Draft lesson.

Unbounded injective replacements and derived functors

An injective object solves extension problems for one degree. A K-injective complex solves the extension problems that remain after all degrees and differentials interact. We construct functorial K-injective replacements for arbitrary complexes in a Grothendieck abelian category, without a boundedness or exact-products assumption. The construction attaches a fixed set of small acyclic monomorphisms and continues through an ordinal of sufficiently large cofinality. We then prove the unbounded derived-functor transformation property, products and coproducts in the derived category, and the exact-adjoint consequence.

We use Chain homotopies, mapping cones and triangles for the cochain Hom differential, cone signs and cohomology exactness. Derived morphisms through quasi-isomorphism roofs, Sections 4–5, proves localization and the K-injective test in all degrees. Triangles after quasi-isomorphism localization proves the derived triangulation. Testing injectivity on generators and constructing enough injectives, Lemma 7.1, supplies the complete cardinal-product and large-cofinality proof. Injective replacements and right derived functors supplies the bounded-below comparison and proves the self-injectivity of the dual-number ring used below.

Throughout, \(\mathcal A\) is a locally small Grothendieck abelian category: it has all small coproducts, filtered colimits are exact, and it has a generator \(G\), meaning \(\operatorname{Hom}(G,-)\) is faithful. We use set-indexed Choice and fixed ambient universes as in Universes and small categories. Functor categories and their chosen universal constructions may be organized in an enlargement; every indexing set and ordinal used for one object is small in the original universe.

1. Size bounds and small epimorphic lifts

First, all small colimits exist. For a small diagram take the coproduct of its objects, and quotient by the image of the map from the coproduct over its arrows whose component is the difference between the target-arrow inclusion and the source inclusion. An arrow from this quotient is exactly a compatible family of arrows from the diagram. This proves the colimit universal property, including the empty diagram.

The evaluation map \(\bigoplus_{f:G\to X}G\to X\) is epic. If its image were proper, the nonzero quotient map from \(X\) would annihilate every map from \(G\), contradicting faithfulness. A subobject \(Y\subset X\) is determined by the subset of \(\operatorname{Hom}(G,X)\) consisting of maps that factor through \(Y\): those maps generate \(Y\) by the same evaluation argument. Thus \(\operatorname{Sub}(X)\) is a small set. Put

Subobjects and quotients have size at most the size of the original object: composition embeds a subobject's subobject set, and inverse image embeds a quotient's subobject set. The function \(c\) is nondecreasing, since padding a coproduct with zero components gives a split inclusion into every larger coproduct.

If \(s(X)\leq\mu\), there is an epimorphism from at most \(\mu\) copies of \(G\) to \(X\). For each proper subobject \(Y\subset X\), faithfulness applied to \(X\to X/Y\) chooses a map \(G\to X\) not factoring through \(Y\). The sum of these maps is epic: otherwise its image is one of the subobjects it was chosen to escape. After padding with zeros, \(X\) is a quotient of \(\bigoplus_\mu G\). Therefore the objects of size at most a fixed \(\mu\) have only a small set of isomorphism classes, represented among the quotients of that one coproduct.

Lemma 1.1 (small lift). For an epimorphism \(B\to N\), there is a subobject \(E\subset B\) mapping onto \(N\), with \(E\) a quotient of at most \(s(N)\) copies of \(G\). In particular

Proof. For every proper subobject \(V\subset N\), the composite \(B\to N/V\) is nonzero. Faithfulness gives a map \(G\to B\) whose composite is nonzero. Let \(E\) be the image of the sum of all these maps. If its image in \(N\) were a proper subobject \(V\), the component chosen for that very \(V\) would contradict it. The asserted presentation and size bound follow from its defining image and the quotient inequality. \(\square\)

Exact filtered colimits preserve kernels, images and cohomology. Indeed apply their exactness to \(0\to Z^nX\to X^n\to B^{n+1}X\to0\) and \(0\to B^nX\to Z^nX\to H^nX\to0\). Their colimits give the same two kernel/image sequences for the colimit complex. Arbitrary coproducts are exact too: they are filtered colimits of finite partial biproducts, whose exactness follows from the component projections and inclusions.

