Draft lesson.

Universal cohomology and acyclic resolutions

A cohomology theory has to compare different resolutions and all their connecting maps. This lesson proves those comparisons from effacement, then proves when a bounded-below complex of acyclic objects computes a derived functor. It also constructs and characterizes the comparison for a composition of functors.

We use cochain complexes and the shift and cone signs fixed in Chain homotopies, mapping cones and triangles. Derived morphisms through quasi-isomorphism roofs, Section 5, supplies the K-injective and K-projective Hom tests. Triangles after quasi-isomorphism localization supplies the localized triangulation, exact-sequence triangles and zero detection. Injective replacements and right derived functors, Sections 2–4, constructs the bounded-below resolution functor, its exact derived equivalence and its universal transformation property. Its Section 6 proves that abelian groups have enough injectives.

All abelian categories and choices are taken in fixed ambient universes. Section 2 needs no enough-injectives hypothesis. In Sections 3–5 the source abelian category has enough injectives; further hypotheses are stated where used. A bounded-below complex may have nonzero terms in arbitrarily high degrees.

1. The data that a comparison must preserve

Let \(\mathcal A,\mathcal B\) be abelian categories. A cohomological delta-functor \(T\) consists of additive functors \(T^n:\mathcal A\to\mathcal B\), for \(n\geq0\), and, for every short exact sequence \(e=(0\to A\to B\to C\to0)\), arrows

Their long sequence starts with \(0\to T^0(A)\to T^0(B)\to T^0(C)\), continues through (1.1) and all the higher terms, and is exact everywhere. The connecting arrows are natural for every commuting morphism of short exact sequences. In particular \(T^0\) is left exact.

A morphism \(t:T\to U\) consists of natural transformations \(t^n:T^n\to U^n\) satisfying, for every \(e\) and \(n\),

We call \(T\) universal when every natural transformation \(T^0\to U^0\), for every delta-functor \(U\), extends uniquely to such a morphism. The target \(U\) is not required to be effaceable.

For \(n>0\), degree \(n\) is effaceable if for every \(A\) there is a monomorphism \(u:A\hookrightarrow M\) such that \(T^n(u)=0\). The choice may depend on both \(A\) and \(n\). Effacement concerns a zero induced arrow, and does not require \(T^n(M)=0\).

2. Effacement gives the complete universal property

Theorem 2.1. If every positive degree of \(T\) is effaceable, then \(T\) is universal.

Proof. Fix a natural transformation \(t^0:T^0\to U^0\). Suppose the transformations through degree \(n\) have already been constructed, are natural, and satisfy (1.2) for connecting arrows landing in those degrees. For \(A\), choose a monomorphism \(u:A\to M\) that effaces degree \(n+1\), and put \(C_u=\operatorname{coker}u\). Exactness gives an epimorphism

Naturality of \(t^n\) for \(M\to C_u\), and the zero consecutive composite in the long sequence for \(U\), show that \(\delta^n_{U,u}t^n_{C_u}\) annihilates this image. In an abelian category an epimorphism is the cokernel of its kernel. There is therefore a unique arrow defined by

We now prove that this definition has all the required compatibilities.

First consider a commuting morphism from the short exact sequence associated to \(u:A\to M\) to that associated to an effacing monomorphism \(v:A'\to M'\), with components \(f:A\to A'\), \(g:M\to M'\) and \(h:C_u\to C_v\). Naturality of the connecting arrows, (2.2) and naturality in degree \(n\) give

Cancel the epimorphism in (2.1). This proves compatibility whenever such a morphism of effacing sequences exists.

To compare two choices \(u:A\to M\) and \(v:A\to N\), take the pushout \(P=M\amalg_A N\). Both maps \(M\to P\) and \(N\to P\) are monic, by the pushout calculation in Relative replacements and finite resolution bounds, Section 1. The common map \(w:A\to P\) is monic and effaces degree \(n+1\): its induced arrow factors through \(T^{n+1}(u)=0\). There are morphisms from both chosen short exact sequences to the sequence for \(w\), each with identity on \(A\). Equation (2.3) then says that both definitions at \(A\) coincide with the definition made using \(w\). Thus (2.2) is independent of the choice.

