projective object

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

### General

+-- {: .num_defn}
###### Definition

An [[object]] $P$ of a [[category]] $C$ is **projective** (with respect to [[epimorphisms]]) if it has the [[left lifting property]] against [[epimorphisms]].

=--

This means that $P$ is projective if for any

1. [[morphism]]$\;f \colon P \to B$

1. [[epimorphism]]$\;q \colon A \twoheadrightarrow B$,

{#DefiningLiftingDiagram} $f$ factors through $q$, making a [[commuting diagram]] of this form:

\begin{tikzcd}[sep=20pt]
  &&
  A
  \ar[dd, ->>, "{ q }"]
  \\
  \\
  P
  \ar[rr, "f"]
  \ar[
    uurr,
    dashed,
    "{ \exists \, \widehat{f} }"
  ]
  &&
  B
\end{tikzcd}

Yet another way to say  this is that:

+-- {: .num_defn #ByCovHomPreservingEpis}
###### Definition

An object $P$ is projective precisely if the [[hom-functor]] $Hom(P,-)$ preserves [[epimorphisms]].

=--

+-- {: .num_remark}
###### Remark

This generalizes the notion of _[[projective modules]]_ over a [[ring]].

=--

+-- {: .num_remark}
###### Remark

There are variations of the definition where "epimorphism" is replaced by some other type of morphism, such as a [[regular epimorphism]] or [[strong epimorphism]] or the left class in some [[orthogonal factorization system]].  In this case one may speak of **regular projectives** and so on.  In a [[regular category]] "projective" almost always means "regular projective."

If the ambient category has an [[initial object]] $\emptyset$, then the projective objects $P$ are those for which $\emptyset \stackrel{\exists!}{\to} P$ is called a [[projective morphism]].

=--

+-- {: .num_remark}
###### Remark

The [[duality|dual]] notion is that of _[[injective objects]]_.

=--

+-- {: .num_remark}
###### Remark
There is also a stronger notion of projective object, where $\hom(C, -)$ preserves coequalizers. See more at [[tiny object]], which can be defined as projective [[connected objects]] with "projective" used in this stronger sense.
=--

+-- {: .num_defn #EnoughProjectives}
###### Definition

A category $C$ has **enough projectives** if for every object $X$ there is an [[epimorphism]] $P\to X$ where $P$ is projective.

Equivalently: if every object admits a _[[projective presentation]]_.

=--

This terminology refers to the existence of [[projective resolutions]], prop. \ref{EnoughIsEnough} below.

### In abelian categories
 {#InAbelianCats}

Projective objects and [[injective objects]] in [[abelian categories]] $\mathcal{A}$ are of central interest in [[homological algebra]]. Here they appear as parts of [[cofibrant resolutions]] and [[fibrant resolutions]], respectively, in the [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$, with respect to one of the two standard [[model structures on chain complexes]].

#### Equivalent characterizations
 {#EquivalentCharacterizationInAbelianCats}

+-- {: .num_prop #EquivalenceOfDefinitionsInAbelian}
###### Proposition

The following are equivalent

1. $X \in \mathcal{A}$ is a projective object (in that it has the [[left lifting property]] against [[epimorphisms]], def. \ref{ByCovHomPreservingEpis});

1. The [[hom-functor]] $Hom(X,-) : \mathcal{A} \to $ [[Ab]] is an [[exact functor]].

=--

+-- {: .num_remark }
###### Remark

For every object $X$, the hom-functor $Hom(X,-)$ is a [[left exact functor]]. So the second statement is equivalently that it is also right exact precisely if $X$ is projective.

=--

+-- {: .proof}
###### Proof
**of prop. \ref{EquivalenceOfDefinitionsInAbelian}**

Let

$$
  0 \to A \stackrel{i}{\hookrightarrow} B \stackrel{p}{\to} C \to 0
$$

be a [[short exact sequence]]. The exactness at $A$ and $B$ together is equivalent to the statement that $i = \ker(p)$. Since as remarked above $Hom(X, -)$ is left exact, it preserves [[kernels]] and so $Hom(X, i) = \ker(Hom(X, p))$, giving exactness of the sequence

$$
0 \to  Hom(X,A)
    \stackrel{Hom(X,i)}{\to}
  Hom(X,B)
   \stackrel{Hom(X,p)}{\to}
  Hom(X,C)
  \,.
$$

Therefore we are reduced to showing that $Hom(X,p)$ is an [[epimorphism]] precisely if $X$ is projective. But this is def. \ref{ByCovHomPreservingEpis}.

=--

#### Projective resolutions

Let $\mathcal{A}$ be an [[abelian category]].

+-- {: .num_defn}
###### Definition

For $N \in \mathcal{A}$ an object, a **[[projective resolution]]** of $N$ is a [[chain complex]] $(Q N)_\bullet \in Ch_\bullet(\mathcal{A})$ equipped with a [[chain map]]

$$
  Q N \to N
$$

(with $N$ regarded as a complex concentrated in degree 0) such that

1. this morphism is a [[quasi-isomorphism]] (this is what makes it a [[resolution]]), which is equivalent to

   $$ \cdots \to (Q N)_1 \to (Q N)_0 \to N $$

   being an [[exact sequence]];

1.  all whose entries $(Q N)_n$ are projective objects.

=--

+-- {: .num_remark}
###### Remark

This means precisely that $Q N \to N$ is an [[cofibrant resolution]] with respect to the standard [[model structure on chain complexes]] (see [here](model%20structure%20on%20chain%20complexes#StandardQuillenOnBounded)) for which the fibrations are the positive-degreewise epimorphisms. Notice that in this model structure every object is fibrant, so that cofibrant resolutions are the only resolutions that need to be considered.

=--

+-- {: .num_prop #EnoughIsEnough}
###### Proposition

If $\mathcal{A}$ has _enough projectives_ in the sense of def. \ref{EnoughProjectives}, then every object has a projective resolution.

=--

## Examples

+-- {: .num_remark #ProjectiveInSetsIsAxiomOfChoice}
###### Remark

The [[axiom of choice]] can be phrased as "all objects of [[Set]] are projective."

See also _[[internally projective object]]_ and _[[COSHEP]]_.

=--

+-- {: .num_example}
###### Example

If $C$ has [[pullback|pullbacks]] and epimorphisms are preserved by pullback, as is the case in a [[pretopos]], then $P$ is projective iff any epimorphism $Q\to P$ is [[split epimorphism|split]].

=--

+-- {: .num_example #ProjectiveObjectsInAbAreFreeGroups}
###### Example

An object in [[Ab]], an [[abelian group]], is projective precisely if it is a [[free group]].

=--

+-- {: .num_example}
###### Example

For $R$ a [[commutative ring]], an object in  $R$[[Mod]], an $R$-[[module]], is projective (a [[projective module]], see there for more details) precisely if it is a [[direct sum|direct summand]] of a [[free module]]. See at _[[projective module]]_ for more on this.

=--

+-- {: .num_example}
###### Example

The projective objects in [[compact topological space|compact]] [[Hausdorff topological spaces]] are precisely the [[extremally disconnected profinite sets]].

=--

## Properties

### Existence of enough projectives
 {#ExistenceOfEnoughProjectives}

We list examples of classes of categories that have enough projective, according to def. \ref{EnoughProjectives}.

+-- {: .num_prop #ModHasEnoughProjectives}
###### Proposition

Assuming the [[axiom of choice]], for $R$ a [[ring]] the category $R$[[Mod]] of [[modules]] over $R$ is has enough projectives.

=--

See at _[[projective module]]_ for more.

+-- {: .num_lemma #FreeModulesAreProjective}
###### Lemma

Assuming the [[axiom of choice]],
a [[free module]] $N \simeq R^{(S)}$ is projective.

=--

+-- {: .proof}
###### Proof

By remark \ref{ProjectiveInSetsIsAxiomOfChoice} and the [[free-forgetful adjunction]].

More explicitly: if $S \in Set$ and $F(S) = R^{(S)}$ is the [[free module]] on $S$, then a module homomorphism $F(S) \to N$ is specified equivalently by a [[function]] $f : S \to U(N)$ from $S$ to the underlying set of $N$, which can be thought of as specifying the images of the unit elements in $R^{(S)} \simeq \oplus_{s \in S} R$ of the ${\vert S\vert}$ copies of $R$.

Accordingly then for $\tilde N \to N$ an epimorphism, the underlying function $U(\tilde N) \to U(N)$ is an epimorphism, and by remark \ref{ProjectiveInSetsIsAxiomOfChoice} the [[axiom of choice]] in [[Set]] says that we have all lifts $\tilde f$ in

$$
  \array{
     && U(\tilde N)
     \\
     & {}^{\tilde f} \nearrow & \downarrow
     \\
     S &\stackrel{f}{\to}& U(N)
  }
  \,.
$$

By [[adjunction]] these are equivalently lifts of module homomorphisms

$$
  \array{
     && \tilde N
     \\
     & \nearrow & \downarrow
     \\
     R^{(S)} &\stackrel{}{\to}& N
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof
**of prop. \ref{ModHasEnoughProjectives}**

For $N \in R Mod$ and $U(N) \in Set$ its underlying set,
consider the $R$-linear map

$$
  \left(
    \oplus_{n \in U(N)} R
  \right)
  \to
  N
$$

out of the [[direct sum]] of ${\vert U(N)\vert}$ copies of $N$,
which sends the unit element $1 \in R_{n}$ of the $n$-labeled copy of $R$ to the corresponding element of $N$ (and is thus fixed on all other elements by $R$-linearity).

This is clearly a [[surjection]] and by lemma \ref{FreeModulesAreProjective} it is a surjection out of a projective object.

