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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{projective resolution} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra} [[!include homological algebra - contents]] \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{projective_and_injective_resolutions}{}\section*{{Projective and injective resolutions}}\label{projective_and_injective_resolutions} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{ResolutionOfObject}{Resolution of an object}\dotfill \pageref*{ResolutionOfObject} \linebreak \noindent\hyperlink{FResolutions}{$F$-Resolutions of an object}\dotfill \pageref*{FResolutions} \linebreak \noindent\hyperlink{ResolutionOfAChainComplex}{Resolution of a chain complex}\dotfill \pageref*{ResolutionOfAChainComplex} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{ExistenceAndConstruction}{Existence and construction of resolutions for objects}\dotfill \pageref*{ExistenceAndConstruction} \linebreak \noindent\hyperlink{ExistenceAndConstructionOfResolutionsOfComplexes}{Existence and construction of resolutions of complexes}\dotfill \pageref*{ExistenceAndConstructionOfResolutionsOfComplexes} \linebreak \noindent\hyperlink{FunctorialResolutions}{Functorial resolutions and derived functors}\dotfill \pageref*{FunctorialResolutions} \linebreak \noindent\hyperlink{DerivedHomFunctor}{Derived Hom-functor/$Ext$-functor and extensions}\dotfill \pageref*{DerivedHomFunctor} \linebreak \noindent\hyperlink{relation_to_syzygies}{Relation to syzygies}\dotfill \pageref*{relation_to_syzygies} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{Lenght1ResolutionsOfAbelianGroups}{Length-1 resolutions}\dotfill \pageref*{Lenght1ResolutionsOfAbelianGroups} \linebreak \noindent\hyperlink{ProjectiveResolutionsForGroupCocycles}{Projective resolutions adapted to abelian group cocycles}\dotfill \pageref*{ProjectiveResolutionsForGroupCocycles} \linebreak \noindent\hyperlink{ProjectiveResolutionsForGroupCohomology}{Projective resolutions adapted to general group cohomology}\dotfill \pageref*{ProjectiveResolutionsForGroupCohomology} \linebreak \noindent\hyperlink{CohomologyOfCyclicGroups}{Cohomology of cyclic groups}\dotfill \pageref*{CohomologyOfCyclicGroups} \linebreak \noindent\hyperlink{RelatedConcepts}{Related concepts}\dotfill \pageref*{RelatedConcepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In the context of [[homological algebra]] a \emph{projective/injective} [[resolution]] of an [[object]] or [[chain complex]] in an [[abelian category]] is a [[resolution]] by a [[quasi-isomorphism|quasi-isomorphic]] chain complex that consists of [[projective objects]] or [[injective objects]], respectively. Under suitable conditions these are precisely the [[cofibrant resolution]] or [[fibrant resolution]] with respect to a standard [[model structure on chain complexes]]. For instance for non-negatively graded chain complexes of abelian groups there is a model structure with [[weak equivalences]] are the quasi-isomorphisms and the [[fibrations]] are the positive-degreewise surjections. Here every object is a [[fibrant object]] and hence no [[fibrant resolution]] is necessary; while the [[cofibrant resolutions]] are precisely the projective resolutions. Dually, for non-negatively graded chain complexes of abelian groups there is a model structure with [[weak equivalences]] are the quasi-isomorphisms and the [[cofibrations]] the positive-degreewise injections. Here every object is a [[cofibrant object]] and hence no [[cofibrant resolution]] is necessary; while the [[fibrant resolutions]] are precisely the projective resolutions. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} We first discuss, as is traditional, projective/injective resolutions of single objects, and then the general cases of projective/injective resolutions of chain complexes. This subsumes the previous case by regarding an object as a chain complex concentrated in degree 0. \hypertarget{ResolutionOfObject}{}\subsubsection*{{Resolution of an object}}\label{ResolutionOfObject} Let $\mathcal{A}$ be an [[abelian category]]. \begin{defn} \label{InjectiveResolution}\hypertarget{InjectiveResolution}{} For $X \in \mathcal{A}$ an [[object]], an \textbf{injective resolution} of $X$ is a [[cochain complex]] $J^\bullet \in Ch^\bullet(\mathcal{A})$ (in non-negative degree) equipped with a [[quasi-isomorphism]] \begin{displaymath} i : X \stackrel{\sim}{\to} J^\bullet \end{displaymath} such that $J^n \in \mathcal{A}$ is an [[injective object]] for all $n \in \mathbb{N}$. \end{defn} \begin{remark} \label{InjectiveResolutionInComponents}\hypertarget{InjectiveResolutionInComponents}{} In components the quasi-isomorphism of def. \ref{InjectiveResolution} is a [[chain map]] of the form \begin{displaymath} \itexarray{ X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots \\ \downarrow^{\mathrlap{i^0}} && \downarrow && && \downarrow \\ J^0 &\stackrel{d^0}{\to}& J^1 &\stackrel{d^1}{\to}& \cdots &\to& J^n &\stackrel{d^n}{\to}&\cdots } \,. \end{displaymath} Since the top complex is concentrated in degree 0, this being a [[quasi-isomorphism]] happens to be equivalent to the sequence \begin{displaymath} 0 \to X \stackrel{i^0}{\to} J^0 \stackrel{d^0}{\to} J^1 \stackrel{d^1}{\to} J^2 \stackrel{d^2}{\to} \cdots \end{displaymath} being an [[exact sequence]]. In this form one often finds the definition of injective resolution in the literature. \end{remark} \begin{defn} \label{ProjectiveResolution}\hypertarget{ProjectiveResolution}{} For $X \in \mathcal{A}$ an [[object]], a \textbf{projective resolution} of $X$ is a [[chain complex]] $J_\bullet \in Ch_\bullet(\mathcal{A})$ (in non-negative degree) equipped with a [[quasi-isomorphism]] \begin{displaymath} p : J_\bullet \stackrel{\sim}{\to} X \end{displaymath} such that $J_n \in \mathcal{A}$ is a [[projective object]] for all $n \in \mathbb{N}$. \end{defn} \begin{remark} \label{ProjectiveResolutionInComponents}\hypertarget{ProjectiveResolutionInComponents}{} In components the quasi-isomorphism of def. \ref{ProjectiveResolution} is a [[chain map]] of the form \begin{displaymath} \itexarray{ \cdots &\stackrel{\partial_n}{\to}& J_n &\stackrel{\partial_{n-1}}{\to}& \cdots &\to& J_1 &\stackrel{\partial_0}{\to}& J_0 \\ && \downarrow && && \downarrow && \downarrow^{\mathrlap{p_0}} \\ \cdots &\to& 0 &\to& \cdots &\to& 0 &\to& X } \,. \end{displaymath} Since the bottom complex is concentrated in degree 0, this being a [[quasi-isomorphism]] happens to be equivalent to the sequence \begin{displaymath} \cdots J_2 \stackrel{\partial_1}{\to} J_1 \stackrel{\partial_0}{\to} J_0 \stackrel{p_0}{\to} X \to 0 \end{displaymath} being an [[exact sequence]]. In this form one often finds the definition of projective resolution in the literature. \end{remark} \hypertarget{FResolutions}{}\subsubsection*{{$F$-Resolutions of an object}}\label{FResolutions} Projective and injective resolutions are typically used for computing the [[derived functor]] of some [[additive functor]] $F \colon \mathcal{A} \to \mathcal{B}$; see at \emph{[[derived functor in homological algebra]]}. While projective resolutions in $\mathcal{A}$ are \emph{sufficient} for computing \emph{every} [[left derived functor]] on $Ch_\bullet(\mathcal{A})$ and injective resolutions are sufficient for computing \emph{every} [[right derived functor]] on $Ch^\bullet(\mathcal{A})$, if one is interested just in a single functor $F$ then such resolutions may be more than \emph{necessary}. A weaker kind of resolution which is still sufficient is then often more convenient for applications. These \emph{$F$-projective resolutions} and \emph{$F$-injective resolutions}, respectively, we discuss here. A special case of both are \emph{$F$-[[acyclic resolutions]]}. $\,$ Let $\mathcal{A}, \mathcal{B}$ be [[abelian categories]] and let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[additive functor]]. \begin{defn} \label{FInjectives}\hypertarget{FInjectives}{} Assume that $F$ is [[left exact functor|left exact]]. An [[additive category|additive]] [[full subcategory]] $\mathcal{I} \subset \mathcal{A}$ is called \textbf{$F$-injective} (or: consisting of $F$-injective objects) if \begin{enumerate}% \item for every object $A \in \mathcal{A}$ there is a [[monomorphism]] $A \to \tilde A$ into an object $\tilde A \in \mathcal{I} \subset \mathcal{A}$; \item for every [[short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $A, B \in \mathcal{I} \subset \mathcal{A}$ also $C \in \mathcal{I} \subset \mathcal{A}$; \item for every [[short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $A\in \mathcal{I} \subset \mathcal{A}$ also $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence in $\mathcal{B}$. \end{enumerate} \end{defn} And dually: \begin{defn} \label{FProjectives}\hypertarget{FProjectives}{} Assume that $F$ is [[right exact functor|right exact]]. An [[additive category|additive]] [[full subcategory]] $\mathcal{P} \subset \mathcal{A}$ is called \textbf{$F$-projective} (or: consisting of $F$-projective objects) if \begin{enumerate}% \item for every object $A \in \mathcal{A}$ there is an [[epimorphism]] $\tilde A \to A$ from an object $\tilde A \in \mathcal{P} \subset \mathcal{A}$; \item for every [[short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $B, C \in \mathcal{P} \subset \mathcal{A}$ also $A \in \mathcal{P} \subset \mathcal{A}$; \item for every [[short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $C\in \mathcal{P} \subset \mathcal{A}$ also $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence in $\mathcal{B}$. \end{enumerate} \end{defn} For instance (\hyperlink{Schapira}{Schapira, def. 4.6.5}). With the $\mathcal{I},\mathcal{P}\subset \mathcal{A}$ as above, we say: \begin{defn} \label{FProjectivesResolution}\hypertarget{FProjectivesResolution}{} For $A \in \mathcal{A}$, \begin{itemize}% \item an \textbf{$F$-injective resolution} of $A$ is a [[cochain complex]] $I^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ and a [[quasi-isomorphism]] \begin{displaymath} A \stackrel{\simeq_{qi}}{\to} I^\bullet \end{displaymath} \item an \textbf{$F$-projective