For an arbitrary \(f:A\to A'\), push an effacing monomorphism \(u:A\to M\) along \(f\), obtaining \(P=M\amalg_A A'\). The resulting map \(A'\to P\) is monic. Choose a monomorphism \(P\to E\) that effaces degree \(n+1\) at \(P\). Its composite \(v:A'\to E\) is monic and effaces that degree at \(A'\). The map \(M\to P\to E\), together with \(f\), gives a morphism from the sequence for \(u\) to that for \(v\). Choice independence and (2.3) prove naturality for this arbitrary \(f\). The new components consequently form a natural transformation between the additive functors in degree \(n+1\).

Finally fix any short exact sequence \(e=(0\to A\xrightarrow{i}B\to C\to0)\). Choose an effacing monomorphism \(j:B\to E\) for degree \(n+1\). The composite \(ji:A\to E\) also effaces that degree. There is a morphism of short exact sequences from \(e\) to \(0\to A\xrightarrow{ji}E\to E/A\to0\), with identity on \(A\), map \(j\) on \(B\), and induced \(h:C\to E/A\). Apply (2.2) for \(ji\), precompose with \(T^n(h)\), and use naturality of both connecting maps and of \(t^n\). The result is exactly (1.2) for \(e\). Thus every connecting square commutes, not only those used to define the new components.

The induction constructs a morphism in all degrees. For uniqueness, any extension has to satisfy (2.2) for every chosen effacement. The epimorphism (2.1), and induction on the degree, force each component. This proves existence and uniqueness. \(\square\)

If two universal delta-functors have an isomorphism between their degree-zero functors, extend that isomorphism and its inverse. Each composite extends an identity, so is an identity by uniqueness. The resulting isomorphism is unique with the specified degree-zero component. With three such comparisons the composite equals the direct comparison, again by uniqueness. Identities, compositions and sums of initial transformations extend to identities, compositions and sums, since each of those operations preserves (1.2).

3. Higher derived functors and dimension shifting

Let \(F:\mathcal A\to\mathcal B\) be additive and left exact, with \(\mathcal A\) having enough injectives. The prerequisite constructs the exact functor \(RF:D^+(\mathcal A)\to D^+(\mathcal B)\), natural maps \(\eta_X:Q_{\mathcal B}F(X)\to RF(Q_{\mathcal A}X)\), and the natural long exact sequences for \(R^nF(A)=H^nRF(A[0])\).

These higher functors, in nonnegative degrees, are a delta-functor with degree zero naturally identified with \(F\). For \(n>0\), an injective embedding \(A\to I\) effaces degree \(n\), since \(R^nF(I)=0\). Theorem 2.1 proves:

Corollary 3.1. The higher right derived functors of a left exact functor form a universal delta-functor. Every transformation \(F\to U^0\) extends uniquely, including all its connecting-map squares. Any effaceable cohomological delta-functor with degree zero identified with \(F\) is uniquely isomorphic to \(R^\bullet F\) with that identification.

For an object \(E\), call it \(F\)-acyclic when \(R^nF(E)=0\) for every \(n>0\). Left exactness and the vanishing of negative derived degrees say equivalently that

Indeed the map identifies degree zero and the source has no other cohomology; a map is invertible in the derived category exactly when it is a quasi-isomorphism.

For a short exact sequence \(0\to A\to E\to C\to0\) with \(E\) acyclic, its long sequence gives

All these maps are the actual connecting maps; their naturality follows from that of the exact-sequence triangle. Repeating (3.2) along an acyclic resolution gives dimension shifting with coherent comparisons, rather than an unrelated choice of isomorphism for each group.

Some useful closure statements follow from the same long sequence. In \(0\to A\to B\to C\to0\), if \(A,B\) are acyclic, so is \(C\); if \(A,C\) are acyclic, so is \(B\). In either case \(0\to F(A)\to F(B)\to F(C)\to0\) is exact. If \(B,C\) are acyclic, then \(A\) is acyclic exactly when \(F(B)\to F(C)\) is epic: the only possible positive obstruction is the cokernel identifying \(R^1F(A)\) in (3.2), and every higher obstruction is zero. Finite biproducts of acyclic objects are acyclic because \(RF\) and cohomology are additive. In particular injectives and the zero object are acyclic.

