for all <math>i\geq 2</math> if <math>R</math> is a [[principal ideal domain]].

*If <math>R</math> is a commutative ring and <math>u</math> in <math>R</math> is not a [[zero divisor]], then
:<math display="block">\operatorname{Ext}_R^i(R/(u),B)\cong\begin{cases} B[u] & i=0\\ B/uB & i=1\\ 0 &\text{otherwise,}\end{cases}</math>
:for any <math>R</math>-module <math>B</math>. Here <math>B[u]</math> denotes the <math>u</math>-torsion subgroup of <math>B</math>, <math>\{x\in B:ux=0\}</math>. Taking <math>R</math> to be the ring <math>\Z</math> of integers, this calculation can be used to compute <math>\operatorname{Ext}^1_{\Z}(A,B)</math> for any [[finitely generated abelian group]] <math>A</math>.

*Generalizing the previous example, one can compute Ext groups when the first module is the quotient of a commutative ring by any [[regular sequence]], using the [[Koszul complex]].<ref>Weibel (1994), section 4.5.</ref> For example, if ''R'' is the [[polynomial ring]] ''k''[''x''<sub>1</sub>,...,''x''<sub>''n''</sub>] over a field ''k'', then Ext{{supsub|*|''R''}}(''k'',''k'') is the [[exterior algebra]] ''S'' over ''k'' on ''n'' generators in Ext<sup>1</sup>. Moreover, Ext{{supsub|*|''S''}}(''k'',''k'') is the polynomial ring ''R''; this is an example of [[Koszul duality]].

*By the general properties of derived functors, there are two basic [[exact sequence]]s for Ext.<ref>Weibel (1994), Definition 2.1.1.</ref> First, a [[short exact sequence]] <math>0\rightarrow K\rightarrow L\rightarrow M\rightarrow 0</math> of ''R''-modules induces a long exact sequence of the form
::<math>0 \to \mathrm{Hom}_R(A,K) \to \mathrm{Hom}_R(A,L) \to \mathrm{Hom}_R(A,M) \to \mathrm{Ext}^1_R(A,K) \to \mathrm{Ext}^1_R(A,L) \to \cdots,</math>
:for any ''R''-module ''A''. Also, a short exact sequence <math>0\rightarrow K\rightarrow L\rightarrow M\rightarrow 0</math> induces a long exact sequence of the form
::<math>0 \to \mathrm{Hom}_R(M,B) \to \mathrm{Hom}_R(L,B) \to \mathrm{Hom}_R(K,B) \to \mathrm{Ext}^1_R(M,B) \to \mathrm{Ext}^1_R(L,B) \to \cdots,</math>
:for any ''R''-module ''B''.

*Ext takes [[direct sum of modules|direct sums]] (possibly infinite) in the first variable and [[direct product#Direct product of modules|product]]s in the second variable to products.<ref>Weibel (1994), Proposition 3.3.4.</ref> That is:
::<math>\begin{align}
\operatorname{Ext}^i_R \left(\bigoplus_\alpha M_\alpha,N \right) &\cong\prod_\alpha \operatorname{Ext}^i_R (M_\alpha,N) \\
\operatorname{Ext}^i_R \left(M,\prod_\alpha N_\alpha \right ) &\cong\prod_\alpha \operatorname{Ext}^i_R (M,N_\alpha)
\end{align}</math>

* Let ''A'' be a finitely generated module over a commutative [[Noetherian ring]] ''R''. Then Ext commutes with [[localization of a ring|localization]], in the sense that for every [[multiplicatively closed set]] ''S'' in ''R'', every ''R''-module ''B'', and every integer ''i'',<ref>Weibel (1994), Proposition  3.3.10.</ref>
::<math>S^{-1} \operatorname{Ext}_R^i(A, B) \cong \operatorname{Ext}_{S^{-1} R}^i \left (S^{-1} A, S^{-1} B \right ).</math>

==Ext and extensions== <!-- "Extension of modules" redirects here -->
===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

:<math>0\to B\to E\to A\to 0.</math>

Two extensions

:<math>0\to B\to E\to A\to 0</math>
:<math>0\to B\to E' \to A\to 0</math>

are said to be '''equivalent''' (as extensions of ''A'' by ''B'') if there is a [[commutative diagram]]:

:[[File:EquivalenceOfExtensions.svg|300px]]

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\to B\to A\oplus B\to A\to 0.</math>

There is a one-to-one correspondence between [[equivalence class]]es of extensions of ''A'' by ''B'' and elements of Ext{{supsub|1|''R''}}(''A'', ''B'').<ref>Weibel (1994), Theorem 3.4.3.</ref> This can be made precise as follows.

