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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-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.
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.
Let be an abelian category.
For an object, an injective resolution of is a cochain complex (in non-negative degree) equipped with a quasi-isomorphism
such that is an injective object for all .
In components the quasi-isomorphism of def. is a chain map of the form
Since the top complex is concentrated in degree 0, this being a quasi-isomorphism happens to be equivalent to the sequence
being an exact sequence. In this form one often finds the definition of injective resolution in the literature.
For an object, a projective resolution of is a chain complex (in non-negative degree) equipped with a quasi-isomorphism
such that is a projective object for all .
In components the quasi-isomorphism of def. is a chain map of the form
Since the bottom complex is concentrated in degree 0, this being a quasi-isomorphism happens to be equivalent to the sequence
being an exact sequence. In this form one often finds the definition of projective resolution in the literature.
Projective and injective resolutions are typically used for computing the derived functor of some additive functor ; see at derived functor in homological algebra. While projective resolutions in are sufficient for computing every left derived functor on and injective resolutions are sufficient for computing every right derived functor on , if one is interested just in a single functor 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 -projective resolutions and -injective resolutions, respectively, we discuss here. A special case of both are -acyclic resolutions.
Let be abelian categories and let be an additive functor.
Assume that is left exact. An additive full subcategory is called -injective (or: consisting of -injective objects) if
for every object there is a monomorphism into an object ;
for every short exact sequence in with also ;
for every short exact sequence in with also is a short exact sequence in .
And dually:
Assume that is right exact. An additive full subcategory is called -projective (or: consisting of -projective objects) if
for every object there is an epimorphism from an object ;
for every short exact sequence in with also ;
for every short exact sequence in with also is a short exact sequence in .
For instance (Schapira, def. 4.6.5).
With the as above, we say:
For ,
an -injective resolution of is a cochain complex and a quasi-isomorphism
an -projective resolution of is a chain complex and a quasi-isomorphism
Let now have enough projectives / enough injectives, respectively.
For an additive functor, let be the full subcategory on the -acyclic objects. Then
if is left exact, then is a subcategory of -injective objects;
if is right exact, then is a subcategory of -projective objects.
Consider the case that is left exact. The other case works dually.
The first condition of def. is satisfied because every injective object is an -acyclic object and by assumption there are enough of these.
For the second and third condition of def. use that there is the long exact sequence of derived functors prop.
For the second condition, by assumption on and and definition of -acyclic object we have and for and hence short exact sequences
which imply that for all , hence that is acyclic.
Similarly, the third condition is equivalent to .
The -projective/injective resolutions by acyclic objects as in example are called -acyclic resolutions.
The above definition of a projective resolution of an object has an immediate generalization to resolutions of chain complexes.
For a chain complex, a projective resolution of is an exact sequence of chain complexes
such that for each the component is a projective resolution of the object , according to def. .
A projective resolution as above may in particular also be regarded as a double complex equipped with a morphism of double complex to regarded as a vertically constant double complex.
For purposes of computations one is often interested in the following stronger notion.
For any chain complex , write , , and for the graded objects of cycles, boundaries and homology groups, respectively, regarded as chain complexes with vanishing differentials.
A projective resolution of a chain complex , def. , is called fully projective (or proper) if furthermore for all the induced sequence of (horizontal) cycles
and (horizontal) boundaries
and (horizontal) homology groups
We first discuss the existence of injective/projective resolutions, and then the functoriality of their constructions.
Let be an abelian category with enough injectives (such as Mod for some ring ).
Then every object has an injective resolution, def. .
Let be the given object. By remark we need to construct an exact sequence of the form
such that all the are injective objects.
This we now construct by induction on the degree .
In the first step, by the assumption of enough enjectives we find an injective object and a monomorphism
hence an exact sequence
Assume then by induction hypothesis that for an exact sequence
has been constructed, where all the are injective objects. Forming the cokernel of yields the short exact sequence
By the assumption that there are enough injectives in we may now again find a monomorphism into an injective object . This being a monomorphism means that
is exact in the middle term. Therefore we now have an exact sequence
which completes the induction step.
The following proposition is formally dual to prop. .
Let be an abelian category with enough projectives (such as Mod for some ring ).