=--

A slightly subtle point is that there is no guarantee that the free module $F U(M)$ is actually projective, unless one assumes some form of the [[axiom of choice]]. Since the axiom of choice is not available in all [[toposes]], one cannot use this procedure in general to construct, say, projective resolutions of [[abelian sheaves]], hence in the abelian category $Ab(E)$ of abelian [[group objects]] in a general [[Grothendieck topos]] $E$ (even though one can construct
[[free resolutions]]), such as needed in general in [[abelian sheaf cohomology]].

There are however weak forms of the axiom of choice that hold in many toposes, such as the [[presentation axiom]], aka _[[COSHEP]]_. We have the following result:

+-- {: .num_prop #EnoughWithCOSHEP}
###### Proposition

Let $E$ be a [[W-pretopos]] that satisfies [[COSHEP]].
Then $Ab(E)$ has enough projectives.

=--

The idea of the proof is that under COSHEP, the underlying object of an abelian group $A$ in $E$ admits an [[epimorphism]] from a projective object $p \colon X \to U(A)$ in $E$. Then the corresponding $F(X) \to A$ is an epimorphism out of a projective in $Ab(E)$, for this map is a composite of epimorphisms

$$F(X) \stackrel{F(p)}{\to} F U(A) \stackrel{\varepsilon_A}{\to} A$$

(the first is epic because left adjoints preserve epis, whereas the second map, the component of the counit $\varepsilon \colon F U \to id$ at $A$, is epic because $U$ is faithful).

## Related concepts

* [[choice object]]

* [[COSHEP]]

* **projective object**, [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

  * [[internally projective object]]

  * [[compact projective object]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

For instance section 4.3 of

* [[Pierre Schapira]], _Categories and homological algebra_ (2011) ([pdf](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/HomAl.pdf))

[[!redirects enough projectives]]
[[!redirects projective objects]]


projective plane

# Projective planes

* table of contents
{: toc}

## Idea

A projective plane is a [[projective space]] of [[dimension]] 2.  However, projective planes over [[fields]] (or more generally [[division rings]] or [[ternary rings]]) can be characterized axiomatically with a small list of "synthetic" axioms.

## Synthetic Definition

A **projective plane** consists of

* a set of "points"
* a set of "lines"
* a relation between points and lines called "incidence"

satisfying the following axioms:

* there exist at least one point and at least one line
* every pair of distinct points lies on exactly one line
* every pair of distinct lines meet in exactly one point
* every line has at least three points
* not all points are on the same line

Some further axioms which may be added are:

* the diagonal points of a [[complete quandrangle]] are not collinear ("Fano's axiom")

* if the lines joining corresponding vertices of two triangles are concurrent, then the intersections of corresponding sides are collinear ([[Desargues' theorem]])

* if the vertices of a hexagon lie alternately on two lines, then the intersections of opposite sides are collinear ([[Pappus' theorem]])

Of these, Pappus' theorem implies Desargues' theorem.

## Analytic projective planes

For any [[field]] $F$, we can construct the projective plane $F P^2$ in several ways:

* As the set of lines through the origin in $F^3$
* As the set of nonzero vectors in $F^3$ modulo scalar multiples (homogeneous coordinates)
* As the set of projections from $F^3$ onto 1-dimensional linear subspaces
* By starting with $F^2$ and adding a "line at infinity"

The resulting plane satisfies Pappus' theorem, hence also Desargues' theorem.  It satisfies Fano's axiom iff the [[characteristic]] of $F$ is $\neq 2$.

These methods also work for any [[division ring]], such as the [[quaternions]].  In this case, Pappus' theorem doesn't hold, but Desargues' theorem still does.

The [[octonions]] are not associative and this breaks the first two methods, but the latter two can be made to work, resulting in the [[Cayley plane]].  In this case Desargues' theorem does not hold in general, but there are special cases of it which do (corresponding to the fact that the octonions are "alternative").

## Equivalence of synthetic and analytic

In any projective plane, we can define a "scalar" to be an ordered set of four collinear points $(A,B,C,D)$ of which no more than two are equal.  Two scalars are considered equal if they are projectively related, i.e. the four lines joining corresponding points are concurrent.  If Desargues' theorem holds, we can define addition and multiplication on the scalars (omitting one of them that acts like $\infty$) making them into a division ring $F$ such that our plane is isomorphic to $F P^2$.  The ring $F$ is commutative (hence a field) iff Pappus' theorem also holds, and has characteristic $\neq 2$ iff Fano's axiom holds.

If Desargues' theorem fails, then we can still construct a sort of algebraic structure on the scalars, called a [[ternary ring]], which suffices to reconstruct our plane.  However, distinct ternary rings can give rise to isomorphic projective planes, in contrast to the situation for fields and division rings.

## Examples

* [[real projective plane]]

* [[complex projective plane]]

* [[quaternionic projective plane]]

* [[octonionic projective plane]]

## Related pages

* [[projective line]]

## References

* A. Kryftis, _A constructive approach to projective and affine planes_ , PhD Cambridge 2015. ([arXiv:1601.04998](http://arxiv.org/abs/1601.04998))
[[!redirects projective planes]]


projective presentation

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### Context
#### Category theory
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[[!include category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _projective presentation_ of an [[object]] is a realization of that object as a suitable [[quotient]] of a [[projective object]].

In [[homological algebra]] projective presentations can sometimes be used in place of genuine [[projective resolutions]] in the computation of [[derived functors]]. See for instance at _[[Ext-functor]]_ for examples.

The [[duality|dual]] notion is that of [[injective presentation]].

## Definition

### In abelian categories

Let $\mathcal{A}$ be an [[abelian category]]. For $X \in \mathcal{A}$ any [[object]], a _projective presentation_ of $X$ is a [[short exact sequence]] of the form

$$
  0 \to N \stackrel{i}{\hookrightarrow}
  P \stackrel{p}{\to}
  X \to 0
  \,,
$$

hence exhibiting $X$ as the [[cokernel]]

$$
  X \simeq coker(N \hookrightarrow P)
$$

such that  $P$ is a [[projective object]].

## Related concepts

* [[projective object]], **projective presentation**, [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

[[!redirects projective presentations]]


projective representation

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### Context
#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

A _projective representation_ of a [[group]] $G$ is a [[representation]] up to a central term: a [[group]] [[homomorphism]] $G\longrightarrow PGL(V)$, to the [[projective general linear group]] of some $\mathbb{K}$-[[vector space]] $V$.

## Properties
 {#Properties}

### The group extension and its cocycle
 {#GroupExtensionAndCocycle}

By construction, there is a [[short exact sequence]]

$$
  1 \to \mathbb{K}^\times \longrightarrow GL(V) \longrightarrow PGL(V)\to 1
$$

which exhibits $GL(V)$ as a [[group extension]] of $PGL(V)$ by the [[group of units]] $\mathbb{K}^\times$ of the [[ground field]]. This is classified by a 2-[[cocycle]] $c$ in [[group cohomology]] $H^2_{Grp}(PGL(V),\mathbb{K}^\times)$.

It is useful to re-express this equivalently in terms of [[homotopy theory]] via the discussion at [[looping and delooping]], by which [[group]] [[homomorphisms]] $\phi \colon G\longrightarrow H$ are equivalently maps $\mathbf{B}\phi \colon \mathbf{B}G\longrightarrow \mathbf{B}H$ between their [[delooping]] [[groupoids]].

In terms of this the above [[group extension]] and its classifying cocycle is exhibited by a [[homotopy fiber sequence]] of [[deloopings]] of the form

$$
 \array{
    \mathbf{B}\mathbb{K}^*& \longrightarrow & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \ast &\longrightarrow&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
  \,.
$$

### Relation to genuine representations
 {#RelationToGenuineRepresentations}

Via the projection $GL(V)\to PGL(V)=GL(V)/\mathbb{K}^\times$, every linear representation of $G$ induces a projective representation.

By the [[universal property]] of the [[homotopy pullback]], the discussion [above](#GroupExtensionAndCocycle) means that the [[obstruction]] to lift a given projective representation $\mathbf{B}\rho \colon \mathbf{B}G\longrightarrow \mathbf{B} PGL(V)$ to a linear representation $\hat \rho$

$$
 \array{
    &  & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    & {}^{\mathllap{\mathbf{B}\hat \rho}}\nearrow & \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
$$

is the class of the 2-[[cocycle]] $c(\rho)\coloneqq c \circ \mathbf{B}\rho$ in  the [[group cohomology]] class $H^2_{Grp}(G,\mathbb{K}^\times)$.

### As twisted linear representations

By the [above](#GroupExtensionAndCocycle) and the discussion at _[group extension -- Central group extensions -- Formulation in homotopy theory](group+extension#FormulationInHomotopyTheory)_ the cocycle map $\mathbf{B}c \colon \mathbf{B}PGL(V)\to \mathbf{B}^2 \mathbb{K}^\times$ of [[homotopy types]] may be represented by a [[zigzag]] ("[[infinity-anafunctor|2-anafunctor]]") of [[crossed complexes]] as

$$
  \array{
    \mathbf{B}(\mathbb{K}^\times \to GL(V)) &\longrightarrow& \mathbf{B}(\mathbb{K}^\times \to 1)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}PGL(V)
  }
  \,.
$$

Here the [[2-groupoid]] $\mathbf{B}(\mathbb{K}^\times \to GL(V))$ looks schematically like

$$
  \left\{
     \array{
       && \ast
       \\
       & {}^{\mathllap{p_1}}\nearrow &\Downarrow^{\mathrlap{k}}& \searrow^{\mathrlap{p_2}}
       \\
       \ast && \underset{k p_1 p_2}{\longrightarrow}&& \ast
     }
  \right\}
$$

This shows that a map $\mathbf{B}G\to \mathbf{B}PGL(V)$ may equivalently be represented by two [[functions]] (not group homomorphisms in general!)