'''Proof.''' Fix a short exact sequence 
:<math>0 \to M \to P \to A \to 0</math>
where <math>P</math> is projective. Applying <math>\operatorname{Hom}(-, B)</math> yields the long exact sequence
:<math>\operatorname{Hom}(P, B) \to \operatorname{Hom}(M, B) \xrightarrow{\delta} \operatorname{Ext}(A, B) \to 0.</math>

Given <math>x \in \operatorname{Ext}(A, B)</math>, choose <math>\beta \in \operatorname{Hom}(M, B)</math> such that <math>\delta(\beta) = x</math>. Consider the pushout of <math>j: M \to P</math> along <math>\beta</math>, given by the cokernel of the map
:<math>M \to P \oplus B, \quad m \mapsto (j(m), -\beta(m)).</math>

Define <math>X</math> as this pushout object. This yields the commutative diagram:
:[[File:ExtDiagram1.svg|300px]]

Here, <math>X \to A</math> is induced by the map <math>P \to A</math>. The bottom row is an extension of <math>A</math> by <math>B</math>, denoted <math>\xi</math>, and the connecting map <math>\delta</math> ensures that <math>\delta(\xi) = x</math>, proving surjectivity.

To show well-definedness on equivalence classes, suppose <math>\beta'</math> is another lift of <math>x</math>. Then there exists <math>f \in \operatorname{Hom}(P, B)</math> such that <math>\beta' = \beta + f \circ j</math>. If <math>X'</math> is the pushout of <math>j</math> and <math>\beta'</math>, then an isomorphism <math>X \cong X'</math> is induced, making the extensions equivalent.

Conversely, given an extension 
:<math>0 \to B \to X \to A \to 0</math>,
the lifting property of <math>P</math> gives a map <math>\tau: P \to X</math> fitting into the diagram
:[[File:ExtDiagram2.svg|300px]]

Here <math>X</math> is the pushout of <math>j</math> and <math>\gamma</math>. This shows that the map is injective.

Thus, the set of equivalence classes of extensions of <math>A</math> by <math>B</math> is naturally isomorphic to <math>\operatorname{Ext}(A, B)</math>. ∎

The trivial extension corresponds to the zero element of Ext{{supsub|1|''R''}}(''A'', ''B'').

===The Baer sum of extensions===
The '''Baer sum''' is an explicit description of the abelian group structure on <math>\operatorname{Ext}_R^1(A,B)</math>, viewed as the set of equivalence classes of extensions of <math>A</math> by <math>B</math>.<ref>Weibel (1994), Corollary 3.4.5.</ref> Namely, given two extensions

:<math>0\to B\xrightarrow[f]{} E \xrightarrow[g]{} A\to 0</math>

and

:<math>0\to B\xrightarrow[f']{} E'\xrightarrow[g']{} A\to 0,</math>

first form the [[Pullback (category theory)|pullback]] over <math>A</math>,

:<math>\Gamma = \left\{ (e, e') \in E \oplus E' \; | \; g(e) = g'(e')\right\}.</math>

Then form the [[quotient module]]

:<math>Y = \Gamma / \{(f(b), -f'(b)) \;|\;b \in B\}.</math>

The Baer sum of <math>E</math> and <math>E'</math> is the extension

:<math>0\to B\to Y\to A\to 0,</math>

where the first map is <math>b \mapsto [(f(b), 0)] = [(0, f'(b))]</math> and the second is <math>(e, e') \mapsto g(e) = g'(e')</math>.

[[Up to]] equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The negative of an extension <math>0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0</math> is the extension involving the same module <math>E</math>, but with the homomorphism <math>B\rightarrow E</math> replaced by its negative.

==Construction of Ext in abelian categories==
[[Nobuo Yoneda]] defined the abelian groups Ext{{su|b='''C'''|p=''n''}}(''A'', ''B'') for objects ''A'' and ''B'' in any [[abelian category]] '''C'''; this agrees with the definition in terms of resolutions if '''C''' has [[projective object#Enough projectives|enough projectives]] or [[injective object#Enough injectives and injective hulls|enough injectives]]. First, Ext{{supsub|0|'''C'''}}(''A'',''B'') = Hom<sub>'''C'''</sub>(''A'', ''B''). Next, Ext{{su|b='''C'''|p=1}}(''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{{su|b='''C'''|p=''n''}}(''A'', ''B'') are defined as equivalence classes of ''n-extensions'', which are exact sequences

:<math>0\to B\to X_n\to\cdots\to X_1\to A\to 0,</math>

under the [[equivalence relation]] generated by the relation that identifies two extensions

:<math>\begin{align}
\xi : 0 &\to B\to X_n\to\cdots\to X_1\to A\to 0 \\
\xi': 0 &\to B\to X'_n\to\cdots\to X'_1\to A\to 0
\end{align}</math>

if there are maps <math>X_m \to X'_m</math> for all ''m'' in {1, 2, ..., ''n''} so that every resulting [[Commutative diagram|square commutes]]
<math display='block'>
\begin{array}{cc cc cc c cc cc cc}
  0 & \longrightarrow & B & \longrightarrow & X_n & \longrightarrow &
  \dots & \longrightarrow & X_1 & \longrightarrow & A & 
  \longrightarrow & 0 
  \\
  && \Bigg\Vert && \Bigg\downarrow \iota_n \! &&&&
  \Bigg\downarrow \iota_1 && \Bigg\Vert && 
  \\
  0 & \longrightarrow & B & \longrightarrow & X'_n & \longrightarrow &
  \dots & \longrightarrow & X'_1 & \longrightarrow & A & 
  \longrightarrow & 0
\end{array}
</math>
that is, if there is a [[chain map]] <math>\iota\colon \xi \to \xi'</math> which is the identity on ''A'' and ''B''.