We also need smallness along increasing unions. If \((B_\alpha)_{\alpha<\theta}\) is a continuous increasing chain and \(f:M\to\operatorname{colim}B_\alpha\), the inverse images \(f^{-1}B_\alpha\) have union \(M\). To verify that assertion from AB5, express the pullback as the kernel of \(M\oplus B_\alpha\to\operatorname{colim}B_\alpha\); taking filtered colimits gives the graph of \(f\), whose projection is all of \(M\). If \(s(M)\leq\lambda\) and \(\operatorname{cf}(\theta)>\lambda\), choose one stage for each distinct inverse-image subobject. Those at most \(\lambda\) stages have a common upper bound below \(\theta\); their union is therefore attained at that stage. Thus \(f\) factors there. Equal factorizations stay equal because the transition maps and their colimit inclusions are monic.

For a complex whose component sizes are at most \(\lambda\), the same factorization holds if additionally \(\operatorname{cf}(\theta)>\aleph_0\). Factor each degree and take a common upper bound for their countably many stages. The component factorizations satisfy the chain-map equations because they do so after the monomorphism into the colimit. This proves smallness for complexes, not just for individual objects.

Fix the following parameters, which depend only on \(G\):

Here \(\theta\) denotes the initial ordinal of the successor cardinal. The linked Lemma 7.1 proves, using the infinite cardinal-product calculation and Choice, that it is a small infinite limit ordinal and that every family of at most \(\lambda\) ordinals below it is bounded below it. In particular \(\operatorname{cf}(\theta)>\lambda\geq\aleph_0\). No fixed point inequality \(c(\kappa)\leq\kappa\) is assumed.

2. A fixed set generates all acyclic monomorphisms

A monomorphism of complexes means a monomorphism in every degree. Call it acyclic if its quotient complex is acyclic; cohomology exactness shows this is equivalent to its being a quasi-isomorphism.

Lemma 2.1. Every nonzero acyclic complex \(M\) contains a nonzero acyclic subcomplex \(N\), bounded above, with \(s(N^n)\leq\kappa\) in every degree.

Proof. Some cycle object of \(M\) is nonzero. Otherwise every differential has zero kernel, while its image is the next zero cycle object, forcing every term to be zero. Choose a nonzero map \(G\to Z^aM\), and let its image be \(N^a\). Set all terms above \(a\) to zero. Since \(M\) is acyclic, the inverse image of \(N^a\) under \(M^{a-1}\to Z^aM\) maps onto it. Lemma 1.1 chooses a subobject \(N^{a-1}\) mapping onto \(N^a\), of size at most \(c(\kappa_0)\leq\kappa_1\). Next lift the kernel of that restricted differential through \(M^{a-2}\to Z^{a-1}M\), using the same lemma. Continue downward. At the \(j\)-th step its size is at most \(\kappa_j\), and

This gives a subcomplex, proves acyclicity in every degree, and gives the uniform bound \(\kappa\). Its nonzero top term proves it is nonzero. Each degree is chosen once in this downward recursion; there is no inference that \(c\) preserves a countable supremum. \(\square\)

Consequently every acyclic complex \(Q\) has a continuous filtration \(0=Q_0\subset Q_1\subset\cdots\subset Q\) by acyclic subcomplexes, whose nonzero successive quotients have the properties of Lemma 2.1. At a successor stage choose such a subcomplex of the nonzero quotient and take its inverse image. At a limit take the union; it is acyclic by AB5. The quotient remains acyclic by cohomology exactness. The process reaches \(Q\): subcomplexes form a small set, since they are particular sequences of subobjects of its terms. A strictly increasing chain cannot have more distinct stages than that set; if the whole complex has not been reached the next step is strict. Taking an ordinal of larger cardinality proves termination at a small stage.