4. Complete bounded-below acyclic computation

Theorem 4.1. If every term of a bounded-below complex \(E\) is \(F\)-acyclic, then its canonical map

is invertible. Consequently a bounded-below acyclic resolution computes the derived functor, including comparisons on morphisms.

Proof. The transformation \(\eta\) respects shifts and cone triangles. To check this from the construction, use \(\eta_E=QF(q_E)\). The comparison of a replacement of \(E[1]\) with \(I_E[1]\) is the unique homotopy class compatible with the shifted replacement map. Applying \(F\) makes the shift square commute. A termwise split short exact sequence represents a cone triangle in the homotopy category. Naturality of \(q\) there, followed by the exact functor \(RF\), makes its image a morphism of distinguished triangles. These observations prove the compatibility needed below without selecting strictly commuting chain replacements.

First suppose \(E\) has finitely many nonzero terms, in degrees \(a\) through \(b\). A one-term complex is a shift of an acyclic object, so (4.1) is invertible by (3.1). For more terms, the termwise split short exact sequence

separates the bottom term from the remaining tail. Here a stupid truncation retains the stated terms and sets the others to zero, with the boundary differential set to zero. The sequence is termwise split, so applying additive \(F\) preserves it, and it gives a triangle already in the homotopy category. Induction makes the two outer components of the triangle morphism \(\eta\) invertible. The two-vertex criterion then makes the middle component invertible.

Now let \(E^k=0\) for \(k<a\), with no upper bound. Fix an integer \(i\geq a\), and use the termwise split sequence

Write its tail as \(L\) and finite head as \(B\). A replacement of \(L\) starts in degree \(i+2\), by the bounded replacement construction. Thus \(RF(QL)\) and \(QF(L)\) have zero cohomology in degrees \(i\) and \(i+1\). The cohomology sequences of (4.3) and its \(RF\) image identify both middle degree-\(i\) objects with the corresponding degree-\(i\) objects of the head. Naturality identifies the middle comparison with the head comparison, which is invertible by the finite case. In degrees \(i<a\), both \(QF(E)\) and its derived value have zero cohomology, by using a replacement starting in degree \(a\). Therefore (4.1) induces isomorphisms in every degree and is invertible. No infinite totalization, infinite exactness assertion, or convergence interchange occurred. \(\square\)

If \(q:A[0]\to E\) is a quasi-isomorphism to such a complex, the induced comparison is

It is invertible. Naturality gives \(\eta_E QF(q)=RF(Qq)\eta_{A[0]}\), so (4.4) has the correct augmentation from \(F(A)[0]\). A quasi-isomorphism \(f:E\to E'\) between bounded-below complexes of acyclic objects is sent by \(F\) to a quasi-isomorphism: in the square for \(\eta\), both vertical maps and \(RF(Qf)\) are invertible, forcing \(QF(f)\) to be invertible. For a map between two augmented resolutions that extends \(g:A\to A'\), the same naturality square proves that the induced map is precisely \(RF(g)\) under (4.4). Two such maps give the same derived comparison even when no chain homotopy between their representatives has been supplied.

There is also a categorical description of all these comparisons. Let \(\mathcal E_F\) be the full additive subcategory of acyclic objects, and let \(K^+(\mathcal E_F)\) consist of bounded-below complexes with terms in it. Finite-biproduct closure makes this homotopy category closed under shifts and cones. Invert its arrows that are quasi-isomorphisms in \(\mathcal A\), obtaining a category \(\mathcal L_F\). Its functor to \(D^+(\mathcal A)\) is an equivalence.

To prove the assertion, include the injective resolution functor \(J:D^+(\mathcal A)\to K^+(\mathcal I)\) into \(\mathcal L_F\); injectives are acyclic, so this is defined. On every \(E\) the replacement \(q_E:E\to I_E\) lies in \(K^+(\mathcal E_F)\) and becomes invertible in \(\mathcal L_F\). Equation \(j(f)q_E=q_{E'}f\) in the homotopy category proves naturality for every original arrow. Naturality extends to the formal inverses by inverting that square, hence to their composites. This gives an isomorphism from identity to the composite through \(J\). In the other direction the derived isomorphisms \(Qq_X\) give identity to \(QJ\). The two composites are thus naturally isomorphic to identity, which proves the equivalence, including full faithfulness. Transporting the distinguished triangles along it makes this an exact equivalence; the images of the original cone triangles are distinguished.