1. $\rho \colon G\to GL(V)$

1.  $\lambda \colon G\times G\to \mathbb{K}^\times$

such that for all $g,h \in G$

1.  $\rho(g)\rho(h)=\lambda(g,h)\rho(g h)$

1. $\lambda$ is a [[group cohomology|group 2-cocycle]] on $G$ with values in $\mathbb{K}^*$ representing the above cohomology class of $c_\rho$.

This is the form in which projective representations are often discussed in the literature.

### As genuine representations after extensions

Alternatively, one may consider the [above](#RelationToGenuineRepresentations) diagram

$$
 \array{
    &  & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    & & \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
$$

and form the further [[pullback]] along $\mathbf{B}\rho$. By the [[pasting law]] this is (the [[delooping]] of) the [[group extension]] $\hat G$ of $G$ which is classified by $c(\rho)$:

$$
 \array{
    \mathbf{B}\hat G  & \stackrel{\mathbf{B}\tilde \rho}{\longrightarrow} & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    \downarrow & & \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
$$

This way the projective representation $\rho$ of $G$ induces a genuine linear representation $\tilde \rho$ of $\hat G$. One finds (this is a special case of the general discussion at [[twisted infinity-bundle]]) that this constitutes an equivalence between projective representations of $G$ and genuine representations of $\hat G$.

## Related concepts

* [[projectively flat connection]]

## References

(...)

Relation to [[twisted equivariant K-theory]]:

* José Manuel Gómez, Johana Ramírez, _A Decomposition of twisted equivariant K-theory_ ([arXiv:2001.02164](https://arxiv.org/abs/2001.02164))

[[!redirects projective representations]]


projective resolution

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### Context
#### Homological algebra
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[[!include homological algebra - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Projective and injective resolutions
* table of contents
{: toc}

## Idea

In the context of [[homological algebra]] a _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.

## 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.

### Resolution of an object
 {#ResolutionOfObject}

Let $\mathcal{A}$ be an [[abelian category]].

+-- {: .num_defn #InjectiveResolution}
###### Definition

For $X \in \mathcal{A}$ an [[object]], an **injective resolution** of $X$ is a [[cochain complex]] $J^\bullet \in Ch^\bullet(\mathcal{A})$ (in non-negative degree) equipped with a [[quasi-isomorphism]]

$$
  i : X \stackrel{\sim}{\to} J^\bullet
$$

such that $J^n \in \mathcal{A}$ is an [[injective object]] for all $n \in \mathbb{N}$.

=--

+-- {: .num_remark #InjectiveResolutionInComponents}
###### Remark

In components the quasi-isomorphism of def. \ref{InjectiveResolution}
is a [[chain map]] of the form

$$
  \array{
    X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots
    \\
    \downarrow^{\mathrlap{i^0}} && \downarrow && && \downarrow
    \\
    J^0 &\stackrel{d^0}{\to}& J^1 &\stackrel{d^1}{\to}& \cdots &\to&
    J^n &\stackrel{d^n}{\to}&\cdots
  }
  \,.
$$

Since the top complex is concentrated in degree 0, this being a [[quasi-isomorphism]] happens to be equivalent to the sequence

$$
  0 \to X \stackrel{i^0}{\to} J^0 \stackrel{d^0}{\to} J^1 \stackrel{d^1}{\to}
  J^2 \stackrel{d^2}{\to}
  \cdots
$$

being an [[exact sequence]]. In this form one often finds the definition of injective resolution in the literature.

=--

+-- {: .num_defn #ProjectiveResolution}
###### Definition

For $X \in \mathcal{A}$ an [[object]], a **projective resolution** of $X$ is a [[chain complex]] $J_\bullet \in Ch_\bullet(\mathcal{A})$ (in non-negative degree) equipped with a [[quasi-isomorphism]]

$$
  p : J_\bullet \stackrel{\sim}{\to} X
$$

such that $J_n \in \mathcal{A}$ is a [[projective object]] for all $n \in \mathbb{N}$.

=--

+-- {: .num_remark #ProjectiveResolutionInComponents}
###### Remark

In components the quasi-isomorphism of def. \ref{ProjectiveResolution}
is a [[chain map]] of the form

$$
  \array{
    \cdots &\stackrel{\partial_n}{\to}& J_n &\stackrel{\partial_{n-1}}{\to}&
    \cdots &\to& J_1 &\stackrel{\partial_0}{\to}& J_0
    \\
    && \downarrow && && \downarrow && \downarrow^{\mathrlap{p_0}}
    \\
    \cdots &\to& 0 &\to& \cdots &\to& 0 &\to& X
  }
  \,.
$$

Since the bottom complex is concentrated in degree 0, this being a [[quasi-isomorphism]] happens to be equivalent to the sequence

$$
  \cdots J_2 \stackrel{\partial_1}{\to} J_1 \stackrel{\partial_0}{\to} J_0 \stackrel{p_0}{\to}
  X \to 0
$$

being an [[exact sequence]]. In this form one often finds the definition of projective resolution in the literature.

=--

### $F$-Resolutions of an object
 {#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 _[[derived functor in homological algebra]]_. While projective resolutions in $\mathcal{A}$ are _sufficient_ for computing _every_ [[left derived functor]] on $Ch_\bullet(\mathcal{A})$ and injective resolutions are sufficient for computing _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 _necessary_. A weaker kind of resolution which is still sufficient is then often more convenient for applications. These _$F$-projective resolutions_ and _$F$-injective resolutions_, respectively, we discuss here. A special case of both are _$F$-[[acyclic resolutions]]_.

$\,$

Let $\mathcal{A}, \mathcal{B}$ be [[abelian categories]] and let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[additive functor]].

+-- {: .num_defn #FInjectives}
###### Definition

Assume that $F$ is [[left exact functor|left exact]]. An [[additive category|additive]] [[full subcategory]] $\mathcal{I} \subset \mathcal{A}$ is called **$F$-injective** (or: consisting of $F$-injective objects) if

1. for every object $A \in \mathcal{A}$ there is a [[monomorphism]] $A \to \tilde A$ into an object $\tilde A \in \mathcal{I} \subset \mathcal{A}$;

1. 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 $C \in \mathcal{I} \subset \mathcal{A}$;

1. 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 $\mathcal{B}$.

=--

And dually:

+-- {: .num_defn #FProjectives}
###### Definition

Assume that $F$ is [[right exact functor|right exact]]. An [[additive category|additive]] [[full subcategory]] $\mathcal{P} \subset \mathcal{A}$ is called **$F$-projective** (or: consisting of $F$-projective objects) if

1. for every object $A \in \mathcal{A}$ there is an [[epimorphism]] $\tilde A \to A$ from an object $\tilde A \in \mathcal{P} \subset \mathcal{A}$;

1. 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 $A \in \mathcal{P} \subset \mathcal{A}$;

1. 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 $\mathcal{B}$.

=--

For instance ([Schapira, def. 4.6.5](#Schapira)).

With the $\mathcal{I},\mathcal{P}\subset \mathcal{A}$ as above, we say:

+-- {: .num_defn #FProjectivesResolution}
###### Definition

For $A \in \mathcal{A}$,

* an **$F$-injective resolution** of $A$ is a [[cochain complex]] $I^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ and a [[quasi-isomorphism]]

  $$
    A \stackrel{\simeq_{qi}}{\to} I^\bullet
  $$

* an **$F$-projective resolution** of $A$ is a [[chain complex]] $Q_\bullet \in Ch_\bullet(\mathcal{P}) \subset Ch^\bullet(\mathcal{A})$ and a [[quasi-isomorphism]]

  $$
    Q_\bullet \stackrel{\simeq_{qi}}{\to} A
    \,.
  $$

=--

Let now $\mathcal{A}$ have enough projectives / enough injectives, respectively.

+-- {: .num_example #FAcyclicObjectsAreFProjectiveObjects}
###### Example

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

* if $F$ is [[left exact functor|left exact]], then $\mathcal{I} \coloneqq Ac$ is a subcategory of $F$-injective objects;

* if $F$ is [[right exact functor|right exact]], then $\mathcal{P} \coloneqq Ac$ is a subcategory of $F$-projective objects.

=--

+-- {: .proof}
###### Proof

Consider the case that $F$ is left 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}

$$
  0 \to A \to B \to C \to R^1 F(A) \to R^1 F(B) \to R^1 F(C) \to R^2 F(A) \to R^2 F(B) \to R^2 F(C) \to \cdot
  \,.
$$

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

$$
  0 \to 0 \to R^n F(C) \to 0
$$

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 $R^1 F(A) \simeq 0$.

=--

+-- {: .num_example #FAcyclicResolution}
###### Example

The $F$-projective/injective resolutions by [[acyclic objects]] as in example \ref{FAcyclicObjectsAreFProjectiveObjects} are called **$F$-acyclic resolutions**.

=--

### Resolution of a chain complex
 {#ResolutionOfAChainComplex}

The above definition \ref{ProjectiveResolution} of a projective resolution of an object has an immediate generalization to resolutions of chain complexes.

+-- {: .num_defn #ProjectiveResolutionOfChainComplex}
###### Definition

For $C_\bullet \in Ch_\bullet(\mathcal{A})$ a [[chain complex]], a **projective resolution** of $C$ is an [[exact sequence]] of chain complexes

$$
  \cdots
  \to Q_{\bullet,2}
  \to Q_{\bullet,1}
  \to  Q_{\bullet,0}
  \to C_\bullet
  \to 0
$$

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}.

=--

+-- {: .num_remark }
###### Remark

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.

=--

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]].