resolution} of $A$ is a [[chain complex]] $Q_\bullet \in Ch_\bullet(\mathcal{P}) \subset Ch^\bullet(\mathcal{A})$ and a [[quasi-isomorphism]] \begin{displaymath} Q_\bullet \stackrel{\simeq_{qi}}{\to} A \,. \end{displaymath} \end{itemize} \end{defn} Let now $\mathcal{A}$ have enough projectives / enough injectives, respectively. \begin{example} \label{FAcyclicObjectsAreFProjectiveObjects}\hypertarget{FAcyclicObjectsAreFProjectiveObjects}{} For $F \colon \mathcal{A} \to \mathcal{B}$ an [[additive functor]], let $Ac \subset \mathcal{A}$ be the [[full subcategory]] on the $F$-[[acyclic objects]]. Then \begin{itemize}% \item if $F$ is [[left exact functor|left exact]], then $\mathcal{I} \coloneqq Ac$ is a subcategory of $F$-injective objects; \item if $F$ is [[right exact functor|right exact]], then $\mathcal{P} \coloneqq Ac$ is a subcategory of $F$-projective objects. \end{itemize} \end{example} \begin{proof} Consider the case that $F$ is right exact. The other case works dually. The first condition of def. \ref{FInjectives} is satisfied because every [[injective object]] is an $F$-[[acyclic object]] and by assumption there are enough of these. For the second and third condition of def. \ref{FInjectives} use that there is the [[long exact sequence]] of [[derived functors]] prop. \ref{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence} \begin{displaymath} 0 \to A \to B \to C \to R^1 F(A) \to R^1 F(B) \to R^1 F(C) \to R^2 F(A) \to R^2 F(B) \to R^2 F(C) \to \cdot \,. \end{displaymath} For the second condition, by assumption on $A$ and $B$ and definition of $F$-[[acyclic object]] we have $R^n F(A) \simeq 0$ and $R^n F(B) \simeq 0$ for $n \geq 1$ and hence short exact sequences \begin{displaymath} 0 \to 0 \to R^n F(C) \to 0 \end{displaymath} which imply that $R^n F(C)\simeq 0$ for all $n \geq 1$, hence that $C$ is acyclic. Similarly, the third condition is equivalent to $R^1 F(A) \simeq 0$. \end{proof} \begin{example} \label{FAcyclicResolution}\hypertarget{FAcyclicResolution}{} The $F$-projective/injective resolutions by [[acyclic objects]] as in example \ref{FAcyclicObjectsAreFProjectiveObjects} are called \textbf{$F$-acyclic resolutions}. \end{example} \hypertarget{ResolutionOfAChainComplex}{}\subsubsection*{{Resolution of a chain complex}}\label{ResolutionOfAChainComplex} The above definition \ref{ProjectiveResolution} of a projective resolution of an object has an immediate generalization to resolutions of chain complexes. \begin{defn} \label{ProjectiveResolutionOfChainComplex}\hypertarget{ProjectiveResolutionOfChainComplex}{} For $C_\bullet \in Ch_\bullet(\mathcal{A})$ a [[chain complex]], a \textbf{projective resolution} of $C$ is an [[exact sequence]] of chain complexes \begin{displaymath} \cdots \to Q_{\bullet,2} \to Q_{\bullet,1} \to Q_{\bullet,0} \to C_\bullet \to 0 \end{displaymath} such that for each $n \in \mathbb{N}$ the component $Q_{n,\bullet} \to C_n$ is a projective resolution of the object $C_n$, according to def. \ref{ProjectiveResolution}. \end{defn} \begin{remark} \label{}\hypertarget{}{} A projective resolution as above may in particular also be regarded as a [[double complex]] $Q_{\bullet, \bullet}$ equipped with a morphism of double complex to $C_\bullet$ regarded as a vertically constant double complex. In other words, a projective resolution of a chain complex in an [[abelian category]] $\mathcal{A}$ is a projective resolution of an object in a [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$. \end{remark} For purposes of computations one is often interested in the following stronger notion. For any chain complex $C_\bullet$, write $Z_\bullet$, $B_\bullet$, and $H_\bullet$ for the graded objects of [[cycles]], [[boundaries]] and [[homology groups]], respectively, regarded as chain complexes with vanishing [[differentials]]. \begin{defn} \label{FullyProjectiveResolutionOfChainComplex}\hypertarget{FullyProjectiveResolutionOfChainComplex}{} A projective resolution $Q_{\bullet,\bullet} \to C_\bullet$ of a chain complex $C_\bullet$, def. \ref{ProjectiveResolutionOfChainComplex}, is called \textbf{fully projective} (or \textbf{proper}) if furthermore for all $n \in \mathbb{N}$ the induced sequence of (horizontal) [[cycles]] \begin{displaymath} \cdots \to Z_{\bullet,2} \to Z_{\bullet,1} \to Z_{\bullet,0} \to Z(C)_\bullet \to 0 \end{displaymath} and (horizontal) [[boundaries]] \begin{displaymath} \cdots \to B_{\bullet,2} \to B_{\bullet,1} \to B_{\bullet,0} \to B(C)_\bullet \to 0 \end{displaymath} and (horizontal) [[homology groups]] \begin{displaymath} \cdots \to H_{\bullet,2} \to H_{\bullet,1} \to H_{\bullet,0} \to H(C)_\bullet \to 0 \end{displaymath} are each projective resolutions, def. \ref{ProjectiveResolution}, themselves. \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{ExistenceAndConstruction}{}\subsubsection*{{Existence and construction of resolutions for objects}}\label{ExistenceAndConstruction} We first discuss the existence of injective/projective resolutions, and then the [[functor|functoriality]] of their constructions. \begin{prop} \label{ExistenceOfInjectiveResolutions}\hypertarget{ExistenceOfInjectiveResolutions}{} Let $\mathcal{A}$ be an [[abelian category]] with [[injective object|enough injectives]] (such as $R$[[Mod]] for some [[ring]] $R$). Then every object $X \in \mathcal{A}$ has an [[injective resolution]], def. \ref{InjectiveResolution}. \end{prop} \begin{proof} Let $X \in \mathcal{A}$ be the given object. By remark \ref{InjectiveResolutionInComponents} we need to construct an [[exact sequence]] of the form \begin{displaymath} 0 \to X \to J^0 \stackrel{d^0}{\to} J^1 \stackrel{d^1}{\to} J^2 \stackrel{d^2}{\to} \cdots \to J^n \to \cdots \end{displaymath} such that all the $J^\cdot$ are [[injective objects]]. This we now construct by [[induction]] on the degree $n \in \mathbb{N}$. In the first step, by the assumption of enough enjectives we find an injective object $J^0$ and a [[monomorphism]] \begin{displaymath} X \hookrightarrow J^0 \end{displaymath} hence an [[exact sequence]] \begin{displaymath} 0 \to X \to J^0 \,. \end{displaymath} Assume then by induction hypothesis that for $n \in \mathbb{N}$ an [[exact sequence]] \begin{displaymath} X \to J^0 \stackrel{d^0}{\to} \cdots \to J^{n-1} \stackrel{d^{n-1}}{\to} J^n \end{displaymath} has been constructed, where all the $J^\cdot$ are injective objects. Forming the [[cokernel]] of $d^{n-1}$ yields the [[short exact sequence]] \begin{displaymath} 0 \to J^{n-1} \stackrel{d^{n-1}}{\to} J^n \stackrel{p}{\to} J^n/J^{n-1} \to 0 \,. \end{displaymath} By the assumption that there are enough injectives in $\mathcal{A}$ we may now again find a monomorphism $J^n/J^{n-1} \stackrel{i}{\hookrightarrow} J^{n+1}$ into an injective object $J^{n+1}$. This being a monomorphism means that \begin{displaymath} J^{n-1} \stackrel{d^{n-1}}{\to} J^n \stackrel{d^n \coloneqq i \circ p}{\longrightarrow} J^{n+1} \end{displaymath} is [[exact sequence|exact]] in the middle term. Therefore we now have an [[exact sequence]] \begin{displaymath} 0 \to X \to J^0 \to \cdots \to J^{n-1} \stackrel{d^{n-1}}{\to} J^n \stackrel{d^{n}}{\to} J^{n+1} \end{displaymath} which completes the [[induction]] step. \end{proof} The following proposition is [[duality|formally dual]] to prop. \ref{ExistenceOfInjectiveResolutions}. \begin{prop} \label{ExistenceOfInjectiveResolutions}\hypertarget{ExistenceOfInjectiveResolutions}{} Let $\mathcal{A}$ be an [[abelian category]] with [[projective object|enough projectives]] (such as $R$[[Mod]] for some [[ring]] $R$). Then every object $X \in \mathcal{A}$ has a [[projective resolution]], def. \ref{ProjectiveResolution}. \end{prop} \begin{proof} Let $X \in \mathcal{A}$ be the given object. By remark \ref{ProjectiveResolutionInComponents} we need to construct an [[exact sequence]] of the form \begin{displaymath} \cdots \stackrel{\partial_2}{\to} J_2 \stackrel{\partial_1}{\to} J_1 \stackrel{\partial_0}{\to} J_0 \to X \to 0 \end{displaymath} such that all the $J_\cdot$ are [[projective objects]]. This we we now construct by [[induction]] on the degree $n \in \mathbb{N}$. In the first step, by the assumption of enough projectives we find a projective object $J_0$ and an [[epimorphism]] \begin{displaymath} J_0 \to X \end{displaymath} hence an [[exact sequence]] \begin{displaymath} J_0 \to X \to 0 \,. \end{displaymath} Assume then by induction hypothesis that for $n \in \mathbb{N}$ an [[exact sequence]] \begin{displaymath} J_n \stackrel{\partial_{n-1}}{\to} J_{n-1} \to \cdots \stackrel{\partial_0}{\to} J_0 \to X \to 0 \end{displaymath} has been constructed, where all the $J_\cdot$ are projective objects. Forming the [[kernel]] of $\partial_{n-1}$ yields the [[short exact sequence]] \begin{displaymath} 0 \to ker(\partial_{n-1}) \stackrel{i}{\to} J_n \stackrel{\partial_{n-1}}{\to} J_{n-1} \to 0 \,. \end{displaymath} By the assumption that there are enough projectives in $\mathcal{A}$ we may now again find an epimorphism $p : J_{n+1} \to ker(\partial_{n-1})$ out of a projective object $J_{n+1}$. This being an epimorphism means that \begin{displaymath} J_{n+1} \stackrel{\partial_{n} \coloneqq i\circ p}{\to} J_n \stackrel{\partial_{n-1}}{\to} \end{displaymath} is [[exact sequence|exact]] in the middle term. Therefore we now have an [[exact sequence]] \begin{displaymath} J_{n+1} \stackrel{\partial_n}{\to} J_n \stackrel{\partial_{n-1}}{\to} \cdots \stackrel{\partial_0}{\to} J_0 \to X \to 0 \,, \end{displaymath} which completes the [[induction]] step. \end{proof} \begin{prop} \label{MapsOutOfExactIntoInjectiveAreNullHomotopic}\hypertarget{MapsOutOfExactIntoInjectiveAreNullHomotopic}{} Let $f^\bullet : X^\bullet \to J^\bullet$ be a [[chain map]] of cochain complexes in non-negative degree, out of an [[exact sequence|exact complex]] $0 \simeq_{qi} X^\bullet$ to a degreewise injective complex $J^\bullet$. Then there is a [[null homotopy]] \begin{displaymath} \eta : 0 \Rightarrow f^\bullet \end{displaymath} \end{prop} \begin{proof} By definition of [[chain homotopy]] we need to construct a sequence of morphisms $(\eta^{n+1} : X^{n+1} \to J^{n})_{n \in \mathbb{N}}$ such that \begin{displaymath} f^n = \eta^{n+1} \circ d^n_X + d^{n-1}_J \circ \eta^n \,. \end{displaymath} for all $n$. We now construct this by [[induction]] over $n$, where we take $\eta^0 \coloneqq 0$. Then in the induction step assume that for given $n \in \mathbb{N}$ we have constructed $\eta^{\bullet \leq n}$ satisfying the above conditions. First define now \begin{displaymath} g^n \coloneqq f^n - d_J^{n-1} \circ \eta^n \end{displaymath} and observe that \begin{displaymath} \begin{aligned} g^n \circ d_X^{n-1} & = f^n \circ d^{n-1}_X - d^{n-1}_J \circ \eta^n \circ d^{n-1}_X \\ & = f^n \circ d^{n-1}_X - d^{n-1}_J \circ f^{n-1} + d^{n-1}_J \circ d^{n-2}_J \circ \eta^{n-1} \\ & = 0 + 0 \\ & 0 \end{aligned} \,. \end{displaymath} This means that $g^n$ factors as \begin{displaymath} X^n \to X^n / im(d^{n-1}_X) \stackrel{g^n}{\to} J^n \,, \end{displaymath} where the first map is the [[projection]] to the [[quotient]]. Observe then that by exactness of $X^\bullet$ the morphism $X^n / im(d^{n-1}_X) \stackrel{d^n_X}{\to} X^{n+1}$ is a [[monomorphism]]. Together this gives us a diagram of the form \begin{displaymath} \itexarray{ X^n / im(d^{n-1}_X) &\stackrel{d^n_X}{\to}& X^{n+1} \\ \downarrow^{\mathrlap{g^n}} & \swarrow_{\mathrlap{\eta^{n+1}}} \\ J^n } \,, \end{displaymath} where the morphism $\eta^{n+1}$ may be found due to the defining [[right lifting property]] of the [[injective object]] $J^n$ against the top monomorphism. Observing that the [[commuting diagram|commutativity]] of this diagram is the chain homotopy condition involving $\eta^n$ and $\eta^{n+1}$, this completes the induction step. \end{proof} \begin{remark} \label{}\hypertarget{}{} Without the assumption above that $J^\bullet$ is injective, such a null-homotopy indeed need not exist. Basic counterexamples are discussed in the section \href{homotopy+category+of+chain+complexes#ChainHomotopiesThatOughtToExistButDoNot}{Chain homotopies that ought to exist but do not} at \emph{[[homotopy category of chain complexes]]}. \end{remark} The formally dual statement of prop \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic} is the following. \begin{prop} \label{MapsProjectiveIntoExactAreNullHomotopic}\hypertarget{MapsProjectiveIntoExactAreNullHomotopic}{} Let $f_\bullet : P_\bullet \to Y_\bullet$ be a [[chain map]] of [[chain complexes]] in non-negative degree, into an [[exact sequence|exact complex]] $0 \simeq_{qi} Y_\bullet$ from a degreewise [[projective object|projective]] complex $P^\bullet$. Then there is a [[null homotopy]] \begin{displaymath} \eta : 0 \Rightarrow f_\bullet \end{displaymath} \end{prop} \begin{proof} This is formally dual to the proof of prop. \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic}. \end{proof} The following proposition says that, when injectively resolving objects, the morphisms between these objects lift to the resolutions, uniquely up to chain homotopy. \begin{prop} \label{InjectiveResolutionOfCodomainRespectsMorphisms}\hypertarget{InjectiveResolutionOfCodomainRespectsMorphisms}{} Let $f : X \to Y$ be a morphism in $\mathcal{A}$. Let \begin{displaymath} i_Y : Y \stackrel{\sim}{\to} Y^\bullet \end{displaymath} be an injective resolution of $Y$ and \begin{displaymath} i_X : X \stackrel{\sim}{\to} X^\bullet \end{displaymath} any [[monomorphism|monomorphism]] that is a [[quasi-isomorphism]] (possibly but not necessarily an injective resolution). Then there is a [[chain map]] $f^\bullet : X^\bullet \to Y^\bullet$ giving a [[commuting diagram]] \begin{displaymath} \itexarray{ X &\stackrel{\sim}{\to}& X^\bullet \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f^\bullet}} \\ Y &\stackrel{\sim}{\to}& Y^\bullet } \,. \end{displaymath} \end{prop} \begin{proof} By definition of [[chain map]] we need to construct [[morphisms]] $(f^n : X^n \to Y^n)_{n \in \mathbb{N}}$ such that for all $n \in \mathbb{N}$ the [[diagrams]] \begin{displaymath} \itexarray{ X^{n} &\stackrel{d^n_X}{\to}& X^{n+1} \\ \downarrow^{\mathrlap{f^n}} && \downarrow^{\mathrlap{f^{n+1}}} \\ Y^{n} &\stackrel{d^n_Y}{\to}& Y^{n+1} } \end{displaymath} [[commuting diagram|commute]] (the defining condition on a [[chain map]]) and such that the diagram \begin{displaymath} \itexarray{ X &\stackrel{i_X}{\to}& X^0 \\ \downarrow^{f} && \downarrow^{\mathrlap{f^0}} \\ Y &\stackrel{i_Y}{\to}& Y^0 } \end{displaymath} commutes in $\mathcal{A}$ (which makes the full diagram in $Ch^\bullet(\mathcal{A})$ commute). We construct these $f^\bullet = (f^n)_{n \in \mathbb{N}}$ by [[induction]]. To start the induction, the morphism $f^0$ in the first diagram above can be found by the defining [[right lifting property]] of the [[injective object]] $Y^0$ against the [[monomorphism]] $i_X$. Assume then that for some $n \in \mathbb{N}$ component maps $f^{\bullet \leq n}$ have been obtained such that $d^k_Y\circ f^k = f^{k+1}\circ d^k_X$ for all $0 \leq k \lt n$ . In order to construct $f^{n+1}$ consider the following diagram, which we will describe/construct stepwise from left to right: \begin{displaymath} \itexarray{ X^n &\stackrel{}{\to}& X^n/im(d^{n-1}_X) &\stackrel{d^n_X}{\hookrightarrow}& X^{n+1} \\ {}^{\mathllap{f^n}}\downarrow & \searrow^{\mathrlap{g^n}} & \downarrow^{\mathrlap{h^n}} & \swarrow_{\mathrlap{f^{n+1}}} \\ Y^n &\underset{d^n_Y}{\to}& Y^{n+1} } \,. \end{displaymath} Here the morphism $f^n$ on the left is given by induction assumption and we define the diagonal morphism to be the composite \begin{displaymath} g^n \coloneqq d^n_Y \circ f^n \,. \end{displaymath} Observe then that by the chain map property of the $f^{\bullet \leq n}$ we have \begin{displaymath} d^n_Y \circ f^n \circ d^{n-1}_X = d^n_Y \circ d^{n-1}_Y \circ f^{n-1} = 0 \end{displaymath} and therefore $g^n$ factors through $X^n/im(d^{n-1}_X)$ via some $h^n$ as indicated in the middle of the above diagram. Finally the morphism on the top right is a monomorphism by the fact that $X^{\bullet}$ is [[exact sequence|exact]] in positive degrees (being [[quasi-isomorphism|quasi-isomorphic]] to a complex concentrated in degree 0) and so a lift $f^{n+1}$ as shown on the far right of the diagram exists by the defining lifting property of the injective object $Y^{n+1}$. The total outer diagram now commutes, being built from commuting sub-diagrams, and this is the required chain map property of $f^{\bullet \leq n+1}$ This completes the induction step. \end{proof} \begin{prop} \label{HomotopyUniquenessOfResolutionOfMorphism}\hypertarget{HomotopyUniquenessOfResolutionOfMorphism}{} The morphism $f_\bullet$ in prop. \ref{InjectiveResolutionOfCodomainRespectsMorphisms} is the unique one up to [[chain homotopy]] making the given diagram commute. \end{prop} \begin{proof} Given two chain maps $g_1^\bullet, g^2_\bullet$ making the diagram commute, a [[chain homotopy]] $g_1^\bullet \Rightarrow g_2^\bullet$ is equivalently a [[null homotopy]] $0 \Rightarrow g_2^\bullet - g_1^\bullet$ of the difference, which sits in a square of the form \begin{displaymath} \itexarray{ X &\underoverset{h^\bullet}{\sim}{\to}& X^\bullet \\ \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{f^\bullet \coloneqq g_2^\bullet - g_1^\bullet}} \\ Y &\stackrel{\sim}{\to}& Y^\bullet } \end{displaymath} with the left vertical morphism being the [[zero morphism]] (and the bottom an injective resolution). Hence we have to show that in such a diagram $f^\bullet$ is null-homotopic. This we may reduce to the statement of prop. \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic} by considering instead of $f^\bullet$ the induced chain map of augmented complexes \begin{displaymath} \itexarray{ 0 &\stackrel{}{\to}& X &\stackrel{h^0}{\to}& X^0 &\stackrel{d^0_X}{\to}& X^1 &\to& \cdots \\ \downarrow^{\mathrlap{f^{-2} = 0}} && \downarrow^{\mathrlap{f^{-1} = 0}} && \downarrow^{f^0} && \downarrow^{f^1} \\ 0 &\to& Y &\to& Y^0 &\stackrel{d^0_J}{\to}& Y^1 &\to& \cdots } \,, \end{displaymath} where the second square from the left commutes due to the commutativity of the original square of chain complexes in degree 0. Since $h^\bullet$ is a [[quasi-isomorphism]], the top chain complex is [[exact sequence|exact]], by remark \ref{InjectiveResolutionInComponents}. Morover the bottom complex consists of [[injective objects]] from the second degree on (the former degree 0). Hence the induction in the proof of prop \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic} implies the existence of a [[null homotopy]] \begin{displaymath} \itexarray{ 0 &\stackrel{}{\to}& X &\stackrel{}{\to}& X^0 &\stackrel{d^0_X}{\to}& X^1 &\to& \cdots \\ \downarrow^{\mathrlap{f^{-2} = 0}} &\swarrow_{\mathrlap{\eta^{-1} = 0}}& \downarrow^{\mathrlap{f^{-1} = 0}} &\swarrow_{\mathrlap{\eta^0 = 0} }& \downarrow^{f^0} &\swarrow_{\mathrlap{\eta^1}}& \downarrow^{f^1} \\ 0 &\to& Y &\to& Y^0 &\stackrel{d^0_Y}{\to}& Y^1 &\to& \cdots } \end{displaymath} starting with $\eta^{-1} = 0$ and $\eta^{0 } = 0$ (notice that the proof prop. \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic} was formulated exactly this way), which works because $f^{-1} = 0$. The de-augmentation $\{f^{\bullet \geq 0}\}$ of this is the desired [[null homotopy]] of $f^\bullet$. \end{proof} Sometimes one needs to construct resolutions of sequences of morphisms in a more controled way, for instance such that some degreewise exactness is preserved: \begin{lemma} \label{ProjectiveResolutionOfExactSequenceByExactSequence}\hypertarget{ProjectiveResolutionOfExactSequenceByExactSequence}{} For $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ a [[short exact sequence]] in an [[abelian category]] with [[projective object|enough projectives]], there exists a [[commuting diagram]] of [[chain complexes]] \begin{displaymath} \itexarray{ 0 &\to& A_\bullet &\to& B_\bullet &\to& C_\bullet &\to& 0 \\ && \downarrow^{\mathrlap{f_\bullet}} && \downarrow^{\mathrlap{g_\bullet}} && \downarrow^{\mathrlap{h_\bullet}} \\ 0 &\to& A &\stackrel{i}{\to}& B &\stackrel{p}{\to}& C &\to& 0 } \end{displaymath} where \begin{itemize}% \item each vertical morphism is a projective resolution; \end{itemize} and in addition \begin{itemize}% \item the top row is again a short exact