Theorem 4.1 makes \(QF\) invert every specified arrow in this category. Its descended functor is identified with \(RF\) by \(\eta\). This describes derived maps between acyclic models without asserting that their ordinary homotopy Hom groups are already the derived Hom groups.

5. The canonical comparison for a composition

Let \(F:\mathcal A\to\mathcal B\) and \(G:\mathcal B\to\mathcal C\) be additive left exact functors. Suppose \(\mathcal A,\mathcal B\) have enough injectives. The last abelian category \(\mathcal C\) need not have enough injectives. All three derived functors \(RF,RG,R(GF)\) therefore exist by the construction already proved.

Theorem 5.1. There is a canonical natural transformation

It is an isomorphism on all of \(D^+(\mathcal A)\) if and only if \(F(I)\) is \(G\)-acyclic for every injective \(I\) of \(\mathcal A\).

Proof. For \(X\in D^+(\mathcal A)\), choose its injective replacement \(I_X\). Define the component by the canonical replacement transformation for \(G\) applied to the bounded-below complex \(F(I_X)\):

An arbitrary derived arrow \(\alpha:X\to Y\) is represented between the replacements by the homotopy class \(J\alpha\). Naturality of \(\eta^G\) for the arrow \(F(J\alpha)\) proves naturality of (5.2) for this arbitrary derived arrow. Thus the definition is not limited to maps between objects in degree zero.

For a bounded-below complex \(X\), naturality of \(\eta^G\) on \(F(q_X)\) gives the compatibility

The full universal transformation property of \(R(GF)\), with target functor \(RG\,RF\), says that (5.3) uniquely determines \(t\). Changes of replacement choices consequently change (5.1) by exactly their canonical comparisons; their cocycle law is preserved. The same construction respects shifts, since both replacement transformations do.

If every \(F(I)\) is \(G\)-acyclic, every term of \(F(I_X)\) is \(G\)-acyclic. Theorem 4.1 makes (5.2) invertible for every \(X\). Conversely evaluate an everywhere invertible comparison at an injective object \(I[0]\). Its normalized replacement is itself, and (5.2) is precisely \(\eta^G_{F(I)[0]}\). Formula (3.1) then says that \(F(I)\) is \(G\)-acyclic. This proves both directions. \(\square\)

For completeness these canonical comparisons are associative even before they are known to be invertible. Let \(H:\mathcal C\to\mathcal D\) also be additive left exact and suppose \(\mathcal C\) has enough injectives. The two transformations from \(R(HGF)\) to \(RH\,RG\,RF\) satisfy

Here each expression includes the usual objectwise associations of functor composition. To check the equality, fix \(X\), set \(I=I_X\), and choose a replacement \(v:F(I)\to J\) in \(\mathcal B\). The first path in (5.4) is

The second path first applies \(QHG(v)\) and then \(\eta^H_{G(J)}\). The square expressing naturality of \(\eta^H\) for the chain map \(G(v)\) says exactly that these paths agree. The descriptions use the same chosen replacements, and their already proved canonical choice comparisons transport the equality to any other choices. This proves (5.4) on every object and hence as a natural transformation. Whenever the comparisons in question are invertible, this supplies the corresponding associativity of their isomorphisms.

6. Two computed examples

Example 6.1 (torsion and its connecting arrow). Fix an integer \(m\geq2\), and put \(F_m(A)=\operatorname{Hom}_{\mathbb Z}(\mathbb Z/m,A)=A[m]\) on abelian groups. The complex \(P\) with \(P^{-1}=P^0=\mathbb Z\), differential \(-m\), and augmentation by reduction in degree zero is a projective resolution of \((\mathbb Z/m)[0]\). Freeness proves projectivity: lift the image of the generator across an epimorphism. The bounded K-projective test applies to \(P\).