The Baer sum of two ''n''-extensions as above is formed by letting <math>X''_1</math> be the [[Pullback (category theory)|pullback]] of <math>X_1</math> and <math>X'_1</math> over ''A'', and <math>X''_n</math> be the [[Pushout (category theory)|pushout]] of <math>X_n</math> and <math>X'_n</math> under ''B''.<ref>Weibel (1994), Vists 3.4.6. Some minor corrections are in the [http://www.math.rutgers.edu/~weibel/Hbook.errors.edition2.pdf errata].</ref> Then the Baer sum of the extensions is

:<math>0\to B\to X''_n\to X_{n-1}\oplus X'_{n-1}\to\cdots\to X_2\oplus X'_2\to X''_1\to A\to 0.</math>

==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''').<ref>Weibel (1994), sections 10.4 and 10.7; Gelfand & Manin (2003), Chapter III.</ref> The objects of the derived category are complexes of objects in '''C'''. Specifically, one has

:<math>\operatorname{Ext}^i_{\mathbf C}(A,B) = \operatorname{Hom}_{D({\mathbf C})}(A,B[i]),</math>

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]]:

:<math>\operatorname{Ext}^i_{\mathbf C}(A,B) \times \operatorname{Ext}^j_{\mathbf C}(B,C) \to \operatorname{Ext}^{i+j}_{\mathbf C}(A,C),</math>

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{{supsub|''i''|''R''}}(''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:

:<math>P\to Q[i]\to T[i+j].</math>

By any of these interpretations, the Yoneda product is associative. As a result, <math>\operatorname{Ext}^*_R(A,A)</math> is a [[graded ring]], for any ''R''-module ''A''. For example, this gives the ring structure on [[group cohomology]] <math>H^*(G, \Z),</math> since this can be viewed as <math>\operatorname{Ext}^*_{\Z[G]}(\Z,\Z)</math>. Also by associativity of the Yoneda product: for any ''R''-modules ''A'' and ''B'', <math>\operatorname{Ext}^*_R(A,B)</math> is a module over <math>\operatorname{Ext}^*_R(A,A)</math>.

==Important special cases==

*[[Group cohomology]] is defined by <math>H^*(G,M)=\operatorname{Ext}_{\Z[G]}^*(\Z, M)</math>, where ''G'' is a group, ''M'' is a [[group representation|representation]] of ''G'' over the integers, and <math>\Z[G]</math> is the [[group ring]] of ''G''.

*For an [[algebra over a field|algebra]] ''A'' over a field ''k'' and an ''A''-[[bimodule]] ''M'', [[Hochschild cohomology]] is defined by

::<math>HH^*(A,M)=\operatorname{Ext}^*_{A\otimes_k A^{\text{op}}} (A, M).</math>

*[[Lie algebra cohomology]] is defined by <math>H^*(\mathfrak g,M)=\operatorname{Ext}^*_{U\mathfrak g}(k,M)</math>, where <math>\mathfrak g</math> is a [[Lie algebra]] over a commutative ring ''k'', ''M'' is a <math>\mathfrak g</math>-module, and <math>U\mathfrak g</math> is the [[universal enveloping algebra]].

*For a [[topological space]] ''X'', [[sheaf cohomology]] can be defined as <math>H^*(X, A) = \operatorname{Ext}^*(\Z_X, A).</math> Here Ext is taken in the abelian category of [[sheaf (mathematics)|sheaves]] of abelian groups on ''X'', and <math>\Z_X</math> is the sheaf of [[locally constant]] <math>\Z</math>-valued functions.

*For a commutative Noetherian [[local ring]] ''R'' with residue field ''k'', <math>\operatorname{Ext}^*_R(k,k)</math> is the universal enveloping algebra of a [[graded Lie algebra]] π*(''R'') over ''k'', known as the '''homotopy Lie algebra''' of ''R''. (To be precise, when ''k'' has [[characteristic of a field|characteristic]] 2, π*(''R'') has to be viewed as an "adjusted Lie algebra".<ref>Sjödin (1980), Notation 14.</ref>) There is a natural homomorphism of graded Lie algebras from the [[André–Quillen cohomology]] ''D''*(''k''/''R'',''k'') to π*(''R''), which is an isomorphism if ''k'' has characteristic zero.<ref>Avramov (2010), section 10.2.</ref>

==See also==
*[[global dimension]]
*[[bar resolution]]
*[[Grothendieck group#Grothendieck group and extensions|Grothendieck group]]
*[[Grothendieck local duality]]

==Notes==
{{reflist|30em}}

==References==
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[[Category:Homological algebra]]
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[[Category:Functors]]