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\begin{document}

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\section*{Ext}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ContravariantExtOnObject}{Contravariant $Ext$ on an ordinary object}\dotfill \pageref*{ContravariantExtOnObject} \linebreak
\noindent\hyperlink{InTermsOfDerivedCategories}{In terms of derived categories}\dotfill \pageref*{InTermsOfDerivedCategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToGroupExtensions}{Relation to extensions}\dotfill \pageref*{RelationToGroupExtensions} \linebreak
\noindent\hyperlink{1extensions_over_single_objects}{1-Extensions over single objects}\dotfill \pageref*{1extensions_over_single_objects} \linebreak
\noindent\hyperlink{higher_extensions_over_general_chain_complexes}{Higher extensions over general chain complexes}\dotfill \pageref*{higher_extensions_over_general_chain_complexes} \linebreak
\noindent\hyperlink{RelationToGroupCohomology}{Relation to group cohomology}\dotfill \pageref*{RelationToGroupCohomology} \linebreak
\noindent\hyperlink{Locatization}{Localization}\dotfill \pageref*{Locatization} \linebreak
\noindent\hyperlink{techniques_for_constructing_}{Techniques for constructing $Ext^n$}\dotfill \pageref*{techniques_for_constructing_} \linebreak
\noindent\hyperlink{yoneda_product}{Yoneda product}\dotfill \pageref*{yoneda_product} \linebreak
\noindent\hyperlink{applications_in_cohomology}{Applications in cohomology}\dotfill \pageref*{applications_in_cohomology} \linebreak
\noindent\hyperlink{universal_coefficient_theorem}{Universal coefficient theorem}\dotfill \pageref*{universal_coefficient_theorem} \linebreak
\noindent\hyperlink{various_notions_of_cohomology_expressed_by_}{Various notions of cohomology expressed by $Ext$}\dotfill \pageref*{various_notions_of_cohomology_expressed_by_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the context of [[homological algebra]] the [[right derived functor]] of the [[hom-functor]] is called the \emph{$Ext$-functor} . It derives its name from the fact that the [[derived hom-functor]] between [[abelian groups]] classifies abelian [[group extensions]] of $A$ by $K$. (This is a special case of the general classification of [[principal ∞-bundles]]/[[∞-group extensions]] by general [[cohomology]]/[[group cohomology]].)

Together with the [[Tor]]-functor it is one of the central objects of interest in homological algebra.

Given an [[abelian category]] $\mathcal{A}$ we may consider the [[hom-functor]] $Hom_{\mathcal{A}} : \mathcal{A}^{op}\times \mathcal{A}\to$[[Ab]] either as a functor in first or in second argument, and compute the corresponding [[right derived functors]].

If they exist, the classical right derived functors of either functor agree and also agree with the [[homology]] of the mixed [[double complex]] obtained by taking simultaneously a [[projective resolution]] of the first contravariant argument and an [[injective resolution]] of the second covariant argument. The last construction is called the \emph{balanced $Ext$.}

Alternatively, one can consider the [[derived category]] $D(\mathcal{A})$ and define

\begin{displaymath}
Ext^p(X,A) \coloneqq Hom_{D(A)}(X,A[p])
\end{displaymath}
or define $Ext^i$-groups as groups of abelian extensions of length $i$, discussed below at \emph{\hyperlink{RelationToGroupExtensions}{Relation to extensions}}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We give the definition following the discussion at \emph{[[derived functors in homological algebra]]}.

\hypertarget{ContravariantExtOnObject}{}\subsubsection*{{Contravariant $Ext$ on an ordinary object}}\label{ContravariantExtOnObject}

Let $\mathcal{A}$ be an [[abelian category]] with [[projective object|enough projectives]]. And let $A \in \mathcal{A}$ be any object. Consider the [[contravariant functor|contravariant]] [[hom-functor]]

\begin{displaymath}
Hom_{\mathcal{A}}(-, A) : \mathcal{A}^{op} \to Ab
  \,.
\end{displaymath}
\begin{remark}
\label{}\hypertarget{}{}
This is a [[left exact functor]].

Therefore to derive it by [[resolutions]] we need to consider [[injective resolutions]] in the [[opposite category]] $\mathcal{A}^{op}$. But these are [[projective resolutions]] in $\mathcal{A}$ itself.