Then every object has a projective resolution, def. .
Let be the given object. By remark we need to construct an exact sequence of the form
such that all the are projective objects.
This we we now construct by induction on the degree .
In the first step, by the assumption of enough projectives we find a projective object and an epimorphism
hence an exact sequence
Assume then by induction hypothesis that for an exact sequence
has been constructed, where all the are projective objects. Forming the kernel of yields the short exact sequence
By the assumption that there are enough projectives in we may now again find an epimorphism out of a projective object . This being an epimorphism means that
is exact in the middle term. Therefore we now have an exact sequence
which completes the induction step.
Let be a chain map of cochain complexes in non-negative degree, out of an exact complex to a degreewise injective complex . Then there is a null homotopy
By definition of chain homotopy we need to construct a sequence of morphisms such that
for all . We now construct this by induction over , where we take .
Then in the induction step assume that for given we have constructed satisfying the above conditions.
First define now
and observe that
This means that factors as
where the first map is the projection to the quotient.
Observe then that by exactness of the morphism is a monomorphism. Together this gives us a diagram of the form
where the morphism may be found due to the defining right lifting property of the injective object against the top monomorphism.
Observing that the commutativity of this diagram is the chain homotopy condition involving and , this completes the induction step.
Without the assumption above that 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 at homotopy category of chain complexes.
The formally dual statement of prop is the following.
Let be a chain map of chain complexes in non-negative degree, into an exact complex from a degreewise projective complex . Then there is a null homotopy
The following proposition says that, when injectively resolving objects, the morphisms between these objects lift to the resolutions, uniquely up to chain homotopy.
Let be a morphism in . Let
be an injective resolution of and
any monomorphism that is a quasi-isomorphism (possibly but not necessarily an injective resolution). Then there is a chain map giving a commuting diagram
By definition of chain map we need to construct morphisms such that for all the diagrams
commute (the defining condition on a chain map) and such that the diagram
commutes in (which makes the full diagram in commute).
We construct these by induction.
To start the induction, the morphism in the first diagram above can be found by the defining right lifting property of the injective object against the monomorphism .
Assume then that for some component maps have been obtained such that for all . In order to construct consider the following diagram, which we will describe/construct stepwise from left to right:
Here the morphism on the left is given by induction assumption and we define the diagonal morphism to be the composite
Observe then that by the chain map property of the we have
and therefore factors through via some as indicated in the middle of the above diagram. Finally the morphism on the top right is a monomorphism by the fact that is exact in positive degrees (being quasi-isomorphic to a complex concentrated in degree 0) and so a lift as shown on the far right of the diagram exists by the defining lifting property of the injective object .
The total outer diagram now commutes, being built from commuting sub-diagrams, and this is the required chain map property of This completes the induction step.
The morphism in prop. is the unique one up to chain homotopy making the given diagram commute.
Given two chain maps making the diagram commute, a chain homotopy is equivalently a null homotopy of the difference, which sits in a square of the form
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 is null-homotopic.
This we may reduce to the statement of prop. by considering instead of the induced chain map of augmented complexes
where the second square from the left commutes due to the commutativity of the original square of chain complexes in degree 0.
Since is a quasi-isomorphism, the top chain complex is exact, by remark . 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 implies the existence of a null homotopy
starting with and (notice that the proof prop. was formulated exactly this way), which works because . The de-augmentation of this is the desired null homotopy of .
Sometimes one needs to construct resolutions of sequences of morphisms in a more controled way, for instance such that some degreewise exactness is preserved:
For a short exact sequence in an abelian category with enough projectives, there exists a commuting diagram of chain complexes
where
and in addition
This appears for instance in (May, lemma 3.4) or (Murfet, cor. 33).
By prop. we can choose and . The task is now to construct the third resolution 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
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)
be given in the first component by the given composite
and in the second component we take
to be given by a lift in
which exists by the left lifting property of the projective object (since is a projective resolution) against the epimorphism of the short exact sequence.
In total this gives in degree 0
Let then the differentials of be given by
where the are constructed by induction as follows. Let be a lift in
which exists since is a projective object and is an epimorphism by being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through as indicated, because the definition of and the chain complex property of gives
Now in the induction step, assuming that has been been found satisfying the chain complex property, let be a lift in
which again exists since is projective. That the bottom morphism factors as indicated is the chain complex property of inside .