The Hom differential convention in the prerequisites gives, for a group in degree zero,

This complex computes \(RF_m(A)\). Here is the comparison that proves that assertion. The augmentation gives a natural chain map \(\operatorname{Hom}(\mathbb Z/m,X)\to\operatorname{Hom}^{\bullet}(P,X)\). The functor on the right preserves shifts and cones, by their two-summand formulas, and sends quasi-isomorphisms to quasi-isomorphisms by the K-projective Hom test in every shifted degree. It therefore descends to the derived category. On a bounded-below injective complex \(I\), the augmentation-induced map is a quasi-isomorphism: its degree-\(n\) cohomology map is precomposition with \(P\to(\mathbb Z/m)[0]\) on \(\operatorname{Hom}_K(-,I[n])\), which is invertible by the K-injective test. The universal derived transformation property thus supplies a natural isomorphism between the descended Hom functor and \(RF_m\); evaluate on replacements to see invertibility on every object. The maps respect shifts and cone projections, so also the exact-sequence connecting arrows. In particular

For \(0\to A\xrightarrow{i}B\xrightarrow{p}C\to0\), a class \(c\in C[m]\) can be lifted to \(b\in B\). There is a unique \(a\in A\) with \(i(a)=mb\). With the fixed cone projection convention the connecting arrow is

Indeed the termwise exact sequence of the two-term complexes (6.1) has a cone cycle \((b,-a)\) mapping to \(c\), and the cone projection maps it to \(-a\). Replacing \(b\) by \(b+i(x)\) changes \(a\) by \(mx\), so the result is independent of the lift. The formula is natural for commuting maps of short exact sequences. For \(0\to\mathbb Z\xrightarrow{m}\mathbb Z\to\mathbb Z/m\to0\), it is minus identity on \(\mathbb Z/m\), and is an isomorphism as required by the long sequence.

Example 6.2 (an acyclic resolution with a noninjective term). For \(F_2\), formula (6.2) says that a group is acyclic exactly when multiplication by two is surjective. Thus both \(\mathbb Z[1/2]\) and \(\mathbb Z[1/2]/\mathbb Z\) are acyclic: divide a representative by two. There is an acyclic resolution

Its first resolution term is not injective. The homomorphism \(3\mathbb Z\to\mathbb Z[1/2]\) taking \(3\) to \(1\) cannot extend to \(\mathbb Z\), since an extension would require an element whose triple is \(1\), and \(1/3\notin\mathbb Z[1/2]\). Applying \(F_2\) to the resolution gives zero in degree zero and the two-element subgroup \(\{[0],[1/2]\}\) in degree one. Theorem 4.1 therefore computes \(RF_2(\mathbb Z)\) with cohomology \(0,\mathbb Z/2\) in degrees zero and one, exactly as (6.2) predicts. The use of acyclic terms genuinely enlarges the available models beyond injective terms.

7. Four graded exercises with complete solutions

Exercise 1 (foundation: why the effacing arrow is monic). Show that allowing an arbitrary arrow in the definition of effacement would make the condition vacuous. Construct a nonuniversal delta-functor for which every positive induced map can be killed by an arrow to zero.

Solution. Any additive functor sends a zero object to a zero object, so \(A\to0\) kills its induced map in every degree. On abelian groups define \(T^0=0\), \(T^1=\operatorname{id}\), and \(T^n=0\) for \(n\geq2\), with every connecting map zero. For a short exact sequence its only nonzero portion is the original short exact sequence in degree one. Thus the long sequence is exact at every position, and the connecting maps are natural. There are two distinct endomorphisms of this delta-functor with the same degree-zero component: take either identity or zero in degree one. Both commute with the zero connecting maps. The unique transformation \(T^0\to T^0\) therefore has more than one extension, disproving universality. A monomorphism that kills the identity functor at a nonzero group cannot exist, so this example does not satisfy the actual hypothesis of Theorem 2.1.

Exercise 2 (intermediate: the truncation must retain one more term). In the proof of Theorem 4.1, explain why using \(\sigma_{\leq i}E\) in place of \(\sigma_{\leq i+1}E\) need not preserve degree-\(i\) cohomology. Give a counterexample with an exact functor.