We next lift a bounded-above small quotient using small complexes on both sides of a monomorphism. Suppose \(A\subset B\) and \(B/A=N\) is acyclic, bounded above, and \(s(N^n)\leq\kappa\). In each degree use Lemma 1.1 to choose \(E^n\subset B^n\) mapping onto \(N^n\), with a presentation by at most \(\kappa\) copies of \(G\). Take \(E^n=0\) above the upper bound of \(N\), and set

Then \(V\) is a subcomplex of \(B\), \(U\) is a subcomplex of \(A\), and \(V\to N\) is epic degree by degree. Its image lies in \(N\), and contains the image of \(E^n\), which is all of \(N^n\). Hence \(V/U=N\). Both complexes are bounded above, with at most one extra upper degree. Every \(V^n\) is a quotient of two coproducts of at most \(\kappa\) generators, and thus of at most \(\kappa\) generators, by the infinite cardinal-product calculation. Therefore \(s(U^n),s(V^n)\leq c(\kappa)\leq\lambda\).

The original inclusion is the pushout of \(U\to V\) along \(U\to A\):

Indeed the sum \(A+V\) is all of \(B\), because it maps onto \(B/A=N\), and the intersection of \(A\) and \(V\) is \(U\). The map from the biproduct to their sum has exactly the difference-of-inclusions kernel \(U\), proving the pushout universal property.

Choose a small set \(\mathcal S\) of representatives of all acyclic monomorphisms \(i_s:U_s\to V_s\) between bounded-above complexes with component sizes at most \(\lambda\). Such a set exists: Section 1 supplies a set of component-object representatives; sequences of those objects and their differentials form a set, as do chain maps between them. The boundedness, monomorphism and acyclicity conditions select a subset.

Theorem 2.2. Every acyclic monomorphism is a transfinite composite of pushouts of members of \(\mathcal S\).

Proof. For \(A\subset B\) with acyclic quotient \(Q\), use its acyclic filtration above. The inverse images in \(B\) give a filtration from \(A\) to \(B\), and each successor quotient is a bounded-above acyclic complex of component size at most \(\kappa\). Apply (2.2)–(2.3) to that successor inclusion. Its small pair is isomorphic to a member of \(\mathcal S\), so the successor is a pushout of such a member. At a limit the inverse image is the union of the earlier inverse images. This follows from AB5's pullback-with-union argument in Section 1, or from taking the colimit of their short exact sequences with the fixed subcomplex \(A\). The filtration terminates at \(B\). \(\square\)

3. Functorial attachment and the terminal lifting property

For a complex \(X\), let \(\Gamma(X)\) be the set of pairs \((s,f)\) with \(s\in\mathcal S\) and a chain map \(f:U_s\to X\). Define \(\mathsf A(X)\) by the pushout

In an abelian category a pushout of a monomorphism is monic, with the same cokernel. This can be checked in the model \((X\oplus\bigoplus V_s)/\operatorname{im}(f,-i_s)\): a kernel arrow from \(X\) would give a source arrow killed by the monic \(\oplus i_s\), and is therefore zero. The quotient by \(X\) is the quotient of the right-hand coproduct by the image of the upper arrow. Consequently

Each summand is acyclic, and exact coproducts preserve cohomology, so \(j_X\) is an acyclic monomorphism. Every \(f\) occurring in its index extends across \(i_s\) to \(\mathsf A(X)\).

A chain map \(u:X\to Y\) sends the label \((s,f)\) to \((s,uf)\), with identity maps on its \(U_s,V_s\). Distinct labels may merge; the coproduct maps then add their components into the same summand. The evaluation square commutes, giving a unique map \(\mathsf A(u)\). Checking on coproduct components and on \(X\) proves identities, compositions and \(\mathsf A(u)j_X=j_Yu\). Thus \(\mathsf A\) and \(j\) are functorial. This construction is not asserted to be additive on chain maps.