To see that the defines this way indeed squares to 0 notice that
This vanishes by the very commutativity of the above diagram.
This establishes 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 is indeed a quasi-isomorphism, consider the homology long exact sequence associated to the short exact sequence of cochain complexes . In positive degrees it implies that the chain homology of indeed vanishes. In degree 0 it gives the short sequence sitting in a commuting diagram
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 is the following.
For a short exact sequence in an abelian category with enough injectives, there exists a commuting diagram of cochain complexes
where
and in addition
To construct this, let for each
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)
be given in the second component by the given composite
and in the first component we take
to be given by a lift in
which exists by the right lifting property of the injective object (since is an injective resolution) against the monomorphism of the short exact sequence.
Let the differentials be given by (…).
This establishes 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 is indeed a quasi-isomorphism, consider the homology long exact sequence associated to the short exact sequence of cochain complexes (…).
If has enough projectives, then every chain complex has a fully projective (proper) resolution, def. .
Notice that for each we have short exact sequences of chains, cycles, boundaries and homology groups as
Now by prop. we find for each projective resolutions of the objects and :
Moreover, by prop. we find for each a projective resolution of the object such that its fits into a short exact sequence of chain complexes with the previous two chosen resolutions:
Analogously, we find for each a projective resolution that sits in a short exact sequence
Using the exactness of these sequences one checks now that
The arrange into a double complex by taking the horizontal differential to be the composite
this double complex is indeed a fully projective resolution of .
We discuss how the injective/projective resolutions constructed in Existence and construction are functorial if regarded in the homotopy category of chain complexes and how this yields the construction of derived functors in homological algebra.
Write
for the full subcategory of the homotopy category of chain complexes on the one bounded above or bounded below, respectively. Write
for the full subcategory on the degreewise injective complexes, and
for the full subcategory on the degreewise projective objects.
If has enough injectives then there exists a functor
together with a natural isomorphisms
and
By prop. every object has an injective resolution. Proposition says that for and two resolutions the there is a morphism in and prop. says that this morphism is unique in . In particular it is therefore an isomorphism in (since the composite with the reverse lifted morphism, also being unique, has to be the identity).
So choose one such injective resolution for each .
Then for any morphism in , proposition again says that it can be lifted to a morphism between and and proposition says that there is a unique such image in for morphism making the given diagram commute.
This implies that this assignment of morphisms is functorial, since then also the composites are unique.
Dually we have:
If has enough projectives then there exists a functor
together with a natural isomorphisms
and
This is sufficient for the definition and construction of (non-total) derived functors in the next definition . 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.
Given an additive functor , it canonically induces a functor
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
If and have enough projectives, then their derived categories are
and
etc. One wants to accordingly derive from a functor between these derived categories. It is immediate to achive this on the domain category, there we can simply precompose and form
But the resulting composite lands in and in general does not factor through the inclusion .
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-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. and def. below do.
Let
be a left exact functor between abelian categories such that has enough injectives. For the th right derived functor of is the composite
where
is the evident prolongation of to ;
is the -chain homology functor. Hence
Dually:
Let
be a right exact functor between abelian categories such that has enough projectives. For the th left derived functor of is the composite
where
is the evident prolongation of to ;
is the -chain homology functor. Hence
We discuss now the basic general properties of such derived functors.
Let a left exact functor in the presence of enough injectives. Then for all there is a natural isomorphism
Dually, of is a right exact functor in the presence of enough projectives, then
We discuss the first statement, the second is formally dual.
By remark an injective resolution is equivalently an exact sequence of the form
If is left exact then it preserves this excact sequence by definition of left exactness, and hence
is an exact sequence. But this means that
Let be abelian categories and assume that has enough injectives.
Let be a left exact functor and let
be a short exact sequence in .
Then there is a long exact sequence of images of these objects under the right derived functors of def.
in .
By lemma we can find an injective resolution
of the given exact sequence which is itself again an exact sequence of cochain complexes.