+-- {: .num_defn #FullyProjectiveResolutionOfChainComplex}
###### Definition

A projective resolution $Q_{\bullet,\bullet} \to C_\bullet$ of a chain complex $C_\bullet$, def. \ref{ProjectiveResolutionOfChainComplex}, is called **fully projective** (or **proper**) if furthermore for all $n \in \mathbb{N}$ the induced sequence of (horizontal) [[cycles]]

$$
  \cdots
   \to
  Z_{\bullet,2}
   \to
  Z_{\bullet,1}
   \to
  Z_{\bullet,0}
   \to
  Z(C)_\bullet
   \to
  0
$$

and (horizontal) [[boundaries]]

$$
  \cdots
   \to
  B_{\bullet,2}
   \to
  B_{\bullet,1}
   \to
  B_{\bullet,0}
   \to
  B(C)_\bullet
   \to
  0
$$

and (horizontal) [[homology groups]]

$$
  \cdots
   \to
  H_{\bullet,2}
   \to
  H_{\bullet,1}
   \to
  H_{\bullet,0}
   \to
  H(C)_\bullet
   \to
  0
$$

are each projective resolutions, def. \ref{ProjectiveResolution}, themselves.

=--

## Properties

### Existence and construction of resolutions for objects
 {#ExistenceAndConstruction}

We first discuss the existence of injective/projective resolutions, and then the [[functor|functoriality]] of their constructions.

+-- {: .num_prop #ExistenceOfInjectiveResolutions}
###### Proposition

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}.

=--

+-- {: .proof}
###### Proof

Let $X \in \mathcal{A}$ be the given object.
By remark \ref{InjectiveResolutionInComponents} we need to
construct an [[exact sequence]] of the form

$$
  0 \to X \to J^0 \stackrel{d^0}{\to} J^1 \stackrel{d^1}{\to}
  J^2 \stackrel{d^2}{\to} \cdots \to J^n \to \cdots
$$

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]]

$$
  X \hookrightarrow J^0
$$

hence an [[exact sequence]]

$$
  0 \to X \to J^0
  \,.
$$

Assume then by induction hypothesis that for $n \in \mathbb{N}$ an [[exact sequence]]

$$
  X \to J^0 \stackrel{d^0}{\to} \cdots \to J^{n-1} \stackrel{d^{n-1}}{\to} J^n
$$

has been constructed, where all the $J^\cdot$ are injective objects. Forming the [[cokernel]] of $d^{n-1}$ yields the [[short exact sequence]]

$$
  0 \to J^{n-1} \stackrel{d^{n-1}}{\to} J^n \stackrel{p}{\to} J^n/J^{n-1} \to 0
  \,.
$$

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

$$
  J^{n-1} \stackrel{d^{n-1}}{\to} J^n \stackrel{d^n \coloneqq i \circ p}{\longrightarrow} J^{n+1}
$$

is [[exact sequence|exact]] in the middle term. Therefore we now have an [[exact sequence]]

$$
  0 \to X \to J^0 \to \cdots \to J^{n-1} \stackrel{d^{n-1}}{\to} J^n \stackrel{d^{n}}{\to} J^{n+1}
$$

which completes the [[induction]] step.

=--

The following proposition is  [[duality|formally dual]] to prop. \ref{ExistenceOfInjectiveResolutions}.

+-- {: .num_prop #ExistenceOfInjectiveResolutions}
###### Proposition

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}.

=--

+-- {: .proof}
###### Proof

Let $X \in \mathcal{A}$ be the given object.
By remark \ref{ProjectiveResolutionInComponents} we need to
construct an [[exact sequence]] of the form

$$
  \cdots \stackrel{\partial_2}{\to}
  J_2 \stackrel{\partial_1}{\to}
  J_1 \stackrel{\partial_0}{\to}
  J_0 \to X \to 0
$$

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]]

$$
  J_0 \to X
$$

hence an [[exact sequence]]

$$
  J_0 \to X \to 0
  \,.
$$

Assume then by induction hypothesis that for $n \in \mathbb{N}$ an [[exact sequence]]

$$
  J_n \stackrel{\partial_{n-1}}{\to}
  J_{n-1}
  \to
  \cdots
  \stackrel{\partial_0}{\to}
  J_0
  \to
  X
  \to
  0
$$

has been constructed, where all the $J_\cdot$ are projective objects. Forming the [[kernel]] of $\partial_{n-1}$ yields the [[short exact sequence]]

$$
  0 \to ker(\partial_{n-1})
  \stackrel{i}{\to}
   J_n \stackrel{\partial_{n-1}}{\to}
  J_{n-1}
  \to
  0
  \,.
$$

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

$$
  J_{n+1} \stackrel{\partial_{n} \coloneqq i\circ p}{\to}
  J_n \stackrel{\partial_{n-1}}{\to}
$$

is [[exact sequence|exact]] in the middle term. Therefore we now have an [[exact sequence]]

$$
  J_{n+1}
    \stackrel{\partial_n}{\to}
  J_n
    \stackrel{\partial_{n-1}}{\to}
  \cdots
   \stackrel{\partial_0}{\to}
  J_0
  \to
  X
  \to
  0
  \,,
$$

which completes the [[induction]] step.

=--

+-- {: .num_prop #MapsOutOfExactIntoInjectiveAreNullHomotopic}
###### Proposition

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]]

$$
  \eta : 0 \Rightarrow f^\bullet
$$

=--

+-- {: .proof}
###### 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

$$
  f^n = \eta^{n+1} \circ d^n_X + d^{n-1}_J \circ \eta^n
  \,.
$$

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

$$
  g^n \coloneqq f^n - d_J^{n-1} \circ \eta^n
$$

and observe that

$$
  \begin{aligned}
    g^n \circ d_X^{n-1}
    & =
    f^n \circ d^{n-1}_X - d^{n-1}_J \circ \eta^n \circ d^{n-1}_X
    \\
    & = f^n \circ d^{n-1}_X - d^{n-1}_J \circ f^{n-1}
        + d^{n-1}_J \circ d^{n-2}_J \circ \eta^{n-1}
    \\
    & = 0 + 0
    \\
    & 0
  \end{aligned}
  \,.
$$

This means that $g^n$ factors as

$$
  X^n \to X^n / im(d^{n-1}_X) \stackrel{g^n}{\to} J^n
  \,,
$$

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

$$
  \array{
    X^n / im(d^{n-1}_X) &\stackrel{d^n_X}{\to}& X^{n+1}
   \\
    \downarrow^{\mathrlap{g^n}} & \swarrow_{\mathrlap{\eta^{n+1}}}
    \\
    J^n   }
  \,,
$$

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.

=--

+-- {: .num_remark }
###### Remark

Without the assumption above that $J^\bullet$ is injective, such a null-homotopy indeed need not exist. Basic counterexamples are discussed in the section [Chain homotopies that ought to exist but do not](homotopy+category+of+chain+complexes#ChainHomotopiesThatOughtToExistButDoNot) at _[[homotopy category of chain complexes]]_.

=--

The formally dual statement of prop \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic} is the following.

+-- {: .num_prop #MapsProjectiveIntoExactAreNullHomotopic}
###### Lemma

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]]

$$
  \eta : 0 \Rightarrow f_\bullet
$$

=--

+-- {: .proof}
###### Proof

This is formally dual to the proof of prop. \ref{MapsOutOfExactIntoInjectiveAreNullHomotopic}.

=--

The following proposition says that, when injectively resolving objects, the morphisms between these objects lift to the resolutions, uniquely up to chain homotopy.

+-- {: .num_prop #InjectiveResolutionOfCodomainRespectsMorphisms}
###### Proposition

Let $f : X \to Y$ be a morphism in $\mathcal{A}$. Let

$$
  i_Y : Y \stackrel{\sim}{\to} Y^\bullet
$$

be an injective resolution of $Y$ and

$$
  i_X : X \stackrel{\sim}{\to} X^\bullet
$$

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]]

$$
  \array{
    X &\stackrel{\sim}{\to}& X^\bullet
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f^\bullet}}
    \\
    Y &\stackrel{\sim}{\to}& Y^\bullet
  }
  \,.
$$

=--

+-- {: .proof}
###### 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]]

$$
  \array{
    X^{n} &\stackrel{d^n_X}{\to}& X^{n+1}
    \\
    \downarrow^{\mathrlap{f^n}} && \downarrow^{\mathrlap{f^{n+1}}}
    \\
    Y^{n} &\stackrel{d^n_Y}{\to}& Y^{n+1}
  }
$$

[[commuting diagram|commute]] (the defining condition on a [[chain map]]) and such that the diagram

$$
  \array{
    X &\stackrel{i_X}{\to}& X^0
    \\
    \downarrow^{f} && \downarrow^{\mathrlap{f^0}}
    \\
    Y &\stackrel{i_Y}{\to}& Y^0
  }
$$

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:

$$
  \array{
    X^n &\stackrel{}{\to}& X^n/im(d^{n-1}_X) &\stackrel{d^n_X}{\hookrightarrow}& X^{n+1}
    \\
    {}^{\mathllap{f^n}}\downarrow & \searrow^{\mathrlap{g^n}}
    & \downarrow^{\mathrlap{h^n}} & \swarrow_{\mathrlap{f^{n+1}}}
    \\
    Y^n &\underset{d^n_Y}{\to}& Y^{n+1}
  }
  \,.
$$

Here the morphism $f^n$ on the left is given by induction assumption and we define the diagonal morphism to be the composite

$$
 g^n \coloneqq d^n_Y \circ f^n
 \,.
$$

Observe then that by the chain map property of the $f^{\bullet \leq n}$ we have

$$
  d^n_Y \circ f^n \circ d^{n-1}_X = d^n_Y \circ d^{n-1}_Y \circ f^{n-1} = 0
$$

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.

=--

+-- {: .num_prop #HomotopyUniquenessOfResolutionOfMorphism}
###### Proposition

The morphism $f_\bullet$ in prop. \ref{InjectiveResolutionOfCodomainRespectsMorphisms} is the unique one up to [[chain homotopy]] making the given diagram commute.