Iterate with all connecting maps and their compositions:

At a successor the connecting map is \(j\); at a limit it is the colimit leg. Their composition laws determine every earlier-to-later map. The preceding naturality and the colimit universal property define each \(X_\alpha\) on arbitrary chain maps, by transfinite induction. All connecting maps are monic quasi-isomorphisms: this holds at successors by (3.2); at limits monicity and cohomology follow by exact filtered colimits. The initial maps are therefore monic quasi-isomorphisms too, and

For \(s\in\mathcal S\), any map \(U_s\to I(X)\) factors through \(X_\alpha\) for some \(\alpha<\theta\), by Section 1's complex smallness bound. Its labelled attachment supplies an extension \(V_s\to X_{\alpha+1}\to I(X)\), with \(\alpha+1<\theta\) since \(\theta\) is a limit ordinal.

This extension property propagates to pushouts and transfinite composites. For a pushout, combine the map on the old target with the chosen extension on the cell using its universal property. At a successor of a transfinite composite perform that extension; at a limit the already compatible maps define a map out of the colimit. Theorem 2.2 now proves that every map into \(I(X)\) extends across every acyclic monomorphism of complexes.

4. K-injectivity, localization and the total derived functor

Theorem 4.1. For a complex \(J\), the extension property against all acyclic monomorphisms is equivalent to: every \(J^n\) is injective and \(J\) is K-injective. In particular (3.3) is a functorial, termwise monic K-injective replacement of every arbitrary complex.

Proof. Let \(D^n(A)\) be the complex with \(A\) in degrees \(n,n+1\), identity differential, and zero elsewhere. A chain map \(D^n(A)\to J\) is determined by an arbitrary map \(A\to J^n\); its next component must be the composite with \(d_J^n\). For every object monomorphism \(A\to B\), the inclusion \(D^n(A)\to D^n(B)\) has acyclic quotient. The extension property thus proves injectivity of \(J^n\).

For an acyclic complex \(N\), its inclusion into the first summand of the identity cone is an acyclic monomorphism. With the chosen cochain signs,

Every map \(N\to J\) extends to this contractible cone and is consequently null homotopic, by composing the displayed contraction with that extension and inclusion. Thus \(J\) is K-injective.

Conversely let \(A\subset B\) have acyclic quotient \(Q\), and let \(f:A\to J\) be a chain map. Termwise injectivity gives a graded extension \(e:B\to J\). Its chain-map defect factors through \(Q\):

where \(\pi:B\to Q\) and \(D\) is the cochain Hom differential. The defect kills \(A\), and applying \(D\) again and cancelling each epimorphic \(\pi^n\) proves the cycle equation. The Hom-complex identity gives \(H^1\operatorname{Hom}^\bullet(Q,J)=\operatorname{Hom}_{K}(Q,J[1])=0\): shifts preserve the K-injective test. Hence \(a=Db\) for a degree-zero graded map \(b:Q\to J\). The map \(e-b\pi\) is a chain map and still extends \(f\). This proves the converse. \(\square\)

Let \(K_{\mathrm{inj}}(\mathcal A)\) be the full homotopy subcategory of K-injective complexes. The unbounded roof test proves that

is fully faithful; Theorem 4.1 proves essential surjectivity. Its Hom sets are small in the original universe: a chain map is an element of a countable product of component Hom sets, and homotopy is an equivalence relation on that set. Computing a derived Hom set using the K-injective target in (4.3) therefore proves local smallness of the unbounded derived category, even when localization was initially taken in an ambient enlargement.

The strict replacement functor \(I\) descends to an additive functor on the homotopy category. If \(u,v\) are homotopic, (3.4) makes \(I(u)-I(v)\) vanish after precomposition with \(q_X\) in \(K\); the K-injective test makes that precomposition bijective. The same argument compares \(I(u+v)\) with \(I(u)+I(v)\), and compares \(I(0)\) with zero. Quasi-isomorphisms are sent to quasi-isomorphisms by (3.4); between K-injectives these are homotopy equivalences, again by full faithfulness in (4.3).