sequence of chain complexes. \end{itemize} \end{lemma} This appears for instance in (\hyperlink{May}{May, lemma 3.4}) or (\hyperlink{Murfet}{Murfet, cor. 33}). \begin{proof} By prop. \ref{ExistenceOfInjectiveResolutions} we can choose $f_\bullet$ and $h_\bullet$. The task is now to construct the third resolution $g_\bullet$ such as to obtain a short exact sequence of chain complexes, hence degreewise a short exact sequence, in the two row. To construct this, let for each $n \in \mathbb{N}$ \begin{displaymath} B_n \coloneqq A_n \oplus C:n \end{displaymath} be the [[direct sum]] and let the top horizontal morphisms be the canonical inclusion and projection maps of the direct sum. Let then furthermore (in [[matrix calculus]] notation) \begin{displaymath} g_0 = \left( \itexarray{ (j_0)_A & (j_0)_B } \right) : A_0 \oplus C_0 \to B \end{displaymath} be given in the first component by the given composite \begin{displaymath} (g_0)_A : A_0 \oplus C_0 \stackrel{}{\to} A_0 \stackrel{f_0}{\to} A \stackrel{i}{\hookrightarrow} B \end{displaymath} and in the second component we take \begin{displaymath} (j_0)_C : A_0 \oplus C_0 \to C_0 \stackrel{\zeta}{\to} B \end{displaymath} to be given by a lift in \begin{displaymath} \itexarray{ && B \\ & {}^{\mathllap{\zeta}}\nearrow & \downarrow^{\mathrlap{p}} \\ C_0 &\stackrel{h_0}{\to}& C } \,, \end{displaymath} which exists by the [[left lifting property]] of the [[projective object]] $C_0$ (since $C_\bullet$ is a projective resolution) against the [[epimorphism]] $p : B \to C$ of the [[short exact sequence]]. In total this gives in degree 0 \begin{displaymath} \itexarray{ A_0 &\hookrightarrow& A_0 \oplus C_0 &\to& C_0 \\ \downarrow^{\mathrlap{f_0}} && {}^{\mathllap{((g_0)_A, (g_0)_C)}}\downarrow &\swarrow_{\zeta}& \downarrow^{\mathrlap{h_0}} \\ A &\stackrel{i}{\hookrightarrow}& B &\stackrel{p}{\to}& C } \,. \end{displaymath} Let then the [[differentials]] of $B_\bullet$ be given by \begin{displaymath} d_k^{B_\bullet} = \left( \itexarray{ d_k^{A_\bullet} & (-1)^k e_k \\ 0 & d_k^{C_\bullet} } \right) : A_{k+1} \oplus C_{k+1} \to A_k \oplus C_k \,, \end{displaymath} where the $\{e_k\}$ are constructed by [[induction]] as follows. Let $e_0$ be a lift in \begin{displaymath} \itexarray{ & && A_0 \\ & & {}^{\mathllap{e_0}}\nearrow & \downarrow^{\mathrlap{f_0}} \\ \zeta \circ d^{C_\bullet}_0 \colon & C_1 &\stackrel{}{\to}& A &\hookrightarrow B& } \end{displaymath} which exists since $C_1$ is a [[projective object]] and $A_0 \to A$ is an epimorphism by $A_\bullet$ being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through $A$ as indicated, because the definition of $\zeta$ and the chain complex property of $C_\bullet$ gives \begin{displaymath} \begin{aligned} p \circ \zeta \circ d^{C_\bullet}_0 &= h_0 \circ d^{C_\bullet}_0 \\ & = 0 \circ h_1 \\ & = 0 \end{aligned} \,. \end{displaymath} Now in the induction step, assuming that $e_{n-1}$ has been been found satisfying the chain complex property, let $e_n$ be a lift in \begin{displaymath} \itexarray{ & && A_n \\ & & {}^{\mathllap{e_{n}}}\nearrow & \downarrow^{\mathrlap{d^{A_\bullet}_{n-1}}} \\ e_{n-1}\circ d_n^{C_\bullet} \colon & C_{n+1} &\stackrel{}{\hookrightarrow}& ker(d^{A_\bullet}_{n-1}) = im(d^{A_\bullet}_{n-1})) &\to& A_{n-1} } \,, \end{displaymath} which again exists since $C_{n+1}$ is projective. That the bottom morphism factors as indicated is the chain complex property of $e_{n-1}$ inside $d^{B_\bullet}_{n-1}$. To see that the $d^{B_\bullet}$ defines this way indeed squares to 0 notice that \begin{displaymath} d^{B_\bullet}_{n} \circ d^{B_\bullet}_{n+1} = \left( \itexarray{ 0 & (-1)^{n}\left(e_{n} \circ d^{C_\bullet}_{n+1} - d^{A_\bullet}_n \circ e_{n+1} \right) \\ 0 & 0 } \right) \,. \end{displaymath} This vanishes by the very commutativity of the above diagram. This establishes $g_\bullet$ such that the above diagram commutes and the bottom row is degreewise a short exact sequence, in fact a [[split exact sequence]], by construction. To see that $g_\bullet$ is indeed a quasi-isomorphism, consider the [[homology long exact sequence]] associated to the short exact sequence of cochain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$. In positive degrees it implies that the chain homology of $B_\bullet$ indeed vanishes. In degree 0 it gives the short sequence $0 \to A \to H_0(B_\bullet) \to B\to 0$ sitting in a commuting diagram \begin{displaymath} \itexarray{ 0 &\to& A &\hookrightarrow& H_0(B_\bullet) &\to& C &\to& 0 \\ \downarrow && \downarrow^{\mathrlap{=}} && \downarrow && \downarrow^{\mathrlap{=}} && \downarrow \\ 0 &\to& A &\hookrightarrow& B &\to& C &\to& 0 \,, } \end{displaymath} where both rows are exact. That the middle vertical morphism is an [[isomorphism]] then follows by the [[five lemma]]. \end{proof} The formally dual statement to lemma \ref{ProjectiveResolutionOfExactSequenceByExactSequence} is the following. \begin{lemma} \label{InjectiveResolutionOfExactSequenceByExactSequence}\hypertarget{InjectiveResolutionOfExactSequenceByExactSequence}{} For $0 \to A \to B \to C \to 0$ a [[short exact sequence]] in an [[abelian category]] with [[injective object|enough injectives]], there exists a [[commuting diagram]] of cochain complexes \begin{displaymath} \itexarray{ 0 &\to& A &\to& B &\to& C &\to& 0 \\ && \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} \\ 0 &\to& A^\bullet &\to& B^\bullet &\to& C^\bullet &\to& 0 } \end{displaymath} where \begin{itemize}% \item each vertical morphism is an injective resolution; \end{itemize} and in addition \begin{itemize}% \item the bottom row is again a short exact sequence of cochain complexes. \end{itemize} \end{lemma} \begin{proof} To construct this, let for each $n \in \mathbb{N}$ \begin{displaymath} B^n \coloneqq A^n \oplus C^n \end{displaymath} be the [[direct sum]] and let the bottom horizontal morphisms be the canonical inclusion and projection maps of the direct sum. Let then furthermore (in [[matrix calculus]] notation) \begin{displaymath} j^0 = \left( \itexarray{ j^0_A \\ j^0_B } \right) : B \to A^0 \oplus C^0 \end{displaymath} be given in the second component by the given composite \begin{displaymath} j^0_B : B \to C \to C^0 \end{displaymath} and in the first component we take \begin{displaymath} j^0_A : B \to A^0 \end{displaymath} to be given by a lift in \begin{displaymath} \itexarray{ A &\to& A^0 \\ \downarrow & \nearrow_{\mathrlap{j^0_A}} \\ B } \,, \end{displaymath} which exists by the [[right lifting property]] of the [[injective object]] $A^0$ (since $A^\bullet$ is an injective resolution) against the [[monomorphism]] $A \to B$ of the [[short exact sequence]]. Let the differentials be given by (\ldots{}). This establishes $j^\bullet$ such that the above diagram commutes and the bottom row is degreewise a short exact sequence, in fact a [[split exact sequence]], by construction. To see that $j^\bullet$ is indeed a quasi-isomorphism, consider the [[homology long exact sequence]] associated to the short exact sequence of cochain complexes $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ (\ldots{}). \end{proof} \hypertarget{ExistenceAndConstructionOfResolutionsOfComplexes}{}\subsubsection*{{Existence and construction of resolutions of complexes}}\label{ExistenceAndConstructionOfResolutionsOfComplexes} \begin{defn} \label{}\hypertarget{}{} If $\mathcal{A}$ has [[projective object|enough projectives]], then every chain complex $C_\bullet \in Ch_\bullet(\mathcal{A})$ has a fully projective (proper) resolution, def. \ref{FullyProjectiveResolutionOfChainComplex}. \end{defn} \begin{proof} Notice that for each $n \in \mathbb{N}$ we have [[short exact sequences]] of [[chains]], [[cycles]], [[boundaries]] and [[homology groups]] as \begin{displaymath} 0 \to B_n(C) \to Z_n(C) \to H_n(C) \to 0 \end{displaymath} \begin{displaymath} 0 \to Z_n(C) \to C_n \to B_{n-1}(C) \to 0 \,. \end{displaymath} Now by prop. \ref{ExistenceOfInjectiveResolutions} we find for each $n \in \mathbb{N}$ projective resolutions of the objects $H_n(C)$ and $B_n(C)$: \begin{displaymath} H_{n,\bullet} \stackrel{\simeq_{qi}}{\to} H_n(C) \end{displaymath} \begin{displaymath} B_{n,\bullet} \stackrel{\simeq_{qi}}{\to} B_n(C) \,. \end{displaymath} Moreover, by prop. \ref{ProjectiveResolutionOfExactSequenceByExactSequence} we find for each $n \in \mathbb{N}$ a projective resolution $Z_{p,\bullet}(C) \stackrel{\simeq_{qi}}{\to} Z_n(C)$ of the object $Z_p(C)$ such that its fits into a [[short exact sequence]] of chain complexes with the previous two chosen resolutions: \begin{displaymath} 0 \to B_{n,\bullet}(C) \to Z_{n,\bullet}(C) \to H_{n,0}(C) \to 0 \,. \end{displaymath} Analogously, we find for each $n$ a projective resolution $C_{n,\bullet} \to C_n$ that sits in a short exact sequence \begin{displaymath} 0 \to Z_{n,\bullet} \to C_{n,\bullet} \to B_{n+1,\bullet} \to 0 \,. \end{displaymath} Using the exactness of these sequences one checks now that \begin{enumerate}% \item The $\{C_{n,\bullet}\}_{n \in \mathbb{N}}$ arrange into a [[double complex]] by taking the horizontal differential to be the composite \begin{displaymath} C_{n,k} \to B_{n+1,k} \hookrightarrow Z_{n+1,k} \to C_{n+1,k} \,; \end{displaymath} \item this [[double complex]] $C_{\bullet,\bullet}$ is indeed a fully projective resolution of $C_\bullet$. \end{enumerate} \end{proof} \hypertarget{FunctorialResolutions}{}\subsubsection*{{Functorial resolutions and derived functors}}\label{FunctorialResolutions} We discuss how the injective/projective resolutions constructed in \emph{\hyperlink{ExistenceAndConstruction}{Existence and construction}} are [[functor|functorial]] if regarded in the [[homotopy category of chain complexes]] and how this yields the construction of \emph{[[derived functors in homological algebra]]}. Write \begin{displaymath} \mathcal{K}^{+}(\mathcal{A}) \hookrightarrow \mathcal{K}(\mathcal{A}) \end{displaymath} for the [[full subcategory]] of the [[homotopy category of chain complexes]] on the one bounded above or bounded below, respectively. Write \begin{displaymath} \mathcal{K}^+(\mathcal{I}_{\mathcal{A}}) \hookrightarrow \mathcal{K}^+(\mathcal{A}) \end{displaymath} for the [[full subcategory]] on the degreewise [[injective object|injective]] complexes, and \begin{displaymath} \mathcal{K}^-(\mathcal{P}_{\mathcal{A}}) \hookrightarrow \mathcal{K}^-(\mathcal{A}) \end{displaymath} for the full subcategory on the degreewise [[projective object|projective]] objects. \begin{theorem} \label{InjectiveResolutionFunctors}\hypertarget{InjectiveResolutionFunctors}{} If $\mathcal{A}$ has \href{injective+object#EnoughInjectives}{enough injectives} then there exists a [[functor]] \begin{displaymath} P : \mathcal{A} \to \mathcal{K}^+(\mathcal{I}_{\mathcal{A}}) \end{displaymath} together with a [[natural isomorphisms]] \begin{displaymath} H^0(-) \circ P \simeq id_{\mathcal{A}} \end{displaymath} and \begin{displaymath} H^{n \geq 1}(-) \circ P \simeq 0 \,. \end{displaymath} \end{theorem} \begin{proof} By prop. \ref{ExistenceOfInjectiveResolutions} every object $X^\bullet \in Ch^\bullet(\mathcal{A})$ has an injective resolution. Proposition \ref{InjectiveResolutionOfCodomainRespectsMorphisms} says that for $X \to X^\bullet$ and $X \to \tilde X^\bullet$ two resolutions the there is a morphism $X^\bullet \to \tilde X^\bullet$ in $\mathcal{K}^+()$ and prop. \ref{HomotopyUniquenessOfResolutionOfMorphism} says that this morphism is unique in $\mathcal{K}^+(\mathcal{A})$. In particular it is therefore an [[isomorphism]] in $\mathcal{K}^+(\mathcal{A})$ (since the composite with the reverse lifted morphism, also being unique, has to be the identity). So choose one such injective resolution $P(X)^\bullet$ for each $X^\bullet$. Then for $f : X \to Y$ any morphism in $\mathcal{A}$, proposition \ref{ExistenceOfInjectiveResolutions} again says that it can be lifted to a morphism between $P(X)^\bullet$ and $P(Y)^\bullet$ and proposition \ref{InjectiveResolutionOfCodomainRespectsMorphisms} says that there is a unique such image in $\mathcal{K}^+(\mathcal{A})$ for morphism making the given diagram commute. This implies that this assignment of morphisms is [[functor|functorial]], since then also the composites are unique. \end{proof} Dually we have: \begin{theorem} \label{ProjectiveResolutionFunctors}\hypertarget{ProjectiveResolutionFunctors}{} If $\mathcal{A}$ has \href{projectice+object#EnoughInjectives}{enough projectives} then there exists a [[functor]] \begin{displaymath} Q : \mathcal{A} \to \mathcal{K}^-(\mathcal{P}_{\mathcal{A}}) \end{displaymath} together with a [[natural isomorphisms]] \begin{displaymath} H_0(-) \circ P \simeq id_{\mathcal{A}} \end{displaymath} and \begin{displaymath} H_{n \geq 1}(-) \circ P \simeq 0 \,. \end{displaymath} \end{theorem} This is sufficient for the definition and construction of (non-total) [[derived functors]] in the next definition \ref{RightDerivedFunctorOfLeftExactFunctor}. But since that definition is but a model and just for a special case of derived functors, the reader might want to keep the following definition and remark in mind, for conceptual orientation. \begin{defn} \label{}\hypertarget{}{} Given an [[additive functor]] $F : \mathcal{A} \to \mathcal{A}'$, it canonically induces a functor \begin{displaymath} Ch_\bullet(F) \colon Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{A}') \end{displaymath} between [[categories of chain complexes]] (its ``prolongation'') by applying it to each [[chain complex]] and to all the diagrams in the definition of a [[chain map]]. Similarly it preserves [[chain homotopies]] and hence it passes to the quotient given by the strong [[homotopy category of chain complexes]] \begin{displaymath} \mathcal{K}(F) \colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{A}') \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} If $\mathcal{A}$ and $\mathcal{A}'$ have [[projective object|enough projectives]], then their [[derived categories]] are \begin{displaymath} \mathcal{D}_\bullet(\mathcal{A}) \simeq \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}}) \end{displaymath} and \begin{displaymath} \mathcal{D}^\bullet(\mathcal{A}) \simeq \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}}) \end{displaymath} etc. One wants to accordingly \emph{derive} from $F$ a functor $\mathcal{D}_\bullet(\mathcal{A}) \to \mathcal{D}_\bullet(\mathcal{A})$ between these derived categories. It is immediate to achive this on the domain category, there we can simply precompose and form \begin{displaymath} \mathcal{A} \to \mathcal{D}_\bullet(\mathcal{A}) \simeq \mathcal{K}(\mathcal{P}_{\mathcal{A}}) \hookrightarrow \mathcal{K}(\mathcal{A}) \stackrel{\mathcal{K}(F)}{\to} \mathcal{K}(\mathcal{A}') \,. \end{displaymath} But the resulting composite lands in $\mathcal{K}(\mathcal{A}')$ and in general does not factor through the inclusion $\mathcal{D}_\bullet(\mathcal{A}') = \mathcal{K}(\mathcal{P}_{\mathcal{A}'}) \hookrightarrow \mathcal{K}(\mathcal{A}')$. By applying a projective resolution functor \emph{on chain complexes}, one can enforce this factorization. However, by definition of [[resolution]], the resulting chain complex is [[quasi-isomorphism|quasi-isomorphic]] to the one obtained by the above composite. This means that if one is only interested in the ``weak chain homology type'' of the chain complex in the image of a [[derived functor]], then forming [[chain homology]] groups of the chain complexes in the images of the above composite gives the desired information. This is what def. \ref{RightDerivedFunctorOfLeftExactFunctor} and def. \ref{LeftDerivedFunctorOfRightExactFunctor} below do. \end{remark} \begin{defn} \label{RightDerivedFunctorOfLeftExactFunctor}\hypertarget{RightDerivedFunctorOfLeftExactFunctor}{} Let \begin{displaymath} F : \mathcal{A} \to \mathcal{A}' \end{displaymath} be a [[left exact functor]] between [[abelian categories]] such that $\mathcal{A}$ has \href{inhective+object#EnoughInjectives}{enough injectives}. For $n \in \mathbb{N}$ the \textbf{$n$th [[right derived functor]]} of $F$ is the composite \begin{displaymath} R^n F : \mathcal{A} \stackrel{P}{\to} K^+(\mathcal{I}_{\mathcal{A}}) \stackrel{\mathcal{K}(F)}{\to} \mathcal{K}^+(\mathcal{A}') \stackrel{H^n(-)}{\to} \mathcal{A}' \,, \end{displaymath} where \begin{itemize}% \item $P$ is the injective resolution functor of theorem \ref{InjectiveResolutionFunctors}; \item $\mathcal{K}(F)$ is the evident prolongation of $F$ to $\mathcal{K}^+(\mathcal{A})$; \item $H^n(-)$ is the $n$-[[chain homology]] functor. Hence \end{itemize} \begin{displaymath} (R^n F)(X^\bullet) \coloneqq H^n(F(P(X)^\bullet)) \,. \end{displaymath} \end{defn} Dually: \begin{defn} \label{LeftDerivedFunctorOfRightExactFunctor}\hypertarget{LeftDerivedFunctorOfRightExactFunctor}{} Let \begin{displaymath} F : \mathcal{A} \to \mathcal{A}' \end{displaymath} be a [[right exact functor]] between [[abelian categories]] such that $\mathcal{A}$ has \href{projective+object#EnoughProjectives}{enough projectives}. For $n \in \mathbb{N}$ the \textbf{$n$th [[left derived functor]]} of $F$ is the composite \begin{displaymath} L_n F : \mathcal{A} \stackrel{Q}{\to} K^-(\mathcal{P}_{\mathcal{A}}) \stackrel{\mathcal{K}(F)}{\to} \mathcal{K}^-(\mathcal{A}') \stackrel{H_n(-)}{\to} \mathcal{A}' \,, \end{displaymath} where \begin{itemize}% \item $Q$ is the projective resolution functor of theorem \ref{ProjectiveResolutionFunctors}; \item $\mathcal{K}(F)$ is the evident prolongation of $F$ to $\mathcal{K}^+(\mathcal{A})$; \item $H_n(-)$ is the $n$-[[chain homology]] functor. Hence \end{itemize} \begin{displaymath} (L_n F)(X_\bullet) \coloneqq H_n(F(Q(X)_\bullet)) \,. \end{displaymath} \end{defn} We discuss now the basic general properties of such derived functors. \begin{prop} \label{BasicPropertiesOfDerivedFunctors}\hypertarget{BasicPropertiesOfDerivedFunctors}{} Let $F \colon \mathcal{A} \to \mathcal{B}$ a [[left exact functor]] in the presence of [[injective object|enough injectives]]. Then for all $X \in \mathcal{A}$ there is a [[natural isomorphism]] \begin{displaymath} R^0F(X) \simeq F(X) \,. \end{displaymath} Dually, of $F$ is a [[right exact functor]] in the presence of [[projective object|enough projectives]], then \begin{displaymath} L_0 F(X) \simeq F(X) \,. \end{displaymath} \end{prop} \begin{proof} We discuss the first statement, the second is formally dual. By remark \ref{InjectiveResolutionInComponents} an injective resolution $X \stackrel{\simeq_{qi}}{\to} X^\bullet$ is equivalently an [[exact sequence]] of the form \begin{displaymath} 0 \to X \hookrightarrow X^0 \to X^1 \to \cdots \,. \end{displaymath} If $F$ is left exact then it preserves this excact sequence by definition of left exactness, and hence \begin{displaymath} 0 \to F(X) \hookrightarrow F(X^0) \to F(X^1) \to \cdots \end{displaymath} is an exact sequence. But this means that \begin{displaymath} R^0 F(X) \coloneqq ker(F(X^0) \to F(X^1)) \simeq F(X) \,. \end{displaymath} \end{proof} \begin{prop} \label{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence}\hypertarget{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence}{} Let $\mathcal{A}, \mathcal{B}$ be [[abelian categories]] and assume that $\mathcal{A}$ has [[injective object|enough injectives]]. Let $F : \mathcal{A} \to \mathcal{B}$ be a [[left exact functor]] and let \begin{displaymath} 0 \to A \to B \to C \to 0 \end{displaymath} be a [[short exact sequence]] in $\mathcal{A}$. Then there is a [[long exact sequence]] of images of these objects under the right derived functors $R^\bullet F(-)$ of def. \ref{RightDerivedFunctorOfLeftExactFunctor} \begin{displaymath} \itexarray{ 0 &\to& R^0F (A) &\to& R^0 F(B) &\to& R^0 F(C) &\stackrel{\delta_0}{\to}& R^1 F(A) &\to& R^1 F(B) &\to& R^1F(C) &\stackrel{\delta_1}{\to}& R^2 F(A) &\to& \cdots \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ 0 &\to& F(A) &\to& F(B) &\to& F(C) } \end{displaymath} in $\mathcal{B}$. \end{prop} \begin{proof} By lemma \ref{InjectiveResolutionOfExactSequenceByExactSequence} we can find an injective resolution \begin{displaymath} 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 \end{displaymath} of the given exact sequence which is itself again an exact sequence of cochain complexes. Since $A^n$ is an [[injective object]] for all $n$, its component sequences $0 \to A^n \to B^n \to C^n \to 0$ are indeed [[split exact sequences]] (see the discussion there). Splitness is preserved by a functor $F$ and so it follows that \begin{displaymath} 0 \to F(\tilde A^\bullet) \to F(\tilde B^\bullet) \to F(\tilde C^\bullet) \to 0 \end{displaymath} is a again short exact sequence of cochain complexes, now in $\mathcal{B}$. Hence we have the corresponding [[homology long exact sequence]] \begin{displaymath} \cdots \to H^{n-1}(F(A^\bullet)) \to H^{n-1}(F(B^\bullet)) \to H^{n-1}(F(C^\bullet)) \stackrel{\delta}{\to} H^n(F(A^\bullet)) \to H^n(F(B^\bullet)) \to H^n(F(C^\bullet)) \stackrel{\delta}{\to} H^{n+1}(F(A^\bullet)) \to H^{n+1}(F(B^\bullet)) \to H^{n+1}(F(C^\bullet)) \to \cdots \,. \end{displaymath} But by construction of the resolutions and by def. \ref{RightDerivedFunctorOfLeftExactFunctor} this is equal to \begin{displaymath} \cdots \to R^{n-1}F(A) \to R^{n-1}F(B) \to R^{n-1}F(C) \stackrel{\delta}{\to} R^{n}F(A) \to R^{n}F(B) \to R^{n}F(C) \stackrel{\delta}{\to} R^{n+1}F(A) \to R^{n+1}F(B) \to R^{n+1}F(C) \to \cdots \,. \end{displaymath} Finally the equivalence of the first three terms with $F(A) \to F(B) \to F(C)$ is given by prop. \ref{BasicPropertiesOfDerivedFunctors}. \end{proof} \begin{remark} \label{}\hypertarget{}{} Prop. \ref{LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence} implies that one way to interpret $R^1 F(A)$ is as a ``measure for how a [[left exact functor]] $F$ fails to be an [[exact functor]]''. For, with $A \to B \to C$ any [[short exact sequence]], this proposition gives the exact sequence \begin{displaymath} 0 \to F(A) \to F(B) \to F(C) \to R^1 F(A) \end{displaymath} and hence $0 \to F(A) \to F(B) \to F(C) \to$ is a short exact sequence itself precisely if $R^1 F(A) \simeq 0$. \end{remark} In fact we even have the following. \begin{prop} \label{}\hypertarget{}{} \begin{displaymath} R^{\geq 1} F = 0 \end{displaymath} and \begin{displaymath} L_{\geq 1} F = 0 \,. \end{displaymath} \end{prop} \begin{proof} Because an [[exact functor]] preserves all [[exact sequences]]. If $Y_\bullet \to A$ is a projective resolution then also $F(Y)_\bullet$ is exact in all positive degrees, and hence $L_{n\geq 1} F(A) ) H_{n \geq}(F(Y)) = 0$. Dually for $R^n F$. \end{proof} We now discuss how the derived functor of an additive functor $F$ may also be computed not necessarily with genuine injective/projective resolutions, but with (just) $F$-injective/$F$-projective resolutions, such as $F$-acyclic resolutions, as defined \hyperlink{FResolutions}{above}. Let $\mathcal{A}$ be an [[abelian category]] with [[injective object|enough injectives]]. Let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[additive functor|additive]] [[left exact functor]] with [[right derived functor]] $R_\bullet F$, def. \ref{RightDerivedFunctorOfLeftExactFunctor}. Finally let $\mathcal{I} \subset \mathcal{A}$ be a subcategory of $F$-injective objects, def. \ref{FInjectives}. \begin{lemma} \label{FPreservesNullnessOfFInjectiveComplexes}\hypertarget{FPreservesNullnessOfFInjectiveComplexes}{} If a [[cochain complex]] $A^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ is [[quasi-isomorphism|quasi-isomorphic]] to 0, \begin{displaymath} X^\bullet \stackrel{\simeq_{qi}}{\to} 0 \end{displaymath} then also $F(X^\bullet) \in Ch^\bullet(\mathcal{B})$ is quasi-isomorphic to 0 \begin{displaymath} F(X^\bullet) \stackrel{\simeq_{qi}}{\to} 0 \,. \end{displaymath} \end{lemma} \begin{proof} Consider the following collection of [[short exact sequences]] obtained from the [[long exact sequence]] $X^\bullet$: \begin{displaymath} 0 \to X^0 \stackrel{d^0}{\to} X^1 \stackrel{d^1}{\to} im(d^1) \to 0 \end{displaymath} \begin{displaymath} 0 \to im(d^1) \to X^2 \stackrel{d^2}{\to} im(d^2) \to 0 \end{displaymath} \begin{displaymath} 0 \to im(d^2) \to X^3 \stackrel{d^3}{\to} im(d^3) \to 0 \end{displaymath} and so on. Going by [[induction]] through this list and using the second condition in def. \ref{FInjectives} we have that all the $im(d^n)$ are in $\mathcal{I}$. Then the third condition in def. \ref{FInjectives} says that all the sequences \begin{displaymath} 0 \to F(im(d^n)) \to F(X^n+1) \to F(im(d^{n+1})) \to 0 \end{displaymath} are [[short exact sequence|exact]]. But this means that \begin{displaymath} 0 \to F(X^0)\to F(X^1) \to F(X^2) \to \cdots \end{displaymath} is exact, hence that $F(X^\bullet)$ is quasi-isomorphic to 0. \end{proof} \begin{theorem} \label{DerivedFFromFInjectiveResolution}\hypertarget{DerivedFFromFInjectiveResolution}{} For $A \in \mathcal{A}$ an object with $F$-injective resolution $A \stackrel{\simeq_{qi}}{\to} I_F^\bullet$, def. \ref{FProjectivesResolution}, we have for each $n \in \mathbb{N}$ an [[isomorphism]] \begin{displaymath} R^n F(A) \simeq H^n(F(I_F^\bullet)) \end{displaymath} between the $n$th right derived functor, def. \ref{RightDerivedFunctorOfLeftExactFunctor} of $F$ evaluated on $A$ and the [[cochain cohomology]] of $F$ applied to the $F$-injective resolution $I_F^\bullet$. \end{theorem} \begin{proof} By prop. \ref{ExistenceOfInjectiveResolutions} we can also find an injective resolution $A \stackrel{\simeq_{qi}}{\to} I^\bullet$. By prop. \ref{InjectiveResolutionOfCodomainRespectsMorphisms} there is a lift of the identity on $A$ to a [[chain map]] $I^\bullet_F \to I^\bullet$ such that the [[diagram]] \begin{displaymath} \itexarray{ A &\stackrel{\simeq_{qi}}{\to}& I_F^\bullet \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} \\ A &\stackrel{\simeq_{qi}}{\to}& I^\bullet } \end{displaymath} [[commuting diagram|commutes]] in $Ch^\bullet(\mathcal{A})$. Therefore by the [[2-out-of-3]] property of [[quasi-isomorphisms]] it follows that $f$ is a quasi-isomorphism Let $Cone(f) \in Ch^\bullet(\mathcal{A})$ be the [[mapping cone]] of $f$ and let $I^\bullet \to Cone(f)$ be the canonical [[chain map]] into it. By the explicit formulas for mapping cones, we have that \begin{enumerate}% \item there is an [[isomorphism]] $F(Cone(f)) \simeq Cone(F(f))$; \item $Cone(f) \in Ch^\bullet(\mathcal{I})\subset Ch^\bullet(\mathcal{A})$ (because $F$-injective objects are closed under [[direct sum]]). \end{enumerate} The first implies that we have a [[homology exact sequence]] \begin{displaymath} \cdots \to H^n(I^\bullet) \to H^n(I_F^\bullet) \to H^n(Cone(f)^\bullet) \to H^{n+1}(I^\bullet) \to H^{n+1}(I_F^\bullet) \to H^{n+1}(Cone(f)^\bullet) \to \cdots \,. \end{displaymath} Observe that with $f^\bullet$ a quasi-isomorphism $Cone(f^\bullet)$ is quasi-isomorphic to 0. Therefore The second item above implies with lemma \ref{FPreservesNullnessOfFInjectiveComplexes} that also $F(Cone(f))$ is quasi-isomorphic to 0. This finally means that the above homology exact sequences consists of exact pieces of the form \begin{displaymath} 0 \to (R^n F(A)\coloneqq H^n(I^\bullet) \stackrel{\simeq}{\to} H^n(I_F^\bullet) \to 0 \,. \end{displaymath} \end{proof} \hypertarget{DerivedHomFunctor}{}\subsubsection*{{Derived Hom-functor/$Ext$-functor and extensions}}\label{DerivedHomFunctor} Consider the [[derived functor]] of the [[hom functor]]. \begin{defn} \label{ExtFunctorAsRightDerivedContravariantHom}\hypertarget{ExtFunctorAsRightDerivedContravariantHom}{} For $A \in \mathcal{A}$, write \begin{displaymath} Ext^n(-,A) \coloneqq R^n Hom(-,A) \end{displaymath} for the [[right derived functor]], def. \ref{RightDerivedFunctorOfLeftExactFunctor}. \end{defn} We discuss the use of projective resolutions in the computation of [[Ext]]-functors and [[group extensions]]. \begin{defn} \label{Extensions}\hypertarget{Extensions}{} Given $A, G \in \mathcal{A}$, an \textbf{[[extension]]} of $G$ by $A$ is a [[short exact sequence]] of the form \begin{displaymath} 0 \to A \to \hat G \to G \to 0 \,. \end{displaymath} Two extensions $\hat G_1$ and $\hat G_2$ are called \emph{equivalent} if there is a morphism $f : \hat G_1 \to \hat G_2$ in $\mathcal{A}$ such that we have a [[commuting diagram]] \begin{displaymath} \itexarray{ && \hat G_1 \\ & \nearrow && \searrow \\ A &&\downarrow^{\mathrlap{f}}&& G \\ & \searrow && \nearrow \\ && \hat G_2 } \,. \end{displaymath} Write $Ext(G,A)$ for the set of [[equivalence classes]] of extensions of $G$ by $A$. \end{defn} \begin{remark} \label{MorphismOfExtensionIsIsomorphism}\hypertarget{MorphismOfExtensionIsIsomorphism}{} By the [[short five lemma]] a morphism $f$ as above is necessarily an [[isomorphism]] and hence we indeed have an [[equivalence relation]]. \end{remark} \begin{defn} \label{MapFromExtensionsToExtGroup}\hypertarget{MapFromExtensionsToExtGroup}{} If $\mathcal{A}$ has [[projective object|enough projectives]], define a function \begin{displaymath} Extr \colon Ext(G,A) \to Ext^1(G,A) \end{displaymath} from the group of extensions, def. \ref{Extensions}, to the first [[Ext functor]] group as follows. Choose any projective resolution $Y_\bullet \stackrel{\simeq_{qi}}{\to} G$, which exists by prop. \ref{ExistenceOfInjectiveResolutions}. Regard then $A \to \hat G \to G\to 0$ as a resolution \begin{displaymath} \itexarray{ \cdots &\to& 0 &\to& 0 &\to& A &\to& \hat G \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots &\to& 0 &\to& 0 &\to& 0 &\to& G } \end{displaymath} of $G$, by remark \ref{ProjectiveResolutionInComponents}. By prop. \ref{InjectiveResolutionOfCodomainRespectsMorphisms} there exists then a [[commuting diagram]] of the form \begin{displaymath} \itexarray{ Y_2 &\to& 0 \\ \downarrow^{\mathrlap{\partial_1^{Y}}} && \downarrow \\ Y_1 &\stackrel{c}{\to}& A \\ \downarrow^{\mathrlap{\partial_0^Y}} && \downarrow \\ Y_0 &\to& \hat G \\ \downarrow && \downarrow \\ G &\stackrel{id}{\to}& G } \end{displaymath} lifting the identity map on $G$ two a [[chain map]] between the two resolutions. By the commutativity of the top square, the morphism $c$ is 1-[[cocycle]] in $Hom(Y_\bullet,N)$, hence defines an element in $Ext^1(G,A) \coloneqq H^1(Hom(Y_\bullet,N))$. \end{defn} \begin{prop} \label{}\hypertarget{}{} The construction of def. \ref{MapFromExtensionsToExtGroup} is indeed well defined in that it is independent of the choice of projective resolution