=--

+-- {: .proof}
###### 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

$$
  \array{
    X &\underoverset{h^\bullet}{\sim}{\to}& X^\bullet
    \\
    \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{f^\bullet \coloneqq g_2^\bullet - g_1^\bullet}}
    \\
    Y &\stackrel{\sim}{\to}& Y^\bullet
  }
$$

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

$$
  \array{
     0
       &\stackrel{}{\to}&
     X
       &\stackrel{h^0}{\to}&
     X^0
       &\stackrel{d^0_X}{\to}&
     X^1
       &\to&
    \cdots
     \\
     \downarrow^{\mathrlap{f^{-2} = 0}}
     &&
     \downarrow^{\mathrlap{f^{-1} = 0}}
     &&
     \downarrow^{f^0}
     &&
     \downarrow^{f^1}
     \\
     0
       &\to&
     Y
       &\to&
     Y^0
       &\stackrel{d^0_J}{\to}&
     Y^1
       &\to&
     \cdots
  }
  \,,
$$

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]]

$$
  \array{
     0
       &\stackrel{}{\to}&
     X
       &\stackrel{}{\to}&
     X^0
       &\stackrel{d^0_X}{\to}&
     X^1
       &\to&
     \cdots
     \\
     \downarrow^{\mathrlap{f^{-2} = 0}}
       &\swarrow_{\mathrlap{\eta^{-1} = 0}}&
     \downarrow^{\mathrlap{f^{-1} = 0}}
       &\swarrow_{\mathrlap{\eta^0 = 0} }&
     \downarrow^{f^0}
       &\swarrow_{\mathrlap{\eta^1}}&
     \downarrow^{f^1}
     \\
     0 &\to& Y &\to& Y^0 &\stackrel{d^0_Y}{\to}& Y^1 &\to& \cdots
  }
$$

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 $f^\bullet$.

=--

Sometimes one needs to construct resolutions of sequences of morphisms in a more controled way, for instance such that some degreewise exactness is preserved:

+-- {: .num_lemma #ProjectiveResolutionOfExactSequenceByExactSequence}
###### Lemma

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]]

$$
  \array{
    0 &\to& A_\bullet &\to& B_\bullet &\to& C_\bullet &\to& 0
    \\
      &&
    \downarrow^{\mathrlap{f_\bullet}}
      &&
    \downarrow^{\mathrlap{g_\bullet}}
      &&
    \downarrow^{\mathrlap{h_\bullet}}
    \\
    0 &\to& A &\stackrel{i}{\to}& B &\stackrel{p}{\to}& C &\to& 0
  }
$$

where

* each vertical morphism is a projective resolution;

and in addition

* the top row is again a short exact sequence of chain complexes.

=--

This appears for instance in ([May, lemma 3.4](#May)) or ([Murfet, cor. 33](#Murfet)).

+-- {: .proof}
###### 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 $n \in \mathbb{N}$

$$
  B_n \coloneqq A_n \oplus C:n
$$

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)

$$
  g_0 =
   \left(
     \array{
        (j_0)_A
        &
        (j_0)_B
      }
  \right)
 : A_0 \oplus C_0 \to B
$$

be given in the first component by the given composite

$$
 (g_0)_A : A_0 \oplus C_0 \stackrel{}{\to} A_0 \stackrel{f_0}{\to} A \stackrel{i}{\hookrightarrow} B
$$

and in the second component we take

$$
  (j_0)_C : A_0 \oplus C_0 \to C_0 \stackrel{\zeta}{\to} B
$$

to be given by a lift in

$$
  \array{
    && B
    \\
    & {}^{\mathllap{\zeta}}\nearrow & \downarrow^{\mathrlap{p}}
    \\
    C_0 &\stackrel{h_0}{\to}& C
  }
  \,,
$$

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

$$
  \array{
    A_0 &\hookrightarrow& A_0 \oplus C_0 &\to& C_0
    \\
    \downarrow^{\mathrlap{f_0}} && {}^{\mathllap{((g_0)_A, (g_0)_C)}}\downarrow &\swarrow_{\zeta}& \downarrow^{\mathrlap{h_0}}
    \\
    A &\stackrel{i}{\hookrightarrow}& B &\stackrel{p}{\to}& C
  }
  \,.
$$

Let then the [[differentials]] of $B_\bullet$ be given by

$$
  d_k^{B_\bullet}
    =
  \left(
    \array{
      d_k^{A_\bullet} & (-1)^k e_k
      \\
      0 & d_k^{C_\bullet}
    }
  \right)
  :
  A_{k+1} \oplus C_{k+1}
  \to
  A_k \oplus C_k
  \,,
$$

where the $\{e_k\}$ are constructed by [[induction]] as follows. Let $e_0$ be a lift in

$$
  \array{
    & && A_0
    \\
    & & {}^{\mathllap{e_0}}\nearrow & \downarrow^{\mathrlap{f_0}}
    \\
    \zeta \circ d^{C_\bullet}_0 \colon & C_1 &\stackrel{}{\to}& A &\hookrightarrow B&
  }
$$

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

$$
  \begin{aligned}
    p \circ \zeta \circ d^{C_\bullet}_0
    &=
    h_0 \circ d^{C_\bullet}_0
    \\
    & = 0 \circ h_1
    \\
    & = 0
  \end{aligned}
  \,.
$$

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

$$
  \array{
    & && A_n
    \\
    & & {}^{\mathllap{e_{n}}}\nearrow & \downarrow^{\mathrlap{d^{A_\bullet}_{n-1}}}
    \\
    e_{n-1}\circ d_n^{C_\bullet} \colon & C_{n+1} &\stackrel{}{\hookrightarrow}& ker(d^{A_\bullet}_{n-1}) = im(d^{A_\bullet}_{n-1})) &\to& A_{n-1}
  }
  \,,
$$

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

$$
  d^{B_\bullet}_{n}
  \circ
  d^{B_\bullet}_{n+1}
  =
  \left(
    \array{
       0 &  (-1)^{n}\left(e_{n} \circ d^{C_\bullet}_{n+1} - d^{A_\bullet}_n \circ e_{n+1} \right)
       \\
       0 & 0
     }
  \right)
  \,.
$$

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

$$
  \array{
    0 &\to& A &\hookrightarrow& H_0(B_\bullet) &\to& C &\to& 0
    \\
    \downarrow && \downarrow^{\mathrlap{=}} && \downarrow && \downarrow^{\mathrlap{=}} && \downarrow
    \\
    0 &\to& A &\hookrightarrow& B &\to& C &\to& 0
    \,,
  }
$$

where both rows are exact. That the middle vertical morphism is an [[isomorphism]] then follows by the [[five lemma]].

=--

The formally dual statement to lemma \ref{ProjectiveResolutionOfExactSequenceByExactSequence} is the following.

+-- {: .num_lemma #InjectiveResolutionOfExactSequenceByExactSequence}
###### Lemma

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

$$
  \array{
    0 &\to& A &\to& B &\to& C &\to& 0
    \\
      &&
    \downarrow^{\mathrlap{}}
      &&
    \downarrow^{\mathrlap{}}
      &&
    \downarrow^{\mathrlap{}}
    \\
    0 &\to& A^\bullet &\to& B^\bullet &\to& C^\bullet &\to& 0
  }
$$

where

* each vertical morphism is an injective resolution;

and in addition

* the bottom row is again a short exact sequence of cochain complexes.

=--

+-- {: .proof}
###### Proof

To construct this, let for each $n \in \mathbb{N}$

$$
  B^n \coloneqq A^n \oplus C^n
$$

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)

$$
  j^0 =
   \left(
     \array{
        j^0_A
        \\
        j^0_B
      }
  \right)
 : B \to A^0 \oplus C^0
$$

be given in the second component by the given composite

$$
  j^0_B : B \to C \to C^0
$$

and in the first component we take

$$
  j^0_A : B \to A^0
$$

to be given by a lift in

$$
  \array{
     A &\to& A^0
     \\
     \downarrow & \nearrow_{\mathrlap{j^0_A}}
     \\
     B
  }
  \,,
$$

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 (...).

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 $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ (...).

=--

### Existence and construction of resolutions of complexes
 {#ExistenceAndConstructionOfResolutionsOfComplexes}

+-- {: .num_defn }
###### Definition

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}.

=--

+-- {: .proof}
###### Proof

Notice that for each $n \in \mathbb{N}$ we have [[short exact sequences]] of [[chains]], [[cycles]], [[boundaries]] and [[homology groups]] as

$$
  0 \to B_n(C) \to Z_n(C) \to H_n(C) \to 0
$$

$$
  0 \to Z_n(C) \to C_n \to B_{n-1}(C) \to 0
  \,.
$$

Now by prop. \ref{ExistenceOfInjectiveResolutions} we find for each $n \in \mathbb{N}$ projective resolutions of the objects $H_n(C)$ and $B_n(C)$:

$$
  H_{n,\bullet} \stackrel{\simeq_{qi}}{\to} H_n(C)
$$

$$
  B_{n,\bullet} \stackrel{\simeq_{qi}}{\to} B_n(C)
  \,.
$$

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:

$$
  0 \to B_{n,\bullet}(C) \to Z_{n,\bullet}(C) \to H_{n,0}(C) \to 0
  \,.
$$

Analogously, we find for each $n$ a projective resolution $C_{n,\bullet} \to C_n$ that sits in a short exact sequence

$$
  0 \to Z_{n,\bullet} \to C_{n,\bullet} \to B_{n+1,\bullet} \to 0
  \,.
$$

Using the exactness of these sequences one checks now that

1. The $\{C_{n,\bullet}\}_{n \in \mathbb{N}}$ arrange into a [[double complex]] by taking the horizontal differential to be the composite

   $$
     C_{n,k} \to B_{n+1,k} \hookrightarrow Z_{n+1,k} \to C_{n+1,k}
     \,;
   $$

1.  this [[double complex]] $C_{\bullet,\bullet}$ is indeed a fully projective resolution of $C_\bullet$.