K-injectives are closed under shifts and cones. For a cone, apply both exact Hom sequences of its cone triangle to every shifted acyclic source; the two known zero groups force the third to be zero. Thus (4.3) is an exact equivalence. It reflects triangles as well: lift an arrow between two K-injective models, take its cone, and compare its image cone triangle to the given derived triangle. The triangle comparison and the two-vertex isomorphism test make the third vertex isomorphic. Shift and comparison coherence follow by full faithfulness.

Let \(F:\mathcal A\to\mathcal B\) be additive, with \(\mathcal B\) abelian and its derived category taken in the common ambient enlargement. Write \(T=Q_{\mathcal B}K(F):K(\mathcal A)\to D(\mathcal B)\). Applying \(F\) to homotopies and to finite cone sums preserves their equations. Since \(I\) sends quasi-isomorphisms to homotopy equivalences, \(TI\) factors through localization and defines

Its definition on derived arrows is equivalently obtained by the unique K-injective-model arrow of (4.3). Exactness follows by applying \(F\) to the K-injective cone triangles just described. The map \(\eta\) is natural by (3.4).

Theorem 4.2 (the transformation property). For every functor \(H:D(\mathcal A)\to D(\mathcal B)\), composition with \(\eta\) gives a bijection

Proof. Given \(\phi:T\to HQ\), put

Its source is \(RF(QX)=T(I(X))\). Naturality for a homotopy-category arrow follows from naturality of \(\phi\) on \(I(u)\) and (3.4). It then holds for inverses of localized quasi-isomorphisms, and hence for every derived arrow by the roof universal property. Naturality of \(\phi\) on \(q_X\) proves \(\alpha_{QX}\eta_X=\phi_X\).

For uniqueness note that \(I(q_X)\) and \(q_{I(X)}\) have the same precomposition with \(q_X\) by (3.4). Their target is K-injective, so they are equal in the homotopy category. Therefore \(RF(Qq_X)=T(q_{I(X)})=\eta_{I(X)}\). If \(\beta\) maps to \(\phi\), naturality on \(Qq_X\) gives \(H(Qq_X)\beta_{QX}=\beta_{QI(X)}\eta_{I(X)}=\phi_{I(X)}\), forcing (4.6). \(\square\)

Different K-injective choices give canonical natural isomorphisms compatible with \(\eta\). The comparison from \(I_X\) to \(J_X\) is the unique homotopy class whose precomposition with \(q_X\) is the other replacement map. Comparing both composites with the identity after that same precomposition makes them homotopy inverse. Comparing a third choice proves the cocycle; comparing a complex arrow proves naturality, and inverting quasi-isomorphisms proves it for derived arrows. Applying \(F\) preserves all these homotopies and inverses. This also identifies (4.4) on \(D^+\) with the earlier bounded-below derived functor, since its bounded injective replacements are K-injective targets for the same comparison. In particular for left exact \(F\), the previously proved identifications of degree-zero and higher right derived functors agree with this unbounded construction.

5. Ordinary products, derived sums and exact adjoints

We first prove existence of ordinary products in \(\mathcal A\), rather than assuming completeness. Fix a small family \((M_i)\), and consider pairs consisting of an object \(X\) and arrows \(f_i:X\to M_i\). A morphism between pairs preserves all those arrows. This category has all small colimits: form the colimit of the objects in \(\mathcal A\), and use the compatible families of arrows to each \(M_i\).

Every such family factors through a quotient \(C\) of \(X\) whose arrows to the \(M_i\) are jointly monic. Namely take the sum \(L\) of all subobjects of \(X\) annihilated by every \(f_i\) and put \(C=X/L\). That sum is indexed by the small subobject set. If a subobject of \(C\) is annihilated by every induced arrow, its inverse image in \(X\) is annihilated by all \(f_i\), hence contained in \(L\); it must be zero. Thus the induced arrows are jointly monic, and

The product here is only a product of small sets. Let \(\mu\) be an infinite cardinal bounding its size. Evaluation makes \(C\) a quotient of \(\bigoplus_\mu G\). The quotients of that fixed object, together with all their families of arrows to the \(M_i\), form a small set of pairs through which every pair maps. Their coproduct \((W,(w_i))\) is consequently weakly terminal in the pair category.