as well as of the choice of chain map between the projective resolutions. \end{prop} \begin{proof} First consider the same projective resolution but another lift $\tilde c$ of the identity. By prop. \ref{HomotopyUniquenessOfResolutionOfMorphism} any other choice $\tilde c$ fitting into a commuting diagram as above is related by a [[chain homotopy]] to $c$. \begin{displaymath} \itexarray{ Y_2 &\to& 0 \\ \downarrow^{\mathrlap{\partial_1^{Y}}} &\nearrow_{\eta_1 = 0}& \downarrow \\ Y_1 &\stackrel{c - \tilde c}{\to}& A \\ \downarrow^{\mathrlap{\partial_0^Y}} &\nearrow_{\eta_0}& \downarrow \\ Y_0 &\to& \hat G \\ \downarrow &\nearrow_{}& \downarrow \\ G &\to& G } \,. \end{displaymath} The chain homotopy condition here says that \begin{displaymath} c - \tilde c = \eta_0 \circ \partial^{Y}_0 \end{displaymath} and hence that in $Hom(Y_\bullet,N)$ we have that $d \eta_0 = c - \tilde c$ is a [[coboundary]]. Therefore for the given choice of resolution $Y_\bullet$ we have obtained a well-defined map \begin{displaymath} Ext(G,A) \to Ext^1(G,A) \,. \end{displaymath} If moreover $Y'_\bullet \stackrel{\simeq_{qi}}{\to} G$ is another projective resolution, with respect to which we define such a map as above, then lifting the identity map on $G$ to a chain map between these resolutions in both directions, by prop. \ref{InjectiveResolutionOfCodomainRespectsMorphisms}, establishes an isomorphism between the resulting maps, and hence the construction is independent also of the choice of resolution. \end{proof} \begin{prop} \label{ExtensionFromAnElementOfExt1}\hypertarget{ExtensionFromAnElementOfExt1}{} Define a function \begin{displaymath} Rec \colon Ext^1(G,A) \to Ext(G,A) \end{displaymath} as follows. For $Y_\bullet \to G$ a projective resolution of $G$ and $[c] \in Ext^1(G,A) \simeq H^1(Hom_{\mathcal{A}}(F_\bullet,A))$ an element of the $Ext$-group, let \begin{displaymath} \itexarray{ Y_2 &\to& 0 \\ \downarrow && \downarrow \\ Y_1 &\stackrel{c}{\to}& A \\ \downarrow \\ Y_0 \\ \downarrow \\ G } \end{displaymath} be a representative. By the commutativity of the top square this restricts to a morphism \begin{displaymath} \itexarray{ Y_1/Y_2 &\stackrel{c}{\to}& A \\ \downarrow \\ Y_0 \\ \downarrow \\ G } \,, \end{displaymath} where now the left column is itself an extension of $G$ by the [[cokernel]] $Y_1/Y_2$ (because by exactness the kernel of $Y_1 \to Y_0$ is the image of $Y_2$ so that the kernel of $Y_1/Y_2 \to Y_0$ is zero). Form then the [[pushout]] of the horizontal map along the two vertical maps. This yields \begin{displaymath} \itexarray{ Y_1/Y_2 &\stackrel{c}{\to}& A \\ \downarrow && \downarrow \\ Y_0 &\to& Y_0 \coprod_{Y_1/Y_2} A \\ \downarrow && \downarrow \\ G &\stackrel{id}{\to}& G } \,. \end{displaymath} Here the bottom right is indeed $G$, by the [[pasting law]] for pushouts and using that the left vertical composite is the [[zero morphism]]. Moreover, the top right morphism is indeed a monomorphism as it is the pushout of a map of modules along an [[injection]]. Similarly the bottom right morphism is an epimorphism. Hence $A \to Y_0 \coprod_{Y_1/Y_2} Y_0 \to G$ is an element in $Ext(G,A)$ which we assign to $c$. \end{prop} \begin{prop} \label{}\hypertarget{}{} The construction of def. \ref{ExtensionFromAnElementOfExt1} is indeed well defined in that it is independent of the choice of projective resolution as well as of the choice of representative of the $Ext$-element. \end{prop} \begin{proof} The coproduct $Y_0 \coprod_{Y_1/Y_2} A$ is equivalently \begin{displaymath} coker(Y_1/Y_2 \stackrel{(incl,-c)}{\to} Y_0 \oplus A) \,. \end{displaymath} For a different representative $\tilde c$ of $[c]$ there is by construction a \begin{displaymath} \itexarray{ Y_1 &\stackrel{\tilde c - c}{\to}& A \\ {}^{\mathllap{\partial_0}}\downarrow & \nearrow_{\lambda} \\ Y_0 } \,. \end{displaymath} Define from this a map between the two cokernels induced by the [[commuting diagram]] \begin{displaymath} \itexarray{ Y_1/Y_2 &\stackrel{id}{\to}& Y_1/Y_2 \\ \downarrow^{\mathrlap{(id,-c)}} && \downarrow^{\mathrlap{(id,-\tilde c)}} \\ Y_0 \oplus A &\stackrel{\left(\itexarray{ id & 0 \\ \lambda & id }\right)}{\to}& Y_0 \oplus A } \,. \end{displaymath} By construction this respects the inclusion of $A \stackrel{(0,id)}{\hookrightarrow} Y_0 \oplus A \to Y_0 \coprod_{Y_1/Y_2} A$. It also manifestly respects the projection to $G$. Therefore this defines a morphism and hence by remark \ref{MorphismOfExtensionIsIsomorphism} even an isomorphism of extensions. \end{proof} \begin{prop} \label{}\hypertarget{}{} The functions \begin{displaymath} Extr \colon Ext(G,A) \leftrightarrow Ext^1(G,A) \colon Rec \end{displaymath} from def. \ref{MapFromExtensionsToExtGroup} to def. \ref{ExtensionFromAnElementOfExt1} are [[inverses]] of each other and hence exhibit a [[bijection]] between extensions of $G$ by $A$ and $Ext^1(G,A)$. \end{prop} \begin{proof} By straightforward unwinding of the definitions. In one direction, starting with a $c \in Ext^1(G,A)$ and constructing the extension by pushout, the resulting pushout diagram \begin{displaymath} \itexarray{ Y_1 &\stackrel{c}{\to}& A \\ \downarrow && \downarrow \\ Y_0 &\to& Y_0 \coprod^c_{Y_1/Y_2} A \\ \downarrow && \downarrow \\ G &\stackrel{id}{\to}& G } \end{displaymath} at the same time exhibits $c$ as the cocycle extracted from the extension $A \to Y_0 \coprod^c_{Y_1/Y_2} A \to G$. Conversely, when starting with an extension $A \to \hat G \to G$ then extracting a $c$ by a choice of projective resolution and constructing from that another extension by pushout, the [[universal property]] of the pushout yields a morphism of exensions, which by remark \ref{MorphismOfExtensionIsIsomorphism} is an isomorphism of extensions, hence an equality in $Ext(G,A)$. \end{proof} \hypertarget{relation_to_syzygies}{}\subsubsection*{{Relation to syzygies}}\label{relation_to_syzygies} (\ldots{}) [[syzygy]] (\ldots{}) \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{Lenght1ResolutionsOfAbelianGroups}{}\subsubsection*{{Length-1 resolutions}}\label{Lenght1ResolutionsOfAbelianGroups} \begin{prop} \label{}\hypertarget{}{} Assuming the [[axiom of choice]], over $R = \mathbb{Z}$ hence in $R Mod =$ [[Ab]] every object $A$ has a [[projective resolution]], even a [[free resolution]], of length 1, hence a [[short exact sequence]] \begin{displaymath} 0 \to F_1 \to F_0 \to A \to 0 \end{displaymath} with $F_1$ and $F_0$ being [[free abelian groups]]. \end{prop} \begin{proof} By the discussion at \href{free+module#SubmodulesOfFreeModules}{free modules - submodules of free modules} a [[subgroup]] of a [[free abelian group]] is again free. Therefore for $p \colon F_0 \to A$ the surjection out of the free group $F_0 \coloneqq F(U(A))$ on the underlying set of $A$, setting $F_1 \coloneqq ker(p)$ yields the desired short exact sequence. \end{proof} The same argument holds true for $R$ any [[principal ideal domain]]. \hypertarget{ProjectiveResolutionsForGroupCocycles}{}\subsubsection*{{Projective resolutions adapted to abelian group cocycles}}\label{ProjectiveResolutionsForGroupCocycles} (\ldots{}) \hypertarget{ProjectiveResolutionsForGroupCohomology}{}\subsubsection*{{Projective resolutions adapted to general group cohomology}}\label{ProjectiveResolutionsForGroupCohomology} Let $G$ be a [[discrete group]]. Write $\mathbb{Z}[G]$ for the [[group ring]] over $G$. Notice from \emph{\href{module#AbelianGroupsWithGAction}{module -- Abelian groups with G-action as modules over the group ring}} that linear $G$-[[actions]] on [[abelian groups]] $A$ are equivalently $\mathbb{Z}[G]$-[[module]] structures in $A$. We discuss how [[cocycles]] in the [[group cohomology]] of $G$ with [[coefficients]] in such a [[module]] $A$ are naturally encoded in morphisms out of projective resolutions of the trivial $\mathbb{Z}[G]$-module. \begin{defn} \label{AugmentationMap}\hypertarget{AugmentationMap}{} Write \begin{displaymath} \epsilon \colon \mathbb{Z}[G] \to \mathbb{Z} \end{displaymath} for the [[homomorphism]] of [[abelian groups]] which forms the sum of $R$-[[coefficients]] of the [[formal linear combinations]] that constitute the group ring \begin{displaymath} \epsilon \colon r \mapsto \sum_{g \in G} r_g \,. \end{displaymath} This is called the [[augmentation]] map. \end{defn} \begin{defn} \label{ProjectiveResolutionForZAsZGModule}\hypertarget{ProjectiveResolutionForZAsZGModule}{} For $n \in \mathbb{N}$ let \begin{displaymath} Q^u_n \coloneqq F(U(G)^{\times^{n}}) \end{displaymath} be the [[free module]] over the [[group ring]] $\mathbb{Z}[G]$ on $n$-[[tuples]] of elements of $G$ (hence $Q^u_0 \simeq \mathbb{Z}[G]$ is the free module on a single generator). For $n \geq 1$ let $\partial_{n-1} \colon Q^u_n \to Q^u_{n-1}$ be given on [[basis]] elements by \begin{displaymath} \partial_{n-1} (g_1, \cdots, g_n) \coloneqq g_1 [g_2, \cdots, g_n] + \sum_{i = 1}^{n-1} (-1)^i [g_1, \cdots, g_i g_{i+1}, g_{i+2}, \cdots, g_n] + (-1)^n [g_1, \cdots, g_{n-1}] \,, \end{displaymath} where in the first summand we have the coefficient $g_1 \in G \hookrightarrow \mathbb{Z}[G]$ times the basis element $[g_2, \cdots, g_n]$ in $F(U(G)^{n-1})$. In particular \begin{displaymath} \partial_0 \colon [g] \mapsto g[*] - [*] = g-e \in \mathbb{Z}[G] \,. \,. \end{displaymath} Write furthermore $Q_n$ for the [[quotient]] module $Q^u_n \to Q^n$ which is the [[cokernel]] of the inclusion of those elements for which one of the $g_i$ is the unit element. \end{defn} \begin{prop} \label{}\hypertarget{}{} The construction in def. \ref{ProjectiveResolutionForZAsZGModule} defines [[chain complexes]] $Q^u_\bullet$ and $Q_\bullet$ of $\mathbb{Z}[G]$-modules. Moreover, with the augmentation map of def. \ref{AugmentationMap} these are projective resolutions \begin{displaymath} \epsilon \colon Q^u_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z} \end{displaymath} \begin{displaymath} \epsilon \colon Q_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z} \end{displaymath} of $\mathbb{Z}$ equipped with the trivial $\mathbb{Z}[G]$-module structure in $\mathbb{Z}[G]$[[Mod]]. \end{prop} \begin{proof} The proof that we have indeed