Draft lesson.

Compatible injective classes and exact refinements

Original text public domain (CC0).

A useful resolution class need not consist of injective objects. What matters is that there are enough objects of the class to replace complexes, and that applying the functor preserves acyclicity of complexes built from them. We prove a criterion using compatible extensions of monomorphisms. Both the middle object and the quotient of each extension must remain in the class. The proof constructs a better exact complex first and then uses a signed cone induction to handle the original complex.

Work in abelian categories \(\mathcal A,\mathcal B\), with the universe and choice convention of Universes and small categories. Let \(F:\mathcal A\to\mathcal B\) be additive and left exact. Let \(\mathcal J\subset\mathcal A\) be a full additive subcategory, understood up to isomorphism, containing zero and closed under finite biproducts. Complexes are cochain complexes. A complex is exact or acyclic when every cohomology object vanishes. The abelian comparison supplies the kernels, cokernels, images and finite pushouts used below.

1. The criterion and the resolution property

Call \(\mathcal J\) cogenerating when every \(A\in\mathcal A\) admits a monomorphism into an object of \(\mathcal J\). We impose the following compatible extension property. For every monomorphism \(e:J_0\hookrightarrow A\), with \(J_0\in\mathcal J\), there are a map \(v:A\to J_1\) and an exact row

such that

The map \(v\) need not be monic. The object \(J_2\) is the actual quotient of \(i\). Requiring only a middle object in the class would omit a hypothesis of the criterion.

We call \(\mathcal J\) an F-injective resolution class when:

1. every bounded-below complex \(X\) admits a quasi-isomorphism \(X\to I\) to a bounded-below complex of objects of \(\mathcal J\);
2. \(F(E)\) is acyclic for every bounded-below acyclic complex \(E\) of objects of \(\mathcal J\).

Here \(F\) on complexes means applying \(F\) to every differential and term. The first property alone concerns replacements. The second concerns the functor being derived.

Theorem 1.1. If \(\mathcal J\) is cogenerating and satisfies (1.1)–(1.2), it is an F-injective resolution class. Consequently its complexes compute an exact universal right derived functor

For every quasi-isomorphism \(s:X\to I\), with \(I\in K^+(\mathcal J)\), the value is canonically \(F(I)\) in \(D^+(\mathcal B)\). The identification respects the derived-functor unit and is independent of the replacement. Sections 2–4 prove the acyclicity assertion; Section 5 proves the conclusion about (1.3).

We shall also prove a simpler sufficient criterion. If \(\mathcal J\) is cogenerating, is closed under cokernels of monomorphisms between its objects, and \(F\) preserves every short exact row whose three terms lie in \(\mathcal J\), then it is an F-injective resolution class. Section 6 proves this directly. The compatible extension criterion can apply even when this cokernel closure fails.

2. Every row beginning in the class is F-exact

Lemma 2.1. Under (1.1)–(1.2), every exact sequence

becomes an exact sequence under \(F\).

Proof. Apply (1.1) to \(e\). Since \(pve=pi=0\), the cokernel property of \(q\) gives a unique \(w:C\to J_2\) with \(wq=pv\). The square

is a pullback. To verify this, form \(P=J_1\times_{J_2}C\). Pullbacks of epimorphisms are epimorphisms in an abelian category, so projection to \(C\) gives an exact row \(0\to J_0\to P\to C\to0\). The pair \((v,q)\) defines \(\phi:A\to P\). Its maps on the displayed kernel and quotient are identities. Its kernel is therefore zero: a kernel map for \(\phi\) factors through \(e\), and \(\phi e\) is the monic kernel inclusion of \(P\to C\). Also \(\operatorname{im}\phi\) contains that kernel and maps onto \(C\), since \(q\) is epic. Taking the quotient by the kernel identifies \(P/\operatorname{im}\phi\) with the quotient of \(C\) by the image of \(q\), which is zero. Thus \(\phi\) is both monic and epic, hence an isomorphism.

An additive left exact functor preserves (2.2). Indeed its pullback object is the kernel of \((p,-w):J_1\oplus C\to J_2\); \(F\) preserves this kernel and the biproduct with its actual legs and minus sign. By (1.2), \(Fp\) is epic. Its pullback \(Fq\) is consequently epic. Left exactness already gives the kernel and monicity at the other two positions, proving

All assertions concern the actual arrows of (2.1). No injectivity of \(J_0\), monicity of \(v\), or membership of \(A,C\) in \(\mathcal J\) was assumed. \(\square\)

3. Refine an exact complex while fixing its first term

Lemma 3.1. Suppose \(X\) is acyclic, \(X^n=0\) for \(n<m\), and \(X^m\in\mathcal J\). Under (1.1)–(1.2) there are an acyclic complex \(Y\) and a chain map \(f:X\to Y\) such that

and every term and cycle object of \(Y\) belongs to \(\mathcal J\). Moreover \(F(Y)\) is acyclic.

