# Ext (math)

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 ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle X}$${\ displaystyle Z}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle {\ mathcal {E}}}$

${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0.}$

On is now an equivalence relation defined. Two exact sequences and are equivalent if there is a morphism , so the diagram ${\ displaystyle {\ mathcal {E}}}$${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$${\ displaystyle 0 \ rightarrow X \ rightarrow Y '\ rightarrow Z \ rightarrow 0}$ ${\ displaystyle g \ colon Y \ to Y '}$

${\ displaystyle {\ begin {matrix} 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \\ && \ downarrow \ operatorname {id} && \ downarrow g && \ downarrow \ operatorname {id} \\ 0 & \ to & X & \ to & Y '& \ to & Z & \ to & 0 \ end {matrix}}}$

commutes. It is the identical morphism. ${\ displaystyle \ operatorname {id}}$

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. ${\ displaystyle g}$${\ displaystyle {\ mathcal {E}}}$${\ displaystyle \ mathrm {Ext} (Z, X)}$

## Functoriality

Morphisms in the Abelian category induce morphisms between the Ext groups in the following way, so that becomes a two-digit functor. ${\ displaystyle \ mathrm {Ext}}$

You can push-out to and from the sequence : ${\ displaystyle g \ colon X \ to X '}$${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$

${\ displaystyle {\ begin {matrix} 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \\ && \ downarrow g && \ downarrow && \\ 0 & \ to & X '& \ to & Y' \ end {matrix}}}$

Because of the universal property of the push-out there is an induced epimorphism from Y 'to Z, so that the following diagram commutes:

${\ displaystyle {\ begin {matrix} 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \\ && \ downarrow g && \ downarrow && \ downarrow \ operatorname {id} \\ 0 & \ to & X '& \ to & Y' & \ to & Z & \ to & 0 \ end {matrix}}}$

The bottom line is also exact and its equivalence class is therefore an element in . ${\ displaystyle \ mathrm {Ext} (Z, X ')}$

If the equivalence class of is mapped to the equivalence class of , a well-defined group homomorphism is obtained . ${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$${\ displaystyle 0 \ rightarrow X '\ rightarrow Y' \ rightarrow Z \ rightarrow 0}$${\ displaystyle \ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} (Z, X ')}$

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}$${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$

${\ displaystyle {\ begin {matrix} &&&& Y '& \ to & Z' & \ to & 0 \\ &&&& \ downarrow && \ downarrow g \\ 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \ end {matrix}} .}$

Because of the universal property of the pull-back, there is an induced monomorphism from X to Y ', so that the following diagram commutes:

${\ displaystyle {\ begin {matrix} 0 & \ to & X & \ to & Y '& \ to & Z' & \ to & 0 \\ && \ downarrow \ operatorname {id} && \ downarrow && \ downarrow g \\ 0 & \ to & X & \ to & Y & \ to & Z & \ to & 0 \ end {matrix}}}$

The top line is also exact and thus defines an element in . ${\ displaystyle \ mathrm {Ext} (Z ', X)}$

If the equivalence class of is mapped onto the equivalence class of , one again obtains a well-defined group homomorphism . ${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$${\ displaystyle 0 \ rightarrow X \ rightarrow Y '\ rightarrow Z' \ rightarrow 0}$${\ displaystyle \ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} (Z ', X)}$

## 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${\ displaystyle \ mathrm {Hom} (-, X)}$

${\ displaystyle \ mathrm {Ext} ^ {n} (Z, X): = R_ {n} \ mathrm {Hom} (-, X) (Z)}$,

that is, one forms the -th legal derivative of and applies the functor thus created . ${\ displaystyle n}$${\ displaystyle \ mathrm {Hom} (-, X)}$${\ displaystyle Z}$

More specifically, this means the following: Let it be and ${\ displaystyle n \ geq 1}$

${\ displaystyle {\ begin {array} {ccccc} \ ldots \ rightarrow & P_ {n} & \ rightarrow & P_ {n-1} & \ rightarrow \ ldots \ rightarrow Z \ rightarrow 0 \\ & \ lambda _ {n} \ downarrow & \ nearrow \ kappa _ {n} \\ & K_ {n} \ end {array}}}$

a projective resolution of having an epimorphism and a monomorphism such that . Next is the induced homomorphism ${\ displaystyle Z}$ ${\ displaystyle \ lambda _ {n}: P_ {n} \ rightarrow K_ {n}}$ ${\ displaystyle \ kappa _ {n}: K_ {n} \ rightarrow P_ {n-1}}$${\ displaystyle (P_ {n} \ rightarrow P_ {n-1}) = \ kappa _ {n} \ circ \ lambda _ {n}}$${\ displaystyle \ kappa _ {n} ^ {*} = \ mathrm {Hom} (\ kappa _ {n}, X)}$

${\ displaystyle \ kappa _ {n} ^ {*}: \ mathrm {Hom} (P_ {n-1}, X) \ rightarrow \ mathrm {Hom} (K_ {n}, X), \, f \ mapsto f \ circ \ kappa _ {n}}$.