Since is an injective object for all , its component sequences are indeed split exact sequences (see the discussion there). Splitness is preserved by a functor and so it follows that
is a again short exact sequence of cochain complexes, now in . Hence we have the corresponding homology long exact sequence
But by construction of the resolutions and by def. this is equal to
Finally the equivalence of the first three terms with is given by prop. .
Prop. implies that one way to interpret is as a “measure for how a left exact functor fails to be an exact functor”. For, with any short exact sequence, this proposition gives the exact sequence
and hence is a short exact sequence itself precisely if .
In fact we even have the following.
Let be an additive functor which is an exact functor. Then
and
Because an exact functor preserves all exact sequences. If is a projective resolution then also is exact in all positive degrees, and hence . Dually for .
We now discuss how the derived functor of an additive functor may also be computed not necessarily with genuine injective/projective resolutions, but with (just) -injective/-projective resolutions, such as -acyclic resolutions, as defined above.
Let be an abelian category with enough injectives. Let be an additive left exact functor with right derived functor , def. . Finally let be a subcategory of -injective objects, def. .
Consider the following collection of short exact sequences obtained from the long exact sequence :
and so on. Going by induction through this list and using the second condition in def. we have that all the are in . Then the third condition in def. says that all the sequences
are exact. But this means that
is exact, hence that is quasi-isomorphic to 0.
For an object with -injective resolution , def. , we have for each an isomorphism
between the th right derived functor, def. of evaluated on and the cochain cohomology of applied to the -injective resolution .
By prop. we can also find an injective resolution . By prop. there is a lift of the identity on to a chain map such that the diagram
commutes in . Therefore by the 2-out-of-3 property of quasi-isomorphisms it follows that is a quasi-isomorphism
Let be the mapping cone of and let be the canonical chain map into it. By the explicit formulas for mapping cones, we have that
there is an isomorphism ;
(because -injective objects are closed under direct sum).
The first implies that we have a homology exact sequence
Observe that with a quasi-isomorphism is quasi-isomorphic to 0. Therefore The second item above implies with lemma that also is quasi-isomorphic to 0. This finally means that the above homology exact sequences consists of exact pieces of the form
Consider the derived functor of the hom functor.
We discuss the use of projective resolutions in the computation of Ext-functors and group extensions.
Given , an extension of by is a short exact sequence of the form
Two extensions and are called equivalent if there is a morphism in such that we have a commuting diagram
Write for the set of equivalence classes of extensions of by .
By the short five lemma a morphism as above is necessarily an isomorphism and hence we indeed have an equivalence relation.
If has enough projectives, define a function
from the group of extensions, def. , to the first Ext functor group as follows. Choose any projective resolution , which exists by prop. . Regard then as a resolution
of , by remark . By prop. there exists then a commuting diagram of the form
lifting the identity map on to a chain map between the two resolutions.
By the commutativity of the top square, the morphism is 1-cocycle in , hence defines an element in .
The construction of def. 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.
First consider the same projective resolution but another lift of the identity. By prop. any other choice fitting into a commuting diagram as above is related by a chain homotopy to .
The chain homotopy condition here says that
and hence that in we have that is a coboundary. Therefore for the given choice of resolution we have obtained a well-defined map
If moreover is another projective resolution, with respect to which we define such a map as above, then lifting the identity map on to a chain map between these resolutions in both directions, by prop. , establishes an isomorphism between the resulting maps, and hence the construction is independent also of the choice of resolution.
Define a function
as follows. For a projective resolution of and an element of the -group, let
be a representative. By the commutativity of the top square this restricts to a morphism
where now the left column is itself an extension of by the cokernel (because by exactness the kernel of is the image of so that the kernel of is zero). Form then the pushout of the horizontal map along the two vertical maps. This yields
Here the bottom right is indeed , 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 is an element in which we assign to .
The construction of def. 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 -element.
The coproduct is equivalently
For a different representative of there is by construction a
Define from this a map between the two cokernels induced by the commuting diagram
By construction this respects the inclusion of . It also manifestly respects the projection to . Therefore this defines a morphism and hence by remark even an isomorphism of extensions.
The functions
from def. to def. are inverses of each other and hence exhibit a bijection between extensions of by and .
By straightforward unwinding of the definitions.
In one direction, starting with a and constructing the extension by pushout, the resulting pushout diagram
at the same time exhibits as the cocycle extracted from the extension .