Proof. Exactness at \(X^m\) makes \(d_X^m:X^m\to X^{m+1}\) monic. Apply (1.1) to this monomorphism. Set \(Y^m=X^m\), obtain \(Y^{m+1}\in\mathcal J\), and let \(Z^{m+2}\in\mathcal J\) be the quotient. The factorization supplied by the criterion gives \(f^{m+1}:X^{m+1}\to Y^{m+1}\) and

For the induction, assume \(Y^m,\ldots,Y^n\), the chain-map components through degree \(n\), and a quotient map \(r:Y^n\twoheadrightarrow Z^{n+1}\) have been constructed. The lower row

is exact, and its terms and all its cycle objects lie in \(\mathcal J\). Here \(n\ge m+1\).

Put \(D=\operatorname{coker}d_X^{n-1}\). Exactness of \(X\) gives the monomorphism \(j:D\hookrightarrow X^{n+1}\) induced by \(d_X^n\). The composite \(rf^n\) kills \(d_X^{n-1}\), by the already established chain equation and (3.3). It therefore factors as \(a:D\to Z^{n+1}\). Form the pushout

with maps \(b:X^{n+1}\to P\), \(e:Z^{n+1}\to P\). Explicitly \(P\) is the cokernel of \((j,-a):D\to X^{n+1}\oplus Z^{n+1}\), and the relation is \(bj=ea\). Since \(j\) is monic, \(e\) is monic. This follows also by the signed cokernel construction: its restriction to the second summand has kernel zero, because \(j\) has kernel zero. Its cokernel agrees with \(\operatorname{coker}j\), by the pushout's universal property.

Apply (1.1) to \(e\). It gives a map \(v:P\to Y^{n+1}\) and an exact row

Define

The quotient factorization \(d_X^n=j\bar q\), where \(\bar q:X^n\to D\), and \(a\bar q=rf^n\) give

Also \(d_Y^nd_Y^{n-1}=0\), since \(r\) kills the preceding image. The kernel of \(d_Y^n\) is the kernel of \(r\), because \(i\) is monic, so exactness at \(Y^n\) persists. The image of \(d_Y^n\) is \(i(Z^{n+1})\), because \(r\) is epic; it is the kernel of \(r'\). This extends (3.3) and its cycle assertion by one degree.

Induction defines a single complex: every term, differential and chain-map component becomes fixed after its stage. There is no infinite colimit or exactness-of-colimits assumption. The resulting \(Y\) is exact at every degree. Its first cycle is zero, its next cycle is \(Y^m\), and the later cycles are the \(Z^n\) just constructed. Each lies in \(\mathcal J\).

Write its cycle rows as

Lemma 2.1 sends every such row to an exact row. Its maps factor \(Fd_Y^n\) as an epimorphism followed by a monomorphism, and identify the latter with the kernel of the next differential. Thus \(\ker Fd_Y^n=\operatorname{im}Fd_Y^{n-1}\) for every \(n\). This proves acyclicity of \(F(Y)\). \(\square\)

4. The signed cone induction for the original complex

We now show that \(F(X)\) is acyclic whenever \(X\in K^+(\mathcal J)\) is acyclic. Its original cycle objects need not belong to \(\mathcal J\); Lemma 3.1 concerns the cycles of a different complex.

Fix a lower bound \(m\). Simultaneously for all such \(X\), prove \(H^b(FX)=0\) by induction on \(b-m\). Degrees \(b<m\) vanish because the terms do. For \(b\ge m\), apply Lemma 3.1 to get \(f:X\to Y\), with \(f^m=1\) and \(FY\) acyclic.

Let \(X_{>m},Y_{>m}\) be the stupid upper tails: their terms vanish in degrees at most \(m\), and agree with the original terms and differentials in higher degrees. The restricted map is \(f_{>m}\). Use the positive cone convention from Chain homotopies, mapping cones and triangles, Sections 1–2 and 6:

Put \(W=\operatorname{Cone}(f_{>m})\). It has \(W^m=X^{m+1}\), vanishes below \(m\), and agrees with \(C\) in degrees greater than \(m\). In particular its lower bound is \(m\), even though both tails vanish through degree \(m\). Its terms lie in \(\mathcal J\).