Let \(\mathcal E\) be its set of endomorphisms in that category, and coequalize each of them with the identity:

The \(w_i\) factor through \(q\), giving the pair \((P,(p_i))\), which is still weakly terminal. Choose a pair morphism \(v:P\to W\) using weak terminality of \(W\). Since \(vq\in\mathcal E\), the coequalizer gives \(qvq=q\), hence \(qv=1_P\). For any endomorphism \(a\) of the pair \(P\), the same argument applied to \(vaq\) gives \(qvaq=q\), and therefore \(a=1_P\). Finally, for two pair morphisms \(u,z:X\to P\), form their pushout with its two legs \(j_1,j_2:P\to Y\). Weak terminality supplies \(h:Y\to P\). Both \(hj_1\) and \(hj_2\) are identities, while \(j_1u=j_2z\); hence \(u=z\). Existence and uniqueness prove that \(P\) is terminal in the pair category, exactly the ordinary product. The argument includes the empty family.

Theorem 5.1. The unbounded derived category has arbitrary small sums and products. Sums are represented by termwise sums of the original complexes. Products are represented by termwise products of K-injective replacements.

Proof. Choose replacements \(X_i\to I_i\). A map or homotopy into a termwise product is exactly a family of maps or homotopies into its factors, by the ordinary product universal property. Consequently \(\prod I_i\) is K-injective: every family of maps from an acyclic source has a family of null homotopies, which defines one homotopy into the product. Similarly a map or homotopy from a termwise sum is a family of maps or homotopies from its summands. Choice lets representatives and homotopies for small families be chosen simultaneously. For a K-injective target \(J\), the unbounded Hom test now gives

Every derived object can be represented by \(J\), so the first formula proves the sum universal property for all targets. The second proves the product property for all sources. The canonical inclusions and replacement-adjusted projections give the specified legs. Products of the injective component objects are injective, since each component of a map extends across an object monomorphism and their extensions assemble using the product. Thus the constructed product model may also be taken termwise injective. \(\square\)

If products in \(\mathcal A\) are exact, a product of quasi-isomorphisms is a quasi-isomorphism: apply the exact product functor to the kernel/image cohomology sequences, or to the acyclic cones. In this additional case the product of the original complexes also represents the derived product. Theorem 5.1 itself uses no exact-products assumption.

Proposition 5.2. Suppose \(\mathcal B\) is Grothendieck, \(L:\mathcal B\to\mathcal A\) is exact, and \(L\) is left adjoint to an additive \(F:\mathcal A\to\mathcal B\). Then \(F\) takes K-injectives to K-injectives, and \(RF\) is right adjoint to the termwise induced \(L:D(\mathcal B)\to D(\mathcal A)\). It preserves all derived products.

Proof. The ordinary adjunction identifies chain maps and chain homotopies \(LN\to I\) with those \(N\to FI\), degree by degree; its naturality makes the differential and homotopy equations correspond. If \(N\) is acyclic then \(LN\) is acyclic by exactness. Thus \(I\) K-injective implies \(FI\) K-injective. Exactness also makes \(L\) preserve quasi-isomorphisms, so it induces the stated derived functor without a replacement. For \(Y\to I_Y\) use both K-injective Hom tests and the homotopy adjunction:

Every bijection is natural, including for derived arrows by localization; hence this is an adjunction with its natural unit and counit. For a product family, apply the bijection and the product universal property: maps from \(X\) into \(RF(\prod Y_i)\) are families of maps from \(LX\) into \(Y_i\), hence families into \(RFY_i\). This proves the product comparison is invertible by its universal property. \(\square\)

6. Two unbounded examples

Example 6.1: all degrees over a field. In vector spaces every acyclic complex is contractible. Choose in each degree a complement \(U^n\) to \(Z^n=B^n\). Its differential is an isomorphism \(U^n\to B^{n+1}\). Define a degree-minus-one homotopy to be its inverse on \(B^{n+1}\) and zero on the complement. Checking on those two summands gives \(dh+hd=1\) in every degree. Every map from that complex into any other complex is therefore null homotopic. Thus every complex of vector spaces is K-injective, with no bounds.