Solution. Cohomology in degree \(i\) involves both the incoming differential and the outgoing differential to degree \(i+1\). Cutting at \(i\) replaces that outgoing differential by zero and can enlarge its kernel. Take \(F=\operatorname{id}\) on abelian groups, \(i=0\), and \(E=(\mathbb Z\xrightarrow{1}\mathbb Z)\) in degrees zero and one. Every object is \(F\)-acyclic, since an exact functor has no positive derived functors, and the complex has zero cohomology. Its head \(\sigma_{\leq0}E=\mathbb Z[0]\) has \(H^0=\mathbb Z\). In the sequence with tail starting at degree one, \(H^1\) of the tail is \(\mathbb Z\), so the cohomology sequence does not identify the middle and head groups in degree zero. Starting the tail at degree \(i+2\), as in (4.3), removes precisely this obstruction.

Exercise 3 (advanced: composition can fail). Let \(F=G=F_m\) on abelian groups, with \(m\geq2\). Although \(GF\) is naturally isomorphic to \(F_m\), show that \(R(GF)\to RG\,RF\) is not everywhere invertible.

Solution. Every element of \(F_m(A)=A[m]\) is killed by \(m\), so evaluation at \([1]\) identifies \(F_m(F_m(A))\) naturally with \(F_m(A)\). Let \(I=\mathbb Q/\mathbb Z\), which is injective by the divisible-group proof in the prerequisite. Its \(m\)-torsion subgroup is \(\{[a/m]:a\in\mathbb Z\}\simeq\mathbb Z/m\). Thus \(R(GF)(I[0])=GF(I)[0]\) has no positive cohomology, whereas

The comparison cannot be invertible because it cannot induce an isomorphism on this degree. Equivalently \(F_m(I)\) is not \(G\)-acyclic, which violates exactly the necessary condition in Theorem 5.1. The example checks the necessity using an actual injective object, rather than the behavior of an arbitrary group.

Exercise 4 (synthesis: all transformations between Ext theories). In an abelian category \(\mathcal A\) with enough injectives, let \(M,N\) be objects. Determine all morphisms of delta-functors from \(R^\bullet\operatorname{Hom}_{\mathcal A}(M,-)\) to \(R^\bullet\operatorname{Hom}_{\mathcal A}(N,-)\). Determine the composition order for endomorphisms when \(M=N\).

Solution. For a degree-zero transformation \(s:\operatorname{Hom}(M,-)\to\operatorname{Hom}(N,-)\), set \(u=s_M(1_M):N\to M\). Naturality for \(f:M\to A\) says \(s_A(f)=fu\). Conversely every such \(u\) defines an additive natural transformation by this formula. Thus the transformations in degree zero are in bijection with \(\operatorname{Hom}(N,M)\). Theorem 2.1 and Corollary 3.1 extend each uniquely to all degrees, and every delta-functor morphism has this form. Under the derived Hom identification, its degree-\(n\) component is precomposition with \(u[0]\) on \(\operatorname{Hom}_{D(\mathcal A)}(M[0],A[n])\). It commutes with connecting arrows because those are obtained by postcomposition with the natural derived connecting morphism; precomposition and postcomposition commute.

For endomorphisms, write \(s_u(f)=fu\). Applying \(s_v\) first and \(s_u\) second gives \(s_us_v(f)=fvu\). Hence composition corresponds to the reversed product of source endomorphisms, while addition corresponds to ordinary addition. The full ring of delta-functor endomorphisms is therefore

This also proves that a specified degree-zero comparison leaves no freedom to alter just one higher Ext component or one connecting square.

References and further reading

• The Stacks Project, cohomological delta-functors, for the terminology and the effaceability criterion. Section 2 above supplies the choice, naturality and connecting-map verifications explicitly.
• The Stacks Project, Leray's acyclicity lemma, for bounded-below computation by acyclic terms.
• The Stacks Project, composition of right derived functors, for the composition criterion and further spectral-sequence developments. This lesson proves the canonical comparison, its criterion and associativity; a construction of the spectral sequence requires additional filtered-complex mathematics.