=--

### Functorial resolutions and derived functors
 {#FunctorialResolutions}

We discuss how the injective/projective resolutions constructed in _[Existence and construction](#ExistenceAndConstruction)_ are [[functor|functorial]] if regarded in the [[homotopy category of chain complexes]] and how this yields the construction of _[[derived functors in homological algebra]]_.

Write

$$
  \mathcal{K}^{+}(\mathcal{A}) \hookrightarrow \mathcal{K}(\mathcal{A})
$$

for the [[full subcategory]] of the [[homotopy category of chain complexes]] on the one bounded above or bounded below, respectively.
Write

$$
  \mathcal{K}^+(\mathcal{I}_{\mathcal{A}})
    \hookrightarrow
  \mathcal{K}^+(\mathcal{A})
$$

for the [[full subcategory]] on the degreewise [[injective object|injective]] complexes, and

$$
  \mathcal{K}^-(\mathcal{P}_{\mathcal{A}})
    \hookrightarrow
  \mathcal{K}^-(\mathcal{A})
$$

for the full subcategory on the degreewise [[projective object|projective]] objects.

+-- {: .num_theorem #InjectiveResolutionFunctors}
###### Theorem

If $\mathcal{A}$ has [enough injectives](injective+object#EnoughInjectives) then there exists a [[functor]]

$$
  P : \mathcal{A} \to \mathcal{K}^+(\mathcal{I}_{\mathcal{A}})
$$

together with a [[natural isomorphisms]]

$$
  H^0(-) \circ P \simeq id_{\mathcal{A}}
$$

and

$$
  H^{n \geq 1}(-) \circ P \simeq 0
  \,.
$$

=--

+-- {: .proof}
###### 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.

=--

Dually we have:

+-- {: .num_theorem #ProjectiveResolutionFunctors}
###### Theorem

If $\mathcal{A}$ has [enough projectives](projectice+object#EnoughInjectives) then there exists a [[functor]]

$$
  Q : \mathcal{A} \to \mathcal{K}^-(\mathcal{P}_{\mathcal{A}})
$$

together with a [[natural isomorphisms]]

$$
  H_0(-) \circ P \simeq id_{\mathcal{A}}
$$

and

$$
  H_{n \geq 1}(-) \circ P \simeq 0
  \,.
$$

=--

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.

+-- {: .num_defn }
###### Definition

Given an [[additive functor]] $F : \mathcal{A} \to \mathcal{A}'$, it canonically induces a functor

$$
  Ch_\bullet(F) \colon Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{A}')
$$

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]]

$$
  \mathcal{K}(F) \colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{A}')
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

If $\mathcal{A}$ and $\mathcal{A}'$ have [[projective object|enough projectives]], then their [[derived categories]] are

$$
  \mathcal{D}_\bullet(\mathcal{A})
   \simeq
  \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}})
$$

and

$$
  \mathcal{D}^\bullet(\mathcal{A}) \simeq \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}})
$$

etc. One wants to accordingly _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

$$
  \mathcal{A}
    \to
  \mathcal{D}_\bullet(\mathcal{A})
   \simeq
  \mathcal{K}(\mathcal{P}_{\mathcal{A}})
   \hookrightarrow
  \mathcal{K}(\mathcal{A})
   \stackrel{\mathcal{K}(F)}{\to}
  \mathcal{K}(\mathcal{A}')
  \,.
$$
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 _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.

=--

+-- {: .num_defn #RightDerivedFunctorOfLeftExactFunctor}
###### Definition

Let

$$
  F : \mathcal{A} \to \mathcal{A}'
$$

be a [[left exact functor]] between [[abelian categories]] such that $\mathcal{A}$ has [enough injectives](inhective+object#EnoughInjectives). For $n \in \mathbb{N}$ the **$n$th [[right derived functor]]** of $F$ is the composite

$$
  R^n F :
  \mathcal{A}
   \stackrel{P}{\to}
  K^+(\mathcal{I}_{\mathcal{A}})
   \stackrel{\mathcal{K}(F)}{\to}
  \mathcal{K}^+(\mathcal{A}')
    \stackrel{H^n(-)}{\to}
  \mathcal{A}'
  \,,
$$

where

* $P$ is the injective resolution functor of theorem \ref{InjectiveResolutionFunctors};

* $\mathcal{K}(F)$ is the evident prolongation of $F$ to $\mathcal{K}^+(\mathcal{A})$;

* $H^n(-)$ is the $n$-[[chain homology]] functor. Hence

$$
  (R^n F)(X^\bullet) \coloneqq H^n(F(P(X)^\bullet))
  \,.
$$

=--

Dually:

+-- {: .num_defn #LeftDerivedFunctorOfRightExactFunctor}
###### Definition

Let

$$
  F : \mathcal{A} \to \mathcal{A}'
$$

be a [[right exact functor]] between [[abelian categories]] such that $\mathcal{A}$ has [enough projectives](projective+object#EnoughProjectives). For $n \in \mathbb{N}$ the **$n$th [[left derived functor]]** of $F$ is the composite

$$
  L_n F :
  \mathcal{A}
   \stackrel{Q}{\to}
  K^-(\mathcal{P}_{\mathcal{A}})
   \stackrel{\mathcal{K}(F)}{\to}
  \mathcal{K}^-(\mathcal{A}')
    \stackrel{H_n(-)}{\to}
  \mathcal{A}'
  \,,
$$

where

* $Q$ is the projective resolution functor of theorem \ref{ProjectiveResolutionFunctors};

* $\mathcal{K}(F)$ is the evident prolongation of $F$ to $\mathcal{K}^+(\mathcal{A})$;

* $H_n(-)$ is the $n$-[[chain homology]] functor. Hence

$$
  (L_n F)(X_\bullet) \coloneqq H_n(F(Q(X)_\bullet))
  \,.
$$

=--

We discuss now the basic general properties of such derived functors.

+-- {: .num_prop #BasicPropertiesOfDerivedFunctors}
###### Proposition

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]]

$$
  R^0F(X) \simeq F(X)
  \,.
$$

Dually, of $F$ is a [[right exact functor]] in the presence of [[projective object|enough projectives]], then

$$
  L_0 F(X) \simeq F(X)
  \,.
$$

=--

+-- {: .proof}
###### 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

$$
  0 \to X \hookrightarrow X^0 \to X^1 \to \cdots
  \,.
$$

If $F$ is left exact then it preserves this excact sequence by definition of left exactness, and hence

$$
  0 \to F(X) \hookrightarrow F(X^0) \to F(X^1) \to \cdots
$$

is an exact sequence. But this means that

$$
  R^0 F(X) \coloneqq ker(F(X^0) \to F(X^1)) \simeq F(X)
  \,.
$$

=--

+-- {: .num_prop #LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence}
###### Proposition

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

$$
  0 \to A \to B \to C \to 0
$$

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}

$$
  \array{
    0 &\to& R^0F (A)  &\to&  R^0 F(B)  &\to&  R^0 F(C)
     &\stackrel{\delta_0}{\to}&
    R^1 F(A) &\to& R^1 F(B) &\to& R^1F(C)
     &\stackrel{\delta_1}{\to}&
    R^2 F(A) &\to& \cdots
    \\
    && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    0 &\to& F(A) &\to& F(B) &\to& F(C)
  }
$$

in $\mathcal{B}$.

=--

+-- {: .proof}
###### Proof

By lemma \ref{InjectiveResolutionOfExactSequenceByExactSequence} we can find an injective resolution

$$
  0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0
$$

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

$$
  0 \to F(\tilde A^\bullet) \to F(\tilde B^\bullet) \to F(\tilde C^\bullet) \to 0
$$

is a again short exact sequence of cochain complexes, now in $\mathcal{B}$. Hence we have the corresponding [[homology long exact sequence]]

$$
  \cdots
   \to
  H^{n-1}(F(A^\bullet)) \to H^{n-1}(F(B^\bullet)) \to H^{n-1}(F(C^\bullet))
    \stackrel{\delta}{\to}
  H^n(F(A^\bullet)) \to H^n(F(B^\bullet)) \to H^n(F(C^\bullet))
    \stackrel{\delta}{\to}
  H^{n+1}(F(A^\bullet)) \to H^{n+1}(F(B^\bullet)) \to H^{n+1}(F(C^\bullet))
   \to
  \cdots
  \,.
$$

But by construction of the resolutions and by def. \ref{RightDerivedFunctorOfLeftExactFunctor} this is equal to

$$
  \cdots
   \to
  R^{n-1}F(A) \to R^{n-1}F(B) \to R^{n-1}F(C)
   \stackrel{\delta}{\to}
  R^{n}F(A) \to R^{n}F(B) \to R^{n}F(C)
    \stackrel{\delta}{\to}
  R^{n+1}F(A) \to R^{n+1}F(B) \to R^{n+1}F(C)
   \to
  \cdots
  \,.
$$

Finally the equivalence of the first three terms with $F(A) \to F(B) \to F(C)$ is given by prop. \ref{BasicPropertiesOfDerivedFunctors}.

=--

+-- {: .num_remark}
###### Remark

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

$$
  0 \to F(A) \to F(B) \to F(C) \to R^1 F(A)
$$

and hence $0 \to F(A) \to F(B) \to F(C) \to $ is a short exact sequence itself precisely if $R^1 F(A) \simeq 0$.

=--

In fact we even have the following.

+-- {: .num_prop }

Let $F$ be an [[additive functor]] which is an [[exact functor]]. Then

$$
  R^{\geq 1} F = 0
$$

and

$$
  L_{\geq 1} F = 0
  \,.
$$

=--

+-- {: .proof}
###### 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 $R^n F$.