a chain complex is much like the proof of the existence of the [[alternating face map complex]] of a [[simplicial group]], because writing \begin{displaymath} \partial^0_n [g_1, \cdots, g_n] \coloneqq g_1 [g_2, \cdots, g_n] \end{displaymath} \begin{displaymath} \partial^i_n [g_1, \cdots, g_n] \coloneqq [g_1, \cdots, g_i g_{i+1}, g_{i+2}, \cdots, g_n] \;\; for 1 \leq i \leq n-1 \end{displaymath} \begin{displaymath} \partial_n [g_1, \cdots, g_n] \coloneqq [g_1, \cdots, g_{n-1}] \end{displaymath} one finds that these satisfy the [[simplicial identities]] and that $\partial_n = \sum_{i = 0}^n (-1)^i \partial^i_n$. That the augmentation map is a [[quasi-isomorphism]] is equivalent, by remark \ref{ProjectiveResolutionInComponents}, to the [[augmentation]] \begin{displaymath} \cdots \stackrel{\partial_2}{\to} \mathbb{Z}[G]^2 \stackrel{\partial_1}{\to} \mathbb{Z}[G] \stackrel{\epsilon}{\to} \mathbb{Z} \to 0 \end{displaymath} being an [[exact sequence]]. In fact we show that it is a [[split exact sequence]] by constructing for the canonical chain map to the 0-complex a [[null homotopy]] $s_\bullet$. To that end, let \begin{displaymath} s_{-1} \colon \mathbb{Z} \to Q^u_0 \end{displaymath} be given by sending $1 \in \mathbb{Z}$ to the single basis element in $Q^u_0 \coloneqq \mathbb{Z}[G][*] \simeq \mathbb{Z}[G]$, and let for $n \in \mathbb{N}$ \begin{displaymath} s_n \colon Q^u_n \to Q^u_{n+1} \end{displaymath} be given on basis elements by \begin{displaymath} s_n(g[g_1, \cdots, g_n]) \coloneqq [g, g_1, \cdots, g_n] \,. \end{displaymath} In the lowest degrees we have \begin{displaymath} \epsilon \circ s_{-1} = id_{\mathbb{Z}} \end{displaymath} because \begin{displaymath} \epsilon(s_{-1}(1)) = \epsilon([*]) = \epsilon(e) = 1 \end{displaymath} and \begin{displaymath} \partial_0 \circ s_0 + s_{-1}\circ \epsilon = id_{Q^u_0} \end{displaymath} because for all $g \in G$ we have \begin{displaymath} \begin{aligned} \partial_0 (s_0(g[*])) + s_{-1}(\epsilon(g[*])) & = \partial_0( [g] ) + s_{-1}(1) \\ & = g[*] - [*] + [*] \\ & = g[*] \end{aligned} \,. \end{displaymath} For all remaining $n \geq 1$ we find \begin{displaymath} \partial_n \circ s_n + s_{n-1} \circ \partial_{n-1} = id_{Q^u_n} \end{displaymath} by a lengthy but straightforward computation. This shows that every cycle is a boundary, hence that we have a resolution. Finally, since the chain complex $Q^u_\bullet$ consists by construction degreewise of [[free modules]] hence in particular of a [[projective module]], it is a projective resolution. \end{proof} \begin{prop} \label{}\hypertarget{}{} For $A$ an [[abelian group]] equipped with a linear $G$-[[action]] and for $n \in \mathbb{N}$, the degree-$n$ [[group cohomology]] $H^n_{grp}(G, A)$ of $G$ with [[coefficients]] in $A$ is equivalently given by \begin{displaymath} \begin{aligned} H^n_{Grp}(G,A) & \simeq Ext^n_{\mathbb{Z}[G]}(\mathbb{Z}, A) \\ & \simeq H^n(Hom_{\mathbb{Z}[G]}(Q^u_n, A)) \\ & \simeq H^n(Hom_{\mathbb{Z}[G]}(Q_n, A)) \,. \end{aligned} \,, \end{displaymath} where on the right we canonically regard $A \in \mathbb{Z}[G]$[[Mod]]. \end{prop} \begin{proof} By the [[free functor]] [[adjunction]] we have that \begin{displaymath} Hom_{\mathbb{Z}[G]}(F^u_n, A) \simeq Hom_{Set}(U(G)^{\times n}, U(A)) \end{displaymath} is the set of [[functions]] from $n$-tuples of elements of $G$ to elements of $A$. It is immediate to check that these are in the [[kernel]] of $Hom_{\mathbb{Z}[G]}(\partial_{n}, A)$ precisely if they are [[cocycles]] in the [[group cohomology]] (by comparison with the explicit formulas there) and that they are group cohomology [[coboundaries]] precisely if they are in the [[image]] of $Hom_{\mathbb{Z}[G]}(\partial_{n-1}, A)$. This establishes the first equivalences. Similarly one finds that $H^n(Hom(F_n, A)))$ is the sub-group of \emph{normalized} cocycles. By the discussion at \emph{[[group cohomology]]} these already support the entire group cohomology (every cocycle is comologous to a normalized one). \end{proof} \hypertarget{CohomologyOfCyclicGroups}{}\subsubsection*{{Cohomology of cyclic groups}}\label{CohomologyOfCyclicGroups} Let $G = C_k$ be a [[cyclic group]] of finite [[order]] $k$, with generator $g$. We discuss the [[group cohomology]] of $G$, as discussed at \emph{\href{http://ncatlab.org/nlab/show/group+cohomology#InTermsOfHomologicalAlgebra}{group cohomology - In terms of homological algebra}}. Define special elements in the [[group algebra]] $\mathbb{Z}G$: \begin{displaymath} N \coloneqq 1 + g + g^2 + \ldots + g^{k-1} \end{displaymath} \begin{displaymath} \, \end{displaymath} \begin{displaymath} D \coloneqq g - 1, \end{displaymath} and denote the corresponding multiplications by these elements by the same letters $N, D \colon \mathbb{Z}G \to \mathbb{Z}G$. Then a very simple and useful projective resolution of the trivial $\mathbb{Z}G$-module $\mathbb{Z}$ is based on an [[exact sequence]] of $\mathbb{Z}G$-[[modules]] \begin{displaymath} \ldots \stackrel{N}{\to} \mathbb{Z}G \stackrel{D}{\to} \mathbb{Z}G \stackrel{N}{\to} \mathbb{Z}G \stackrel{D}{\to} \mathbb{Z}G \to \mathbb{Z} \to 0 \end{displaymath} where the last map $\mathbb{Z}G \to \mathbb{Z}$ is induced from the trivial group homomorphism $G \to 1$, hence is the map that forms the sum of all [[coefficients]] of all group elements. It follows from this resolution that the [[cochain cohomology|cohomology groups]] $H^n(C_k, A)$ for a $C_k$-[[module]] $A$ are periodic of order 2: \begin{displaymath} H^{n+2}(C_k, A) \cong H^n(C_k, A) \end{displaymath} for $n \geq 1$. More precisely, \begin{prop} \label{}\hypertarget{}{} For $G = C_k$, we have \begin{itemize}% \item $H^0(G, A) = A^G = \ker(D) \colon A \to A$, \item $H^{2 j + 1}(G, A) = \ker(N)/im(D)$ for $j \geq 0$, \item $H^{2 j}(G, A) = \ker(D)/im(N)$ for $j \geq 1$. \end{itemize} \end{prop} A well-known calculation in the cohomology of cyclic groups is \textbf{[[Hilbert's Theorem 90]]}. \begin{theorem} \label{}\hypertarget{}{} Suppose $K$ be a finite [[Galois extension]] of a [[field]] $k$, with a cyclic [[Galois group]] $G = \langle g \rangle$ of order $n$. Regard the [[multiplicative group]] $K^\ast$ as a $G$-module. Then $H^1(G, K^\ast) = 0$. \end{theorem} \begin{proof} Let $\sigma \in \mathbb{Z}G$, and denote the action of $\sigma$ on an element $\beta \in K$ by exponential notation $\beta^\sigma$. The action of the element $N \in \mathbb{Z}G$ is \begin{displaymath} \beta^N = \beta^{1 + g + \ldots + g^{n-1}} = \beta \cdot \beta^g \cdot \ldots \beta^{g^{n-1}} \end{displaymath} which is precisely the \emph{norm} $N(\beta)$. We are to show that if $N(\beta) = 1$, then there exists $\alpha \in K$ such that $\beta = \alpha/g(\alpha)$. By the lemma that follows, the homomorphisms $1, g, \ldots, g^{n-1}: K^\ast \to K^\ast$ are, when considered as elements in a vector space of $K$-valued functions, $K$-linearly independent. It follows in particular that \begin{displaymath} 1 + \beta g + \beta^{1+g}g^2 + \ldots + \beta^{1 + g + \ldots + g^{n-2}}g^{n-1} \end{displaymath} is not identically zero, and therefore there exists $\theta \in K^\ast$ such that the element \begin{displaymath} \alpha = \theta + \beta \theta^g + \beta^{1+g}\theta^{g^2} + \ldots + \beta^{1 + g + \ldots + g^{n-2}}\theta^{g^{n-1}} \end{displaymath} is non-zero. Using the fact that $N(\beta) = 1$, one may easily calculate that $\beta \alpha^g = \alpha$, as was to be shown. \end{proof} The next result may be thought of as establishing ``independence of characters'' (where ``[[group character|characters]]'' are valued in the [[multiplicative group]] of a field): \begin{lemma} \label{}\hypertarget{}{} Let $K$ be a [[field]], let $G$ be a [[monoid]], and let $\chi_1, \ldots, \chi_n \colon G \to K^\ast$ be distinct monoid [[homomorphisms]]. Then the [[functions]] $\chi_i$, considered as functions valued in $K$, are $K$-[[linear independence|linearly independent]]. \end{lemma} \begin{proof} A single $\chi \colon G \to K^\ast$ obviously forms a linearly independent set. Now suppose we have an equation \begin{displaymath} a_1 \chi_1 + \ldots + a_n \chi_n = 0 \end{displaymath} where $a_i \in K$, and assume $n$ is as small as possible. In particular, no $a_i$ is equal to $0$, and $n \geq 2$. Choose $g \in G$ such that $\chi_1(g) \neq \chi_2(g)$. Then for all $h \in G$ we have \begin{displaymath} a_1 \chi_1(g h) + \ldots + a_n \chi_n(g h) = 0 \end{displaymath} so that \begin{displaymath} a_1 \chi_1(g) \chi_1 + \ldots + a_n \chi_n(g)\chi_n = 0. \end{displaymath} Dividing equation 2 by $\chi_1(g)$ and subtracting from it equation 1, the first term cancels, and we are left with a shorter relation \begin{displaymath} (a_2\frac{\chi_2(g)}{\chi_1(g)} - a_2)\chi_2 + \ldots = 0 \end{displaymath} which is a [[contradiction]]. \end{proof} \hypertarget{RelatedConcepts}{}\subsection*{{Related concepts}}\label{RelatedConcepts} \begin{itemize}% \item [[projective object]], [[projective presentation]], [[projective cover]], \textbf{projective resolution} \begin{itemize}% \item [[projective module]] \end{itemize} \item [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]] \begin{itemize}% \item [[injective module]] \end{itemize} \item [[free object]], [[free resolution]] \begin{itemize}% \item [[free module]] \end{itemize} \item flat object, [[flat resolution]] \begin{itemize}% \item [[flat module]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} For instance section 4.5 of \begin{itemize}% \item [[Pierre Schapira]], \emph{Categories and homological algebra} (2011) (\href{http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf}{pdf}) \end{itemize} or sections 3.1 and 4.2 in \begin{itemize}% \item [[Peter May]], \emph{Notes on Tor and Ext} (\href{http://www.math.uchicago.edu/~may/MISC/TorExt.pdf}{pdf}) \end{itemize} or section 4 of \begin{itemize}% \item [[Daniel Murfet]], \emph{Derived functors} (\href{http://therisingsea.org/notes/DerivedFunctors.pdf}{pdf}) \end{itemize} [[!redirects projective resolutions]] [[!redirects injective resolution]] [[!redirects injective resolutions]] [[!redirects acyclic resolution]] [[!redirects acyclic resolutions]] \end{document}