There is an inclusion \(i:W\to C\). At degree \(m\) it is \(x\mapsto(0,x)\), and above that it is the identity. We give a homotopy inverse explicitly, retaining a general invertible \(u=f^m:X^m\to Y^m\). Set

The only new chain equations occur in degrees \(m-1,m\). In degree \(m-1\), the image under \(d_C\) is \((uz,-d_X^mz)\), killed by \(p^m\). In degree \(m\), the chain equation \(f^{m+1}d_X^m=d_Y^mu\) gives

Here \(d_X^{m+1}d_X^m=0\) removed the extra lower coordinate. We have \(pi=1_W\). Define the degree-minus-one family \(h\) by

In degree \(m-1\), \(hd_C\) is the identity. In degree \(m\), \(d_Ch\) is \((y,-d_X^mu^{-1}y)=1-ip\); at every other degree both sides are zero. Therefore

Thus \(i\) is a homotopy equivalence. The cone cohomology sequence shows \(C\) is acyclic, because \(X,Y\) are. Consequently \(W\) is acyclic too.

Additivity of \(F\) suffices to transport (4.1)–(4.5): it preserves finite biproducts, zero maps, negatives, compositions and sums. In particular \(FC\) identifies with \(\operatorname{Cone}(Ff)\), and \(FW\to FC\) is a homotopy equivalence. The cone cohomology sequence, using the acyclicity of \(FY\), gives

The complex \(W\) is another bounded-below acyclic complex of class objects with the same lower bound \(m\). The degree \(b-1\) is earlier in the simultaneous induction, so the last object vanishes. This proves \(H^b(FX)=0\) and completes the induction. No acyclicity of \(FX\) was used to construct \(Y\) or to establish the homotopy equivalence.

5. Replacements and the universal derived functor

Cogeneration provides the first resolution-class property by the explicit pushout construction of Relative replacements and finite resolution bounds, Sections 1–2. For every \(X\) vanishing below \(m\), it constructs a quasi-isomorphism \(X\to I\), with \(I\) also vanishing below \(m\) and all its terms in \(\mathcal J\). That theorem uses only cogeneration and finite abelian operations. It makes no assumption about \(F\).

Section 4 proves the second resolution-class property. To identify precisely how these imply (1.3), put

The full subcategory \(\mathcal I\) is triangulated: shifts and cones use only translated terms and finite biproducts in \(\mathcal J\), and the cone triangles have the inherited structure. The functor \(K^+(F)\) is exact on homotopy categories by the same signed formulas as (4.1); applying it to a chain homotopy gives a chain homotopy. Left exactness on abelian rows is not needed for this particular assertion. The localization \(Q_{\mathcal B}\) is exact, hence so is \(G\).

Every source object has an outgoing quasi-isomorphism to \(\mathcal I\). If an acyclic object lies in \(\mathcal I\), Section 4 makes its image under \(G\) zero. These are exactly the hypotheses of the fully proved adapted triangulated localization theorem, Section 3, with the two null systems the acyclic complexes. That theorem constructs the universal ordinary localization after every further functor, gives its actual translation comparison, and proves that it carries all three maps of each distinguished triangle to a distinguished triangle. It does not require colimits in the target derived category.

More explicitly, if \(\mathcal N\) is the acyclic class in \(K^+(\mathcal A)\), the replacement theorem identifies

The functor \(G|_{\mathcal I}\) inverts its denominators: their cones are acyclic class complexes and become zero under \(G\). Descend \(G|_{\mathcal I}\) through the left side of (5.2) and compose with the specified exact inverse equivalence. This gives (1.3) and its unit \(\eta:G\to R^+FQ_{\mathcal A}\).

For an outgoing replacement \(s:X\to I\), naturality specifies the computation isomorphism by

The adapted theorem proves that common outgoing denominator refinements, followed by a further class replacement, compare any two such choices; it does not require a functorial choice of \(I\). Its full universal property and the unit (5.3) give the unique comparison between the resulting functors. This proves the value assertion and finishes Theorem 1.1.

6. A quotient-closed criterion and two examples

Corollary 6.1. Suppose \(\mathcal J\) is cogenerating. Suppose the cokernel of every monomorphism between its objects belongs to \(\mathcal J\), and \(F\) preserves every short exact row with three class objects. Then \(\mathcal J\) is an F-injective resolution class, with the universal computation of Section 5.

Proof. Cogeneration again supplies replacements. For a bounded-below exact class complex \(E\), zero below \(m\), its cycle \(Z^m(E)\) is zero and belongs to \(\mathcal J\). If \(Z^n(E)\in\mathcal J\), the exact row

shows \(Z^{n+1}(E)\in\mathcal J\) by the assumed cokernel closure. Induction puts every cycle in the class. Applying the assumed exactness of \(F\) to (6.1) proves \(FE\) acyclic by the same image/kernel factorization used in (3.8). Section 5 now applies. Alternatively, for a monomorphism \(J_0\to A\), choose a monomorphism \(A\to J_1\) by cogeneration. Its composite has cokernel \(J_2\in\mathcal J\), and its three-object row is F-exact. This verifies (1.1)–(1.2) and provides a second derivation from Theorem 1.1. \(\square\)

Example 6.2 (ordinary injectives). If \(\mathcal A\) has enough injectives, let \(\mathcal J\) be its injective objects. Given \(e:J_0\hookrightarrow A\), injectivity extends \(1_{J_0}\) to \(v:A\to J_0\). In (1.1) take \(J_1=J_0\), \(J_2=0\), \(i=1\). This row is split and every additive \(F\) preserves it. For a left exact \(F\), Theorem 1.1 applies directly. More generally every bounded-below exact complex of injectives is contractible by the component homotopy construction in Relative replacements, Section 4; any additive functor preserves that contraction. Thus the injective resolution-class assertion itself holds even for additive \(F\) that are not left exact.