\end{remark}
\begin{defn}
\label{OfSingleObjByProjResolution}\hypertarget{OfSingleObjByProjResolution}{}
For $X \in \mathcal{A}$ any object and $((Q X) \to X) \in Ch_{\bullet \geq 0}(\mathcal{A})$ a [[projective resolution]], and for $n \in \mathbb{N}$, the \textbf{$n$th $Ext$-group} of $X$ with [[coefficients]] in $A$ is the degree-$n$ [[cochain cohomology]]

\begin{displaymath}
Ext^n(X,A) 
   \coloneqq 
  H^n ( Hom_{\mathcal{A}}((Q X)_\bullet, A))
\end{displaymath}
of the [[cochain complex]] $Hom((Q X)_\bullet, A)$.

\end{defn}
The following proposition expands a bit on the meaning of this definition. Write

\begin{displaymath}
[-,-] : Ch_{\bullet}(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A})
  \to 
  Ch_\bullet(Ab)
\end{displaymath}
for the [[internal hom of chain complexes|enriched hom of chain complexes]].

\begin{prop}
\label{}\hypertarget{}{}
The $n$th Ext-group is canonically identified with the 0-th [[homology]] of this enriched hom from the resolution $Q X$ of $X$ to the $n$-fold [[delooping]]/[[suspension]] chain complex of $A$ $\mathbf{B}^n A = A[n]$ (concentrated on $A$ in degree $n$):

\begin{displaymath}
Ext^n(X,A)
  \simeq
  H_0 [(Q X), A[n] ]
  \,;
\end{displaymath}
or equivalently, if we think of degree [[chain homology]] as the 0th [[homotopy group]] (under [[Dold-Kan correspondence]]) and write the $n$-fold [[suspension]]/[[delooping]] of $A$ as $\mathbf{B}^n A$:

\begin{displaymath}
Ext^n(X,A)
  \simeq
  \pi_0 [(Q X), \mathbf{B}^n A ]
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is a special case of the general discussion at [[cochain cohomology]].

By the discussion at \emph{[[internal hom of chain complexes]]}, the 0-[[cycles]] of $[(Q X), \mathbf{B}^n A ]$ are [[chain maps]] of the form

\begin{displaymath}
\itexarray{
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow
    \\
    (Q X)_{n} &\stackrel{f_{n}}{\to}&  A
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow
    \\
    (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_1}} && \downarrow
    \\
    (Q X)_1 &\stackrel{f_1}{\to}& 0    
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_0}} && \downarrow
    \\
    (Q X)_0 &\stackrel{f_0}{\to}& 0
  }
  \,.
\end{displaymath}
By the definition of chain maps this are precisely those morphisms $f_n : (Q X)_n \to A$ such that

\begin{displaymath}
d^n f_n \coloneqq f_n \circ \partial^{Q X}_n = 0
\end{displaymath}
which exhibits $f_n$ as a degree-$n$ [[cochain]] in the [[cochain complex]] $Hom((Q X)_\bullet, A)$.

Similarly, the $0$-[[boundaries]] in $[(Q X), \mathbf{B}^n A]$ come from [[chain homotopies]] $\lambda : 0 \Rightarrow f$:

\begin{displaymath}
\itexarray{
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} 
     &\nearrow_{\mathrlap{\lambda_{n+1}}}& \downarrow
    \\
    (Q X)_{n} &\stackrel{f_{n}}{\to}&  A
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} 
    &\nearrow_{\mathrlap{\lambda_{n}}}& \downarrow
    \\
    (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_1}} 
    &\nearrow_{\mathrlap{\lambda_{2}}}& \downarrow
    \\
    (Q X)_1 &\stackrel{f_1}{\to}& 0    
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_0}} 
    &\nearrow_{\mathrlap{\lambda_{1}}}& \downarrow
    \\
    (Q X)_0 &\stackrel{f_0}{\to}& 0
  }
  \,.
\end{displaymath}
in that

\begin{displaymath}
f_n = \lambda_n \circ \partial^{Q X}_{n-1}
  \,.
\end{displaymath}
This are precisely the degree-$n$ [[coboundaries]] in $Hom((Q X)_\bullet, A)$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This perspective on the $Ext^n$-group as being the [[homotopy classes]] of maps out of (a resolution of) $X$ to $\mathbf{B}^n A$ is made more manifest in the discussion \hyperlink{InTermsOfDerivedCategories}{in terms of derived categories} below. It connects $Ext$-groups and their \hyperlink{RelationToGroupExtensions}{relation to extensions} to the general context of \emph{[[cohomology]]} and \emph{[[∞-group extensions]]}. See at \emph{[[abelian sheaf cohomology]]} for more on this.