Then

${\ displaystyle \ mathrm {Ext} ^ {n} (Z, X) \ cong \ mathrm {coker} (\ kappa _ {n} ^ {*}) = \ mathrm {Hom} (K_ {n}, X) / \ kappa _ {n} ^ {*} (\ mathrm {Hom} (P_ {n-1}, X))}$.

The elements from are therefore certain equivalence classes of elements from . ${\ displaystyle \ mathrm {Ext} ^ {n} (Z, X)}$${\ displaystyle \ mathrm {Hom} (K_ {n}, X)}$

Finally, it should be noted that one can swap the roles of and also one receives ${\ displaystyle X}$${\ displaystyle Z}$

${\ displaystyle \ mathrm {Ext} ^ {n} (Z, X) \ cong R_ {n} \ mathrm {Hom} (Z, -) (X)}$.

## Relationship between Ext and Ext 1

This section aims to explain how the constructs and defined above are related. We construct a picture . ${\ displaystyle \ mathrm {Ext}}$${\ displaystyle \ mathrm {Ext} ^ {1}}$${\ displaystyle \ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} ^ {1} (Z, X)}$

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 ${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$${\ displaystyle \ mathrm {Ext} (Z, X)}$${\ displaystyle 0 \ rightarrow K \ rightarrow P \ rightarrow Z \ rightarrow 0}$${\ displaystyle P}$${\ displaystyle P}$

${\ displaystyle {\ begin {array} {ccccccc} 0 \ rightarrow & K & \ rightarrow & P & \ rightarrow & Z & \ rightarrow 0 \\ & \ downarrow \ psi && \ downarrow \ varphi && \ Vert \\ 0 \ rightarrow & X & \ rightarrow & Y & \ rightarrow & Z & \ rightarrow 0 \ end {array}}}$

to construct. Then there is a homomorphism whose equivalence class defines an element as described above . ${\ displaystyle \ psi \ in \ mathrm {Hom} (K, X)}$${\ displaystyle \ mathrm {Ext} ^ {n} (Z, X)}$${\ displaystyle \ mathrm {Ext} ^ {1} (Z, X)}$

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. ${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$${\ displaystyle \ mathrm {Ext} (Z, X)}$${\ displaystyle \ psi}$${\ displaystyle \ mathrm {Ext} ^ {1} (Z, X)}$${\ displaystyle \ mathrm {Ext} (Z, X) \ rightarrow \ mathrm {Ext} ^ {1} (Z, X)}$

Therefore one can identify with , that is , in this sense it can be defined as the first legal derivative of the -function. ${\ displaystyle \ mathrm {Ext}}$${\ displaystyle \ mathrm {Ext} ^ {1}}$${\ displaystyle \ mathrm {Ext}}$${\ displaystyle \ mathrm {Hom}}$

## Long exact sequence

The Hom functor is left exact, that is, for a short exact sequence

${\ displaystyle 0 \ rightarrow X \ rightarrow Y \ rightarrow Z \ rightarrow 0}$

and another object (module) has an exact sequence ${\ displaystyle A}$

${\ displaystyle 0 \ rightarrow \ mathrm {Hom} (A, X) \ rightarrow \ mathrm {Hom} (A, Y) \ rightarrow \ mathrm {Hom} (A, Z)}$,

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 ${\ displaystyle \ mathrm {Ext} ^ {n}}$${\ displaystyle n = 0}$${\ displaystyle \ mathrm {Ext} ^ {0} = \ mathrm {Hom}}$

${\ displaystyle 0 \ rightarrow \ mathrm {Hom} (A, X) \ rightarrow \ mathrm {Hom} (A, Y) \ rightarrow \ mathrm {Hom} (A, Z)}$
${\ displaystyle \ rightarrow \ mathrm {Ext} ^ {1} (A, X) \ rightarrow \ mathrm {Ext} ^ {1} (A, Y) \ rightarrow \ mathrm {Ext} ^ {1} (A, Z ) \ rightarrow \ mathrm {Ext} ^ {2} (A, X) \ rightarrow \ ldots}$.

A long exact sequence is obtained analogously

${\ displaystyle 0 \ rightarrow \ mathrm {Hom} (Z, A) \ rightarrow \ mathrm {Hom} (Y, A) \ rightarrow \ mathrm {Hom} (X, A)}$
${\ displaystyle \ rightarrow \ mathrm {Ext} ^ {1} (Z, A) \ rightarrow \ mathrm {Ext} ^ {1} (Y, A) \ rightarrow \ mathrm {Ext} ^ {1} (X, A ) \ rightarrow \ mathrm {Ext} ^ {2} (Z, A) \ rightarrow \ ldots}$.

In this sense, the Ext functors close the gap created by the lack of accuracy of the Hom functor.

## Individual evidence

1. Sergei I. Gelfand & Yuri Ivanovich Manin: Homological Algebra , Springer, Berlin, 1999, ISBN 978-3-540-65378-3
2. ^ Charles A. Weibel: An introduction to homological algebra , Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1999, ISBN 978-0-521-55987-4
3. ^ Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , sentence 3.13
4. Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , Theorem 4.5
5. Saunders Mac Lane : Homology , Springer Grundlehren der Mathematischen Wissenschaften Volume 114 (1967), chap. III, Theorem 3.4 and Theorem 9.1