Ext is a bifunctor that plays a central role in homological algebra .
definition
Let be an Abelian category , for example the category of the modules of a ring , which is the standard example according to Mitchell's embedding theorem . To two objects and from is the class of the short exact sequences of the form
![{\ mathcal {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd)
![{\ mathcal {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![{\ mathcal {E}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0)
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0.](https://wikimedia.org/api/rest_v1/media/math/render/svg/770674a567325a012dea73db16d0a956e1aaae06)
On is now an equivalence relation defined. Two exact sequences and are equivalent if there is a morphism , so the diagram
![{\ mathcal {E}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0)
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![{\ displaystyle g \ colon Y \ to Y '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbbaaee5e7df403f409390ef37dcb66163c0cef)
![\ begin {matrix} 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \\ & & \ downarrow \ operatorname {id} && \ downarrow g && \ downarrow \ operatorname {id} \ \ 0 & \ to & X & \ to & Y '& \ to & Z & \ to & 0 \ end {matrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac01cf3aeba5d674363f755b6d56e874e760e1b4)
commutes. It is the identical morphism.
![\ operatorname {id}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d05c35f453dbbea25a4ed64dce60be8931827d34)
From the five lemma it follows immediately that if there is such a morphism , it must be an isomorphism . The class modulo of this equivalence relation is a set and is denoted by. A group structure can be defined on this set.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![{\ mathcal {E}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0)
![\ mathrm {Ext} (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc0c56b54ffaae8d86470e6c3c5560a15fa7565)
Functoriality
Morphisms in the Abelian category induce morphisms between the Ext groups in the following way, so that becomes a two-digit functor.
![\ mathrm {Ext}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d62c5fa7a8cff017d6febd4d585be1f0bcd2799)
You can push-out to and from the sequence :
![{\ displaystyle g \ colon X \ to X '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36bf51ea216a2f013de144a61ba0e1650195f950)
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![\ begin {matrix} 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \\ & & \ downarrow g && \ downarrow && \\ 0 & \ to & X '& \ to & Y '\ end {matrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74f68b04a3103df2cad479c731ccca0bd6ed357)
Because of the universal property of the push-out there is an induced epimorphism from Y 'to Z, so that the following diagram commutes:
![\ begin {matrix} 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \\ & & \ downarrow g && \ downarrow && \ downarrow \ operatorname {id} \\ 0 & \ to & X '& \ to & Y' & \ to & Z & \ to & 0 \ end {matrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f86a2a7be6b38e4537f319e2a96cb0ef28247375)
The bottom line is also exact and its equivalence class is therefore an element in .
![\ mathrm {Ext} (Z, X ')](https://wikimedia.org/api/rest_v1/media/math/render/svg/0274f74e81264c3aa712b97dfee6830251be6c7c)
If the equivalence class of is mapped to the equivalence class of , a well-defined group homomorphism is obtained .
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![0 \ rightarrow X '\ rightarrow Y' \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/069e148b415582116b9bbc581dfdcb37ce9cee36)
![\ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} (Z, X ')](https://wikimedia.org/api/rest_v1/media/math/render/svg/c21e2c04438171b12ca3539a296945576c37ad01)
This also works dual with morphisms from Z 'to Z. The following pull-back can be made to the sequence :
![{\ displaystyle g \ colon Z '\ to Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f41b7b83ac5a143d444aaca07595eab23021e315)
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![\ begin {matrix} & & & & Y '& \ to & Z' & \ to & 0 \\ & & && \ downarrow && \ downarrow g \\ 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \ end {matrix}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/d70f3cc3055b2ac7edff1677e79c7f73673eab27)
Because of the universal property of the pull-back, there is an induced monomorphism from X to Y ', so that the following diagram commutes:
![\ begin {matrix} 0 & \ to & X & \ to & Y '& \ to & Z' & \ to & 0 \\ & & \ downarrow \ operatorname {id} && \ downarrow && \ downarrow g \\ 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \ end {matrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ca2073f7d763c810e0fd4462016757d4bc3645)
The top line is also exact and thus defines an element in .
![\ mathrm {Ext} (Z ', X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b458ddaf0a4b6606abb959bc41f274240febfae7)
If the equivalence class of is mapped onto the equivalence class of , one again obtains a well-defined group homomorphism .
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![0 \ rightarrow X \ rightarrow Y '\ rightarrow Z' \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a359a90b019e066976773e75aac0ed615af768)
![\ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} (Z ', X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb06ab3138bc0c1d0540a6cadf27a6765f3827b)
Ext as the derivative of the Hom functor
Another way of defining it uses the derived functors of Hom . The construction defined above can be identified with the first right derivative of the Hom functor.