\end{remark}
\hypertarget{InTermsOfDerivedCategories}{}\subsubsection*{{In terms of derived categories}}\label{InTermsOfDerivedCategories}

(\ldots{})

(\hyperlink{KashiwaraShapira}{Kashiwara-Shapira})

(\ldots{})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToGroupExtensions}{}\subsubsection*{{Relation to extensions}}\label{RelationToGroupExtensions}

We discuss how the group $Ext^n(X,A)$ is identified with the group of [[extensions]] of $X$ by $\mathbf{B}^{n-1} A = A[n-1]$. In particular for $n = 1$ and $\mathcal{A} =$ [[Ab]] this means that $Ext^1(X,A)$ classified ordinary [[group extensions]] of $X$ by $A$.

This is the relation that the name ``$Ext$'' derives from. At \emph{[[infinity-group extension]]} is discussed how this relation is a special case of the more general relation that identifies [[derived hom-spaces]] $\mathbf{H}(X,\mathbf{B}^{n+1} A)$ with $\mathbf{B}^n A$-[[principal ∞-bundles]] over $X$.

\hypertarget{1extensions_over_single_objects}{}\paragraph*{{1-Extensions over single objects}}\label{1extensions_over_single_objects}

\begin{defn}
\label{}\hypertarget{}{}
For $X,A \in \mathcal{A}$ two objects, an \textbf{[[extension]]} of $X$ by $A$ is a [[short exact sequence]]

\begin{displaymath}
0 \to A \to P \to X \to 0
  \,.
\end{displaymath}
An [[homomorphism]] of two such extensions $P_1$ and $P_2$ is a [[morphism]] $P_1 \to P_2$ in $\mathcal{A}$ fitting into a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
     && P_1
     \\
     & \nearrow & & \searrow
     \\
     A && \downarrow && X
     \\
     & \searrow & & \nearrow
     \\
     && P_2
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
All these homomophisms are necessarily [[isomorphisms]], by the [[short five lemma]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
Write $Ext(X,A)$ for the set of [[isomorphism classes]] of such extensions.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Under [[Baer sum]] $Ext(X,A)$ becomes an [[abelian group]].

\end{prop}
\begin{defn}
\label{ExtractCocycleFromExtension}\hypertarget{ExtractCocycleFromExtension}{}
Define a morphism

\begin{displaymath}
ExtractCocycle : Extensions(X,A) \to Ext^1(X,A)
\end{displaymath}
by the following construction:

choose a [[projective presentation]] $N \hookrightarrow Q \to X$ of $X$. Then for $A \to P \to X$ an extension consider the diagram

\begin{displaymath}
\itexarray{
    N &\to& Q &\to& X
    \\
    \downarrow^{\mathrlap{\sigma|_N}} && \downarrow^{\mathrlap{\sigma}} && \downarrow^{\mathrlap{=}}
    \\
    A &\to& P &\to& X
  }
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\sigma$ is any choice of lifts of $Q \to A$ through $P \to A$, which exists by definition since $P$ is a [[projective object]],


\item $\sigma|_N$ is the induces morphism on the fibers, which exists by the exactness of the two sequences.



\end{itemize}
By prop. \ref{Ext1FromProjectivePresentation} $\sigma|_N$ represents an element in $[\sigma|_N] \in Ext^1(X,A)$. Let this be the image of the map to be defined:

\begin{displaymath}
ExtractCocycle : (A \to P \to X) \mapsto [\sigma|_N]
  \,.
\end{displaymath}
This definition is independent of the choice of $P$ and $\sigma$ involved.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The map from def. \ref{ExtractCocycleFromExtension} is a [[natural isomorphism]] of abelian groups

\begin{displaymath}
Ext(X,A) \stackrel{\simeq}{\to} Ext^1(X,A) 
  \,.
\end{displaymath}
\end{prop}
\hypertarget{higher_extensions_over_general_chain_complexes}{}\paragraph*{{Higher extensions over general chain complexes}}\label{higher_extensions_over_general_chain_complexes}

(\ldots{})

Given $[g] \in \mathbb{R}Hom(X, A[n])$.