For a general complex choose complements also to \(B^n\) in \(Z^n\). It is a direct sum in each degree of its cohomology representative and the two summands of the preceding contractible construction. Its inclusion and projection to the zero-differential cohomology complex are homotopy inverse. For zero-differential complexes a chain map is a degree-wise family, and a homotopy cannot change that family. Hence

This is the bijection induced by cohomology; the splittings prove its bijectivity, so it is natural regardless of their choices. If \(E_n=k[-n]\), the zero-differential complex having one copy of \(k\) in every integer degree is both \(\bigoplus_nE_n\) and \(\prod_nE_n\): in each degree the family has only one nonzero component. Its endomorphism ring is \(\prod_{n\in\mathbf Z}k\), with coordinatewise composition. These are genuinely unbounded objects and Hom groups.

Example 6.2: injective terms do not suffice. Let \(R=k[\varepsilon]/(\varepsilon^2)\). The earlier injective-replacement lesson proves \(R\) is injective as a left module, by its natural identification with \(\operatorname{Hom}_k(R,k)\) and the restriction adjunction. Consider

Its kernels and images are both \(k\varepsilon\), so it is acyclic, and every term is injective. But it is not contractible. An \(R\)-linear component of a homotopy is multiplication by some \(a_n\in R\); the contraction equation would give

impossible because the right side lies in the proper ideal \((\varepsilon)\). Its identity map is consequently a nonzero homotopy class from an acyclic source to itself, so \(J\) is not K-injective. This verifies directly why the bounded-below termwise-injective theorem cannot simply lose its bound.

7. Graded exercises with solutions

Exercise 1 (routine). Explain the countable-degree cofinality condition when factoring complex maps through a union. Give a bounded-above complex map which factors in every degree through a finite stage of an increasing countable chain, but does not factor as a complex through any one finite stage.

Solution. Section 1 first gives one factorization stage for each integer degree. A single complex map needs one stage above all those countably many stages. Cofinality greater than \(\aleph_0\) supplies it, and monicity into the union verifies the differential equations at the common stage. For the counterexample, let \(A_j\) be the zero-differential vector-space complex with one copy of \(k\) in degrees \(-j,-j+1,\ldots,0\), and zero elsewhere. Include each into the next in the evident components. Its union \(A\) has one copy of \(k\) in every degree \(n\leq0\). The identity map of \(A\) factors in degree \(n\) through \(A_j\) for every \(j\geq-n\), but does not factor through any fixed \(A_j\): in degree \(-j-1\) its target there is zero while the identity component is nonzero. Thus bounded-above domains still need the countable-degree bound. Our choice \(\operatorname{cf}(\theta)>\lambda\geq\aleph_0\) supplies it uniformly.

Exercise 2 (intermediate). In a nonzero Grothendieck category show that \(c(\mu)>\mu\) for every infinite \(\mu\). Explain why this does not obstruct (1.3) and the cell construction, and why replacing it by an assumed fixed point would be wrong.

Solution. The generator is nonzero, since faithfulness from a zero generator would equate every identity map with zero. For every subset \(S\subset\mu\), its coordinate subcoproduct \(\bigoplus_SG\) is a split subobject of \(\bigoplus_\mu G\). Distinct subsets give distinct subobjects: projection to a coordinate in their difference is nonzero on one subcoproduct and zero on the other. Hence there are at least \(2^\mu\) subobjects. Cantor's diagonal argument gives \(c(\mu)\geq2^\mu>\mu\), so no proposed infinite \(c\)-fixed upper bound can exist.