=--

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 [above](#FResolutions).

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}.

+-- {: .num_lemma #FPreservesNullnessOfFInjectiveComplexes}
###### Lemma

If a [[cochain complex]] $A^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ is [[quasi-isomorphism|quasi-isomorphic]] to 0,

$$
  X^\bullet \stackrel{\simeq_{qi}}{\to} 0
$$

then also $F(X^\bullet) \in Ch^\bullet(\mathcal{B})$ is quasi-isomorphic to 0

$$
  F(X^\bullet) \stackrel{\simeq_{qi}}{\to} 0
  \,.
$$

=--

+-- {: .proof}
###### Proof

Consider the following collection of [[short exact sequences]] obtained from the [[long exact sequence]] $X^\bullet$:

$$
  0 \to X^0 \stackrel{d^0}{\to} X^1 \stackrel{d^1}{\to} im(d^1) \to 0
$$

$$
  0 \to im(d^1) \to X^2 \stackrel{d^2}{\to} im(d^2) \to 0
$$

$$
  0 \to im(d^2) \to X^3 \stackrel{d^3}{\to} im(d^3) \to 0
$$

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

$$
  0 \to F(im(d^n)) \to F(X^n+1) \to F(im(d^{n+1})) \to 0
$$

are [[short exact sequence|exact]]. But this means that

$$
  0 \to F(X^0)\to F(X^1) \to F(X^2) \to \cdots
$$

is exact, hence that $F(X^\bullet)$ is quasi-isomorphic to 0.

=--

+-- {: .num_theorem #DerivedFFromFInjectiveResolution}
###### Theorem

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]]

$$
  R^n F(A) \simeq H^n(F(I_F^\bullet))
$$

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 $I_F^\bullet$.

=--

+-- {: .proof}
###### 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]]

$$
  \array{
     A &\stackrel{\simeq_{qi}}{\to}& I_F^\bullet
     \\
     \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}}
     \\
     A &\stackrel{\simeq_{qi}}{\to}& I^\bullet
  }
$$

[[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

1. there is an [[isomorphism]] $F(Cone(f)) \simeq Cone(F(f))$;

1. $Cone(f) \in Ch^\bullet(\mathcal{I})\subset Ch^\bullet(\mathcal{A})$ (because $F$-injective objects are closed under [[direct sum]]).

The first implies that we have a [[homology exact sequence]]

$$
  \cdots \to H^n(I^\bullet) \to H^n(I_F^\bullet) \to H^n(Cone(f)^\bullet)
   \to H^{n+1}(I^\bullet) \to H^{n+1}(I_F^\bullet) \to H^{n+1}(Cone(f)^\bullet) \to \cdots
  \,.
$$

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

$$
  0 \to (R^n F(A)\coloneqq H^n(I^\bullet) \stackrel{\simeq}{\to} H^n(I_F^\bullet) \to 0
  \,.
$$

=--

### Derived Hom-functor/$Ext$-functor and extensions
 {#DerivedHomFunctor}

Consider the [[derived functor]] of the [[hom functor]].

+-- {: .num_defn #ExtFunctorAsRightDerivedContravariantHom}
###### Definition

For $A \in \mathcal{A}$, write

$$
  Ext^n(-,A) \coloneqq R^n Hom(-,A)
$$

for the [[right derived functor]], def. \ref{RightDerivedFunctorOfLeftExactFunctor}.

=--

We discuss the use of projective resolutions in the computation of [[Ext]]-functors and [[group extensions]].

+-- {: .num_defn #Extensions}
###### Definition

Given $A, G \in \mathcal{A}$, an **[[extension]]** of $G$ by $A$
is a [[short exact sequence]] of the form

$$
  0 \to A \to \hat G \to G \to 0
  \,.
$$

Two extensions $\hat G_1$ and $\hat G_2$ are called _equivalent_ if there is a morphism $f : \hat G_1 \to \hat G_2$ in $\mathcal{A}$ such that we have a [[commuting diagram]]

$$
  \array{
     && \hat G_1
     \\
     & \nearrow && \searrow
    \\
    A &&\downarrow^{\mathrlap{f}}&& G
    \\
    & \searrow && \nearrow
    \\
    && \hat G_2
  }
  \,.
$$

Write $Ext(G,A)$ for the set of [[equivalence classes]] of extensions of $G$ by $A$.

=--

+-- {: .num_remark #MorphismOfExtensionIsIsomorphism}
###### Remark

By the [[short five lemma]] a morphism $f$ as above is necessarily an [[isomorphism]] and hence we indeed have an [[equivalence relation]].

=--

+-- {: .num_defn #MapFromExtensionsToExtGroup}
###### Definition

If $\mathcal{A}$ has [[projective object|enough projectives]], define a function

$$
  Extr \colon Ext(G,A) \to Ext^1(G,A)
$$

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

$$
  \array{
     \cdots &\to& 0 &\to& 0 &\to& A &\to& \hat G
     \\
     && \downarrow && \downarrow && \downarrow && \downarrow
     \\
     \cdots &\to& 0 &\to& 0 &\to& 0 &\to& G
  }
$$

of $G$, by remark \ref{ProjectiveResolutionInComponents}. By prop. \ref{InjectiveResolutionOfCodomainRespectsMorphisms} there exists then a [[commuting diagram]] of the form

$$
  \array{
     Y_2 &\to& 0
     \\
     \downarrow^{\mathrlap{\partial_1^{Y}}} && \downarrow
     \\
     Y_1 &\stackrel{c}{\to}& A
     \\
     \downarrow^{\mathrlap{\partial_0^Y}} && \downarrow
     \\
     Y_0 &\to& \hat G
     \\
     \downarrow && \downarrow
     \\
     G &\stackrel{id}{\to}& G
  }
$$

lifting the identity map on $G$ to 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 $Ext^1(G,A) \coloneqq H^1(Hom(Y_\bullet,N))$.

=--

+-- {: .num_prop }
###### Proposition

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.

=--

+-- {: .proof}
###### 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 $c$.

$$
  \array{
     Y_2 &\to& 0
     \\
     \downarrow^{\mathrlap{\partial_1^{Y}}} &\nearrow_{\eta_1 = 0}& \downarrow
     \\
     Y_1 &\stackrel{c - \tilde c}{\to}& A
     \\
     \downarrow^{\mathrlap{\partial_0^Y}} &\nearrow_{\eta_0}& \downarrow
     \\
     Y_0 &\to& \hat G
     \\
     \downarrow &\nearrow_{}& \downarrow
     \\
     G &\to& G
  }
  \,.
$$

The chain homotopy condition here says that

$$
  c - \tilde c = \eta_0 \circ \partial^{Y}_0
$$

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

$$
  Ext(G,A) \to Ext^1(G,A)
  \,.
$$

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.

=--

+-- {: .num_prop #ExtensionFromAnElementOfExt1}
###### Definition

Define a function

$$
  Rec \colon Ext^1(G,A) \to Ext(G,A)
$$

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

$$
  \array{
    Y_2 &\to& 0
    \\
    \downarrow && \downarrow
    \\
    Y_1 &\stackrel{c}{\to}& A
    \\
    \downarrow
    \\
    Y_0
    \\
    \downarrow
    \\
    G
  }
$$

be a representative. By the commutativity of the top square this restricts to a morphism

$$
  \array{
     Y_1/Y_2 &\stackrel{c}{\to}& A
     \\
     \downarrow
     \\
     Y_0
     \\
     \downarrow
     \\
     G
  }
  \,,
$$

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

$$
  \array{
    Y_1/Y_2 &\stackrel{c}{\to}& A
    \\
    \downarrow && \downarrow
    \\
    Y_0 &\to& Y_0 \coprod_{Y_1/Y_2} A
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{id}{\to}& G
  }
  \,.
$$

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 $c$.

=--

+-- {: .num_prop }
###### Proposition

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.

=--

+-- {: .proof}
###### Proof

The coproduct $Y_0 \coprod_{Y_1/Y_2} A$ is equivalently

$$
  coker(Y_1/Y_2 \stackrel{(incl,-c)}{\to} Y_0 \oplus A)
  \,.
$$

For a different representative $\tilde c$ of $[c]$ there is by construction a

$$
  \array{
     Y_1 &\stackrel{\tilde c - c}{\to}& A
     \\
     {}^{\mathllap{\partial_0}}\downarrow & \nearrow_{\lambda}
     \\
     Y_0
  }
  \,.
$$

Define from this a map between the two cokernels induced by the [[commuting diagram]]

$$
  \array{
    Y_1/Y_2 &\stackrel{id}{\to}& Y_1/Y_2
    \\
    \downarrow^{\mathrlap{(id,-c)}} && \downarrow^{\mathrlap{(id,-\tilde c)}}
    \\
    Y_0 \oplus A
      &\stackrel{\left(\array{ id & 0 \\ \lambda & id }\right)}{\to}&
    Y_0 \oplus A
  }
  \,.
$$

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.

=--

+-- {: .num_prop }
###### Proposition

The functions

$$
  Extr \colon Ext(G,A) \leftrightarrow Ext^1(G,A) \colon Rec
$$

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 $Ext^1(G,A)$.

=--

+-- {: .proof}
###### 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

$$
  \array{
     Y_1 &\stackrel{c}{\to}& A
     \\
     \downarrow && \downarrow
     \\
     Y_0 &\to& Y_0 \coprod^c_{Y_1/Y_2} A
     \\
     \downarrow && \downarrow
     \\
     G &\stackrel{id}{\to}& G
  }
$$

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 $Ext(G,A)$.

=--

### Relation to syzygies

(...) [[syzygy]] (...)

## Examples

### Length-1 resolutions
 {#Lenght1ResolutionsOfAbelianGroups}

+-- {: .num_prop }
###### Proposition

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]]

$$
  0 \to F_1 \to F_0 \to A \to 0
$$

with $F_1$ and $F_0$ being [[free abelian groups]].