Let $Q \to X$ be a [[projective resolution]]. Let $g : Q \to A[n]$ be a representative of $[g]$.\newline Consider the [[pullback]]

\begin{displaymath}
\itexarray{
    P &\to& cone(0 \to A[n])
    \\
    \downarrow && \downarrow
    \\
    Q &\stackrel{g}{\to}& A[n]
    \\
    \downarrow
    \\
    X
  }
\end{displaymath}
(\ldots{})

\hypertarget{RelationToGroupCohomology}{}\subsubsection*{{Relation to group cohomology}}\label{RelationToGroupCohomology}

For $G$ a [[discrete group]] with $\mathbb{Z}[G]$ its [[group ring]], over the [[integers]], and for $N$ a linear $G$-[[representation]], hence a $\mathbb{Z}[G]$-[[module]], the [[group cohomology]] of $G$ with [[coefficients]] in $N$ is

\begin{displaymath}
Ext^\bullet_{\mathbb{Z}[G]Mod}(\mathbb{Z}, N)
  \,.
\end{displaymath}
(\ldots{})

\hypertarget{Locatization}{}\subsubsection*{{Localization}}\label{Locatization}

(\ldots{})

\hypertarget{techniques_for_constructing_}{}\subsubsection*{{Techniques for constructing $Ext^n$}}\label{techniques_for_constructing_}

We discuss some facts helpful for the construction of $Ext^n$-groups in certain situations.

\begin{prop}
\label{}\hypertarget{}{}
If $X \in \mathcal{A}$ is a [[projective object]], then

\begin{displaymath}
Ext^n(X, -)  = 0
\end{displaymath}
is the [[zero object|zero]]-[[functor]] for all $n \geq 1$.

\end{prop}
\begin{proof}
The covariant [[hom-functor]] $Hom(X,-)$ is generally a [[left exact functor]]. By the construction of $Ext^n$ via [[projective resolutions]], def. \ref{OfSingleObjByProjResolution}, it is sufficient to show that it is also a [[right exact functor]] if $P$ is projective. In fact, this is one of the equivalent characterizations of \emph{[[projective objects]]} (ee the section \href{projective+object#EquivalentCharacterizationInAbelianCats}{projective object -- in abelian categories -- equivalent characterizations} for details).

\end{proof}
\begin{prop}
\label{Ext1FromProjectivePresentation}\hypertarget{Ext1FromProjectivePresentation}{}
For $X, A \in \mathcal{A}$ two objects, and

\begin{displaymath}
0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0
\end{displaymath}
a [[short exact sequence]] with $P$ a [[projective object]], hence exhibiting a [[projective presentation]] $X \simeq coker(N \hookrightarrow P)$ of $X$, there is an [[exact sequence]]

\begin{displaymath}
0 
   \to 
  Hom(X,A)
    \stackrel{Hom(p,A)}{\to}
  Hom(P, A)
    \stackrel{Hom(i,A)}{\to}
  Hom(N,A)
    \to
  Ext^1(X,A)
    \to 
  0
\end{displaymath}
exhibiting $Ext^1(X,A)$ as the [[cokernel]] of $Hom(i,A)$.

\end{prop}
\hypertarget{yoneda_product}{}\subsubsection*{{Yoneda product}}\label{yoneda_product}

The [[Yoneda product]] is a pairing

\begin{displaymath}
Ext^n(A,M) \otimes Ext^m(A,N) \to Ext^{n+m}(A,M\otimes_A N).
\end{displaymath}
(\ldots{})

\hypertarget{applications_in_cohomology}{}\subsection*{{Applications in cohomology}}\label{applications_in_cohomology}

A [[derived hom-functor]] such as the $Ext$ on chain complexes compute general notions of \emph{[[cohomology]]} (see the discussion there). Here we list some specific incarnations of the $Ext$-construction in the context of cohomology.