More precisely, one considers an Abelian category with a sufficient number of projective objects (i.e. each object is the quotient of a projective object) and defines
the contravariant functor![\ mathrm {Hom} (-, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a922a677d0c45e2086665bf60d4b4bb7a5cdee07)
-
,
that is, one forms the -th legal derivative of and applies the functor thus created .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![\ mathrm {Hom} (-, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a922a677d0c45e2086665bf60d4b4bb7a5cdee07)
![Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd)
More specifically, this means the following: Let it be and
![n \ geq 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe)
![{\ displaystyle {\ begin {array} {ccccc} \ ldots \ rightarrow & P_ {n} & \ rightarrow & P_ {n-1} & \ rightarrow \ ldots \ rightarrow Z \ rightarrow 0 \\ & \ lambda _ {n} \ downarrow & \ nearrow \ kappa _ {n} \\ & K_ {n} \ end {array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe4e4f53c8697ddead33b7c7d7368e9eb7b7a37)
a projective resolution of having an epimorphism and a monomorphism such that . Next is the induced homomorphism
![\ kappa_n: K_n \ rightarrow P_ {n-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e159328b43a00a23dbda84a96f95460fec465338)
![{\ displaystyle (P_ {n} \ rightarrow P_ {n-1}) = \ kappa _ {n} \ circ \ lambda _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3552d40dd2116fcaf95cf2bbdd61282836e48a7a)
![\ kappa_n ^ * = \ mathrm {Hom} (\ kappa_n, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/92fcba686311d65e9f74c7690906b7809264b6b9)
-
.
Then
-
.
The elements from are therefore certain equivalence classes of elements from .
![\ mathrm {Ext} ^ n (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/45eda596d58681c7baab8ae890c1757bd196439f)
![\ mathrm {Hom} (K_n, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbdc06c1ef9ea44210031ea9ece77d9206084936)
Finally, it should be noted that one can swap the roles of and also one receives
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd)
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.
Relationship between Ext and Ext 1
This section aims to explain how the constructs and defined above are related. We construct a picture .
![\ mathrm {Ext}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d62c5fa7a8cff017d6febd4d585be1f0bcd2799)
![\ mathrm {Ext} ^ 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/a15be2d0ea3753a49ba765084700b13295ca4a99)
![\ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} ^ 1 (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b988daf87e3c22f596e71941b788e8bca6cb01e5)
Be a short exact sequence that defines an element from . Next is a short exact sequence with projective . Using the projectivity of , one can create a commutative diagram
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![\ mathrm {Ext} (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc0c56b54ffaae8d86470e6c3c5560a15fa7565)
![0 \ rightarrow K \ rightarrow P \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/95adf5a574c18983d2534b940ad77cd6ed1a7603)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![\ begin {array} {ccccccc} 0 \ rightarrow & K & \ rightarrow & P & \ rightarrow & Z & \ rightarrow 0 \\ & \ downarrow \ psi & & \ downarrow \ varphi & & \ Vert \\ 0 \ rightarrow & X & \ rightarrow & Y & \ rightarrow & Z & \ rightarrow 0 \ end {array}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27619489f890919fe5f6155b2b7d37034968cd84)
to construct. Then there is a homomorphism whose equivalence class defines an element as described above .
![\ psi \ in \ mathrm {Hom} (K, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7355caafe7b589f7e341024e8484348ec04db6c4)
![\ mathrm {Ext} ^ n (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/45eda596d58681c7baab8ae890c1757bd196439f)
![\ mathrm {Ext} ^ 1 (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb09b08fb23116071e882967ba4d66b6e465233e)
If one forms the equivalence class of in the equivalence class of in starting, we obtain a well-defined figure , it can be shown from that it is a group isomorphism is.
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
![\ mathrm {Ext} (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc0c56b54ffaae8d86470e6c3c5560a15fa7565)
![\ psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![\ mathrm {Ext} ^ 1 (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb09b08fb23116071e882967ba4d66b6e465233e)
![\ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} ^ 1 (Z, X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b988daf87e3c22f596e71941b788e8bca6cb01e5)
Therefore one can identify with , that is , in this sense it can be defined as the first legal derivative of the -function.
![\ mathrm {Ext}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d62c5fa7a8cff017d6febd4d585be1f0bcd2799)
![\ mathrm {Ext} ^ 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/a15be2d0ea3753a49ba765084700b13295ca4a99)
![\ mathrm {Ext}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d62c5fa7a8cff017d6febd4d585be1f0bcd2799)
![\ mathrm {hom}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eea3aafc91ebbd147d45c3c69e88431c48cbe9f8)
Long exact sequence
The Hom functor is left exact, that is, for a short exact sequence
![0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a9a19757dfb0c000a99d0ae4d97a8a87aacf9)
and another object (module) has an exact sequence
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
-
,
and in general this cannot be continued exactly with 0. Because of left-hand precision, the 0th derivative of the Hom functor agrees with Hom, that is, if one extends the above definition of to , one has . The long exact sequence for derived additive functors therefore gives the following exact sequence
![\ mathrm {Ext} ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/8953905d9d0ef86217cde0c3728c4f00b72c4860)
![n = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae)
![\ mathrm {Ext} ^ 0 = \ mathrm {Hom}](https://wikimedia.org/api/rest_v1/media/math/render/svg/721150a5670c2117e30b3dad4aaa93762dbb7a30)
-
-
.
A long exact sequence is obtained analogously
-
-
.
In this sense, the Ext functors close the gap created by the lack of accuracy of the Hom functor.
Individual evidence
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↑ Sergei I. Gelfand & Yuri Ivanovich Manin: Homological Algebra , Springer, Berlin, 1999, ISBN 978-3-540-65378-3
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^ Charles A. Weibel: An introduction to homological algebra , Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1999, ISBN 978-0-521-55987-4
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^ Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , sentence 3.13
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↑ Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , Theorem 4.5
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↑ Saunders Mac Lane : Homology , Springer Grundlehren der Mathematischen Wissenschaften Volume 114 (1967), chap. III, Theorem 3.4 and Theorem 9.1