The construction needs no such bound. The downward acyclic construction has its \(j\)-th term bounded by the separately iterated \(\kappa_j\), so every term is bounded by their supremum \(\kappa\). It never applies \(c\) to that supremum and asserts the result stays below it. Lifting a small acyclic quotient to a pair uses the larger bound \(\lambda=\max(\kappa,c(\kappa))\). The set of cell domains has component sizes at most this \(\lambda\), and the still larger cofinality of \(\theta=\lambda^+\) proves their complex smallness. Those are three different bounds serving three proved purposes; all exist by the universe and cardinal arguments already supplied.

Exercise 3 (advanced). Apply \(F=\operatorname{Hom}_R(k,-)\), where \(k=R/(\varepsilon)\), to the acyclic complex in Example 6.2. Compute the result, and construct an explicit valid K-injective replacement showing what the right derived value must be.

Solution. A homomorphism \(k\to R\) is determined by the image of one, which must be annihilated by \(\varepsilon\). Thus each term is \(k\varepsilon\simeq k\), and multiplication by \(\varepsilon\) induces zero on it. Therefore

The second assertion follows because \(J\) is acyclic and hence is the zero derived object. It can also be checked through an explicit replacement. The first-summand inclusion \(J\to C_{1_J}\) is termwise monic and a quasi-isomorphism since its quotient is the acyclic shift \(J[1]\). Formula (4.1) contracts that cone, making it K-injective: every map from any source into a contractible complex is null homotopic by its target contraction. Its terms are finite sums of injectives. An additive functor preserves the contracting equation, so \(F(C_{1_J})\) is contractible. This gives the zero derived value directly. Using only termwise injectivity of \(J\) would instead give the wrong nonzero group in every degree.

Exercise 4 (challenging). For the exact adjunction \(L\dashv F\) of Proposition 5.2, prove termwise injectivity of \(FI\) when \(I\) is termwise injective, identify the derived unit and counit through (5.4), and prove that the derived product is independent of all chosen K-injective models. Specify the extra condition under which original-complex products compute it.

Solution. An object map \(A\to FI^n\) corresponds to \(LA\to I^n\). For \(A\subset B\), exactness makes \(LA\subset LB\), so injectivity of \(I^n\) extends that map. Adjunction converts the extension to \(B\to FI^n\). Together with Proposition 5.2's acyclic-source test this proves the required termwise-injective K-injective target model.

The derived unit at \(X\) is the arrow corresponding under (5.4) to \(1_{LX}\); the counit at \(Y\) corresponds to \(1_{RFY}\). These correspondences are natural and inverse. To check their triangle equations, apply the adjunction bijection: the composites correspond respectively to the same identity arrows, by its naturality. This proves both equations, with no restriction to bounded objects. Comparing different replacements of \(Y\) by the unique homotopy equivalences of Section 4 makes these bijections, and hence the unit and counit, independent of choices.

For a family \(X_i\), comparisons between two K-injective model families are componentwise homotopy equivalences. Their products and the products of their inverse maps exist by the ordinary product property. The component homotopies assemble to homotopies of these products. Hence the two product models are homotopy equivalent, and the comparisons preserve their projection arrows to every \(X_i\) in the derived category. They are the unique isomorphisms with those projection properties, so all identity, composition and cocycle compatibilities follow. Exactness of products in the original abelian category is sufficient for \(\prod X_i\to\prod I_i\) to be a quasi-isomorphism, by the final argument after Theorem 5.1. Without that additional hypothesis, the proved prescription is to use the K-injective models; no exactness of the original product functor was used in constructing them.

8. Further reading

• The Stacks Project, injectives in Grothendieck categories, for size and cofinality arguments.
• The Stacks Project, K-injectives in Grothendieck categories, for unbounded replacements.
• The Stacks Project, additional Grothendieck-category results, for representability and derived products and coproducts.

The theorem proved here covers arbitrary complexes. The filtered convergence conditions in Filtered complexes and Cartan–Eilenberg resolutions remain conditions on a spectral-sequence calculation; existence of an unbounded replacement does not by itself prove convergence of an arbitrary unbounded filtration.