Conversely, when starting with an extension then extracting a 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 is an isomorphism of extensions, hence an equality in .
(…) syzygy (…)
Assuming the axiom of choice, over hence in Ab every object has a projective resolution, even a free resolution, of length 1, hence a short exact sequence
with and being free abelian groups.
By the discussion at free modules - submodules of free modules a subgroup of a free abelian group is again free. Therefore for the surjection out of the free group on the underlying set of , setting yields the desired short exact sequence.
The same argument holds true for any principal ideal domain.
(…)
Let be a discrete group. Write for the group ring over . Notice from module – Abelian groups with G-action as modules over the group ring that linear -actions on abelian groups are equivalently -module structures in .
We discuss how cocycles in the group cohomology of with coefficients in such a module are naturally encoded in morphisms out of projective resolutions of the trivial -module.
Write
for the homomorphism of abelian groups which forms the sum of -coefficients of the formal linear combinations that constitute the group ring
This is called the augmentation map.
For let
be the free module over the group ring on -tuples of elements of (hence is the free module on a single generator).
For let be given on basis elements by
where in the first summand we have the coefficient times the basis element in .
In particular
Write furthermore for the quotient module which is the cokernel of the inclusion of those elements for which one of the is the unit element.
The construction in def. defines chain complexes and of -modules. Moreover, with the augmentation map of def. these are projective resolutions
of equipped with the trivial -module structure in Mod.
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
one finds that these satisfy the simplicial identities and that .
That the augmentation map is a quasi-isomorphism is equivalent, by remark , to the augmentation
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 . To that end, let
be given by sending to the single basis element in , and let for
be given on basis elements by
In the lowest degrees we have
because
and
because for all we have
For all remaining we find
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 consists by construction degreewise of free modules hence in particular of a projective module, it is a projective resolution.
For an abelian group equipped with a linear -action and for , the degree- group cohomology of with coefficients in is equivalently given by
where on the right we canonically regard Mod.
By the free functor adjunction we have that
is the set of functions from -tuples of elements of to elements of . It is immediate to check that these are in the kernel of 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 . This establishes the first equivalences.
Similarly one finds that 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).
Let be a cyclic group of finite order , with generator . We discuss the group cohomology of , as discussed at group cohomology - In terms of homological algebra.
Define special elements in the group algebra :
and denote the corresponding multiplications by these elements by the same letters .
Then a very simple and useful projective resolution of the trivial -module is based on an exact sequence of -modules
where the last map is induced from the trivial group homomorphism , hence is the map that forms the sum of all coefficients of all group elements.
It follows from this resolution that the cohomology groups for a -module are periodic of order 2:
for . More precisely,
For , we have
,
for ,
for .
A well-known calculation in the cohomology of cyclic groups is Hilbert's Theorem 90.
Suppose be a finite Galois extension of a field , with a cyclic Galois group of order . Regard the multiplicative group as a -module. Then .
Let , and denote the action of on an element by exponential notation . The action of the element is
which is precisely the norm . We are to show that if , then there exists such that . By the lemma that follows, the homomorphisms are, when considered as elements in a vector space of -valued functions, -linearly independent. It follows in particular that
is not identically zero, and therefore there exists such that the element
is non-zero. Using the fact that , one may easily calculate that , as was to be shown.
The next result may be thought of as establishing “independence of characters” (where “characters” are valued in the multiplicative group of a field):
Let be a field, let be a monoid, and let be distinct monoid homomorphisms. Then the functions , considered as functions valued in , are -linearly independent.
A single obviously forms a linearly independent set. Now suppose we have an equation
where , and assume is as small as possible. In particular, no is equal to , and . Choose such that . Then for all we have
so that
Dividing equation 2 by and subtracting from it equation 1, the first term cancels, and we are left with a shorter relation
which is a contradiction.
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution
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, pdf)
Pierre Schapira, section 4.5 of: Categories and homological algebra (2011) (pdf)
Peter May, sections 3.1 and 4.2 in: Notes on Tor and Ext (pdf)
Daniel Murfet, section 4 of: Derived functors (pdf)
Last revised on October 19, 2023 at 08:56:53. See the history of this page for a list of all contributions to it.