=--

+-- {: .proof}
###### Proof

By the discussion at [free modules - submodules of free modules](free+module#SubmodulesOfFreeModules) 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.

=--

The same argument holds true for $R$ any [[principal ideal domain]].

### Projective resolutions adapted to abelian group cocycles
 {#ProjectiveResolutionsForGroupCocycles}

(...)

### Projective resolutions adapted to general group cohomology {#ProjectiveResolutionsForGroupCohomology}

Let $G$ be a [[discrete group]]. Write $\mathbb{Z}[G]$ for the [[group ring]] over $G$. Notice from _[module -- Abelian groups with G-action as modules over the group ring](module#AbelianGroupsWithGAction)_ 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.

+-- {: .num_defn #AugmentationMap}
###### Definition

Write

$$
  \epsilon \colon \mathbb{Z}[G] \to \mathbb{Z}
$$

for the [[homomorphism]] of [[abelian groups]] which forms the sum of $R$-[[coefficients]] of the [[formal linear combinations]] that constitute the group ring

$$
  \epsilon \colon r \mapsto \sum_{g \in G} r_g
  \,.
$$

This is called the [[augmentation]] map.

=--

+-- {: .num_defn #ProjectiveResolutionForZAsZGModule}
###### Definition

For $n \in \mathbb{N}$ let

$$
  Q^u_n \coloneqq F(U(G)^{\times^{n}})
$$

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

$$
  \partial_{n-1} (g_1, \cdots, g_n)
  \coloneqq
  g_1 [g_2, \cdots, g_n]
  +
  \sum_{i = 1}^{n-1} (-1)^i [g_1, \cdots, g_i g_{i+1}, g_{i+2}, \cdots, g_n]
  +
  (-1)^n [g_1, \cdots, g_{n-1}]
  \,,
$$

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

$$
  \partial_0 \colon [g] \mapsto g[*] - [*] =  g-e \in \mathbb{Z}[G]
  \,.
  \,.
$$

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.

=--

+-- {: .num_prop }
###### Proposition

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

$$
  \epsilon \colon Q^u_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z}
$$

$$
  \epsilon \colon Q_\bullet \stackrel{\simeq_{qi}}{\to} \mathbb{Z}
$$

of $\mathbb{Z}$ equipped with the trivial $\mathbb{Z}[G]$-module structure in $\mathbb{Z}[G]$[[Mod]].

=--

+-- {: .proof}
###### 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

$$
  \partial^0_n [g_1, \cdots, g_n] \coloneqq g_1 [g_2, \cdots, g_n]
$$

$$
  \partial^i_n [g_1, \cdots, g_n] \coloneqq [g_1, \cdots, g_i g_{i+1}, g_{i+2}, \cdots, g_n]
  \;\;
  for 1 \leq i \leq n-1
$$

$$
  \partial_n [g_1, \cdots, g_n] \coloneqq [g_1, \cdots, g_{n-1}]
$$

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]]

$$
  \cdots
  \stackrel{\partial_2}{\to} \mathbb{Z}[G]^2
  \stackrel{\partial_1}{\to}
  \mathbb{Z}[G] \stackrel{\epsilon}{\to} \mathbb{Z} \to 0
$$

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

$$
  s_{-1} \colon \mathbb{Z} \to Q^u_0
$$

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 $n \in \mathbb{N}$

$$
  s_n \colon Q^u_n \to Q^u_{n+1}
$$

be given on basis elements by

$$
  s_n(g[g_1, \cdots, g_n]) \coloneqq [g, g_1, \cdots, g_n]
  \,.
$$

In the lowest degrees we have

$$
  \epsilon \circ s_{-1} = id_{\mathbb{Z}}
$$

because

$$
  \epsilon(s_{-1}(1)) = \epsilon([*]) = \epsilon(e) = 1
$$

and

$$
  \partial_0 \circ s_0 + s_{-1}\circ \epsilon = id_{Q^u_0}
$$

because for all $g \in G$ we have

$$
  \begin{aligned}
    \partial_0 (s_0(g[*])) + s_{-1}(\epsilon(g[*]))
    & =
    \partial_0( [g] ) + s_{-1}(1)
    \\
    & = g[*] - [*] + [*]
    \\
    & =
    g[*]
  \end{aligned}
  \,.
$$

For all remaining $n \geq 1$ we find

$$
  \partial_n \circ  s_n + s_{n-1} \circ \partial_{n-1}
  =
  id_{Q^u_n}
$$

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.

=--

+-- {: .num_prop }
###### Propoition

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

$$
  \begin{aligned}
    H^n_{Grp}(G,A) & \simeq Ext^n_{\mathbb{Z}[G]}(\mathbb{Z}, A)
    \\
    & \simeq H^n(Hom_{\mathbb{Z}[G]}(Q^u_n, A))
    \\
    & \simeq H^n(Hom_{\mathbb{Z}[G]}(Q_n, A))
    \,.
  \end{aligned}
  \,,
$$

where on the right we canonically regard $A \in \mathbb{Z}[G]$[[Mod]].

=--

+-- {: .proof}
###### Proof

By the [[free functor]] [[adjunction]] we have that

$$
  Hom_{\mathbb{Z}[G]}(F^u_n, A)
  \simeq
  Hom_{Set}(U(G)^{\times n}, U(A))
$$

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 _normalized_ cocycles. By the discussion at _[[group cohomology]]_ these already support the entire group cohomology (every cocycle is comologous to a normalized one).

=--

### Cohomology of cyclic groups
 {#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 _[group cohomology - In terms of homological algebra](http://ncatlab.org/nlab/show/group+cohomology#InTermsOfHomologicalAlgebra)_.

Define special elements in the [[group algebra]] $\mathbb{Z}G$:

$$N \coloneqq 1 + g + g^2 + \ldots + g^{k-1}$$

$$\,$$

$$D \coloneqq g - 1,$$

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]]

$$\ldots \stackrel{N}{\to} \mathbb{Z}G \stackrel{D}{\to} \mathbb{Z}G \stackrel{N}{\to} \mathbb{Z}G \stackrel{D}{\to} \mathbb{Z}G \to \mathbb{Z} \to 0$$

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:

$$
  H^{n+2}(C_k, A) \cong H^n(C_k, A)
$$

for $n \geq 1$. More precisely,

+-- {: .num_prop}
###### Proposition
For $G = C_k$, we have

* $H^0(G, A) = A^G = \ker(D) \colon A \to A$,

* $H^{2 j + 1}(G, A) = \ker(N)/im(D)$ for $j \geq 0$,

* $H^{2 j}(G, A) = \ker(D)/im(N)$ for $j \geq 1$.
=--

A well-known calculation in the cohomology of cyclic groups is **[[Hilbert's Theorem 90]]**.

+-- {: .num_theorem}
###### Theorem
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 $H^1(G, K^\ast) = 0$.

=--

+-- {: .proof}
###### 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

$$\beta^N = \beta^{1 + g + \ldots + g^{n-1}} = \beta \cdot \beta^g \cdot \ldots \beta^{g^{n-1}}$$

which is precisely the _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

$$1 + \beta g + \beta^{1+g}g^2 + \ldots + \beta^{1 + g + \ldots + g^{n-2}}g^{n-1}$$

is not identically zero, and therefore there exists $\theta \in K^\ast$ such that the element

$$\alpha = \theta + \beta \theta^g + \beta^{1+g}\theta^{g^2} + \ldots + \beta^{1 + g + \ldots + g^{n-2}}\theta^{g^{n-1}}$$

is non-zero. Using the fact that $N(\beta) = 1$, one may easily calculate that $\beta \alpha^g = \alpha$, as was to be shown.
=--

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):

+-- {: .num_lemma}
###### Lemma

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]].

=--

+-- {: .proof}
###### Proof
A single $\chi \colon G \to K^\ast$ obviously forms a linearly independent set. Now suppose we have an equation

\[a_1 \chi_1 + \ldots + a_n \chi_n = 0\]

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

$$a_1 \chi_1(g h) + \ldots + a_n \chi_n(g h) = 0$$

so that

\[a_1 \chi_1(g) \chi_1 + \ldots + a_n \chi_n(g)\chi_n = 0.\]

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

$$(a_2\frac{\chi_2(g)}{\chi_1(g)} - a_2)\chi_2 + \ldots = 0$$

which is a [[contradiction]].

=--

## Related concepts
 {#RelatedConcepts}

* [[projective object]], [[projective presentation]], [[projective cover]], **projective resolution**

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

* {#HiltonStammbach71} [[Peter Hilton]],  [[Urs Stammbach]], p. 129 of: _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 ([doi:10.1007/978-1-4419-8566-8](https://link.springer.com/book/10.1007/978-1-4419-8566-8), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf))

* {#Schapira} [[Pierre Schapira]], section 4.5 of: _Categories and homological algebra_ (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))

* {#May} [[Peter May]], sections 3.1 and 4.2 in: _Notes on Tor and Ext_ ([pdf](http://www.math.uchicago.edu/~may/MISC/TorExt.pdf))

* {#Murfet} [[Daniel Murfet]], section 4 of: _Derived functors_ ([pdf](http://therisingsea.org/notes/DerivedFunctors.pdf))

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projective set

The nLab recognizes two distinct usages of the term "projective set".

* A [[projective object]] in a [[category of sets]]; see also [[choice object]].

* A set of real numbers definable by a [[formula]] of [[second-order arithmetic]] with real parameters (where "real numbers" has some latitude of meaning among [[descriptive set theory|descriptive set theorists]], referring to structures that are [[Borel set|Borel isomorphic]] to the standard [[real numbers]]).

This article will touch upon the latter usage, as that is what is more commonly understood among mathematicians.

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