\hypertarget{universal_coefficient_theorem}{}\subsubsection*{{Universal coefficient theorem}}\label{universal_coefficient_theorem}

The \emph{[[universal coefficient theorem]]} identifies, under suitable conditions, [[cohomology]] to the [[duality|dual]] of [[homology]] up to $Ext^1$-groups.

\hypertarget{various_notions_of_cohomology_expressed_by_}{}\subsubsection*{{Various notions of cohomology expressed by $Ext$}}\label{various_notions_of_cohomology_expressed_by_}

Various notions of [[cohomology groups]] in the context of [[algebra]] can be expressed as $Ext$-groups, for instance:

\begin{itemize}%
\item For $G$ a [[discrete group]] with $\mathbb{Z}[G]$ its [[group ring]], over the [[integers]], and for $N$ a linear $G$-[[representation]], hence a $\mathbb{Z}[G]$-[[module]], the [[group cohomology]] of $G$ with [[coefficients]] in $N$ is

\begin{displaymath}
Ext^\bullet_{\mathbb{Z}[G]Mod}(\mathbb{Z}, N)
  \,.
\end{displaymath}

\item For $A$ an [[associative algebra]] over some [[field]] $k$ and $N$ an $A$-[[bimodule]], hence an $A \otimes A^{op}$-[[module]],

\begin{displaymath}
Ext^\bullet_{(A \otimes A^{op})Mod}(A, N)
\end{displaymath}
is the [[Hochschild cohomology]] of $A$ with [[coefficients]] in $N$.


\item For $\mathfrak{g}$ a [[Lie algebra]] with [[universal enveloping algebra]] $\mathcal{U}(\mathfrak{g})$ and $N$ a Lie algebra module, hence an $\mathcal{U}(\mathfrak{g})$-module, the [[Lie algebra cohomology]] of $\mathfrak{g}$ with [[coefficients]] in $N$ is

\begin{displaymath}
Ext^\bullet_{\mathcal{U}(\mathfrak{g}) Mod}(\mathcal{U}(\mathfrak{g}), N)
  \,.
\end{displaymath}


\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include homotopy-homology-cohomology]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Standard texbook accounts include (see also most references at \emph{[[homological algebra]]})

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}, Cambridge Studies in Adv. Math. 38, CUP 1994

\end{itemize}
\begin{itemize}%
\item [[Henri Cartan]], [[Samuel Eilenberg]], \emph{Homological algebra}, Princeton Univ. Press 1956.


\item S. I . Gelfand, [[Yuri Manin]], \emph{Methods of homological algebra}



\end{itemize}
A systematic discussion from the point of view of [[derived categories]] is in

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{[[Categories and Sheaves]]}, Springer (2000)

\end{itemize}
Lecture notes include

\begin{itemize}%
\item Kiyoshi Igusa, \emph{25 The Ext Functor} (\href{http://people.brandeis.edu/~igusa/Math101b/Ext.pdf}{pdf})

\end{itemize}
section 4 of

\begin{itemize}%
\item [[Peter May]], \emph{Notes on Tor and Ext} (\href{http://www.math.uchicago.edu/~may/MISC/TorExt.pdf}{pdf})

\end{itemize}
as well as

\begin{itemize}%
\item [[Patrick Morandi]], \emph{Ext Groups and Ext Functors}, (\href{http://sierra.nmsu.edu/morandi/oldwebpages/math683fall2002/Ext.pdf}{pdf})

(warning: the last section on resolutions for cocycles for group (abelian) exensions is not correct)



\end{itemize}
Original articles include

\begin{itemize}%
\item [[Saunders MacLane]], \emph{Group Extensions by primary abelian groups}, Transactions of the American Mathematical Society Vol. 95, No. 1 (Apr., 1960), pp. 1-16 (\href{http://www.jstor.org/stable/1993327}{JSTOR})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Ext_functor}{Ext functor}}

\end{itemize}
[[!redirects Ext group]] [[!redirects Ext-group]] [[!redirects Ext-groups]] [[!redirects Ext groups]] [[!redirects Ext functor]] [[!redirects Ext functors]] [[!redirects Ext-functor]] [[!redirects Ext-functors]]



\end{document}