Example 6.3 (a proper class with no cokernel closure). In finite-dimensional vector spaces over a field \(k\), let \(\mathcal J\) contain zero and all objects of dimension at least two. It is full, additive and cogenerating. Take \(F=1\). Every monomorphism \(J_0\hookrightarrow A\) splits: extend a basis of its image to a basis of \(A\), and project to that image. The resulting \(v:A\to J_0\) verifies (1.1), with \(J_1=J_0,J_2=0\). The theorem applies. Nevertheless the inclusion \(k^2\hookrightarrow k^3\) has one-dimensional cokernel, outside \(\mathcal J\), so Corollary 6.1's class-closure hypothesis fails. This example shows why Section 4 cannot assume that the original exact class complex already has all its cycles in the class.

7. Four graded exercises with complete solutions

Exercise 1 — The enlargement map may lose a direction (introductory)

Over \(k\), let \(J_0=k^2\), \(A=k^3\), and let \(e(a,b)=(a,b,0)\). Use \(J_1=k^2,J_2=0\) to construct (1.1). Describe the pullback square (2.2), its actual inverse comparison, and the resulting exact row for \(F=1\). Is the enlargement map \(A\to J_1\) monic?

Solution. Take \(v(a,b,c)=(a,b)\), \(i=1\), and \(p:k^2\to0\). The quotient \(C\) is \(k\), with \(q(a,b,c)=c\), and \(w:C\to0\) is zero. The pullback is \(k^2\times_0 k=k^2\oplus k\). The comparison sends \((a,b,c)\) to \(((a,b),c)\), with inverse \(((a,b),c)\mapsto(a,b,c)\). Its projection to \(C\) is surjective with kernel the displayed copy of \(k^2\), so the exact row is \(0\to k^2\xrightarrow e k^3\xrightarrow q k\to0\). However \(\ker v\) is the third coordinate line. Compatibility \(ve=i\), and the pullback argument, do not require \(v\) to be monic.

Exercise 2 — An exact class complex with a cycle outside the class (intermediate)

For Example 6.3, construct a bounded-below exact complex of class objects having a one-dimensional cycle. Explain why Theorem 1.1 still applies and why directly invoking Corollary 6.1 would fail.

Solution. Put \(X^0=k^2,X^1=k^3,X^2=k^3,X^3=k^2\), zero otherwise. Define \(d^0(a,b)=(a,b,0)\), \(d^1(a,b,c)=(c,0,0)\), and \(d^2(a,b,c)=(b,c)\). The composites vanish. At degree zero the first map is monic. At degree one its image is the first two coordinate plane, the kernel of \(d^1\). At degree two the image of \(d^1\) is the first coordinate line, the kernel of \(d^2\). Finally \(d^2\) is epic, so the complex is exact. Every nonzero term has dimension at least two, but \(Z^2(X)\simeq k\) is outside \(\mathcal J\). The splitting argument of Example 6.3 verifies the compatible extension hypothesis for every relevant monomorphism, so Theorem 1.1 applies. The simpler criterion cannot be invoked because cokernels of class monomorphisms can be one-dimensional. The refined complex of Lemma 3.1 has all its cycles in the class; that conclusion was never asserted for this original \(X\).

Exercise 3 — The lowest cone term and its contraction (advanced)

Let \(X=Y=(k\xrightarrow1 k)\) in degrees zero and one, and let \(f=1_X\). Compute \(C=\operatorname{Cone}(f)\), the cone \(W\) of the tails above zero, and the maps \(i,p,h\) of Section 4. Check all their nonzero chain and homotopy equations. What lower bound does \(W\) actually have?

Solution. The nonzero terms of \(C\) are \(C^{-1}=k,C^0=k\oplus k,C^1=k\). Its differentials are \(d_C^{-1}(z)=(z,-z)\) and \(d_C^0(y,x)=y+x\); their composite is zero. Both tails have only \(k\) in degree one, so \(W^0=k,W^1=k\), with \(d_W^0=1\). Thus \(W\) starts in degree zero. The inclusion is \(i^0(x)=(0,x)\), \(i^1=1\), and \(i^{-1}=0\). The projection is \(p^{-1}=0,p^0(y,x)=x+y,p^1=1\). Consequently \(p^0d_C^{-1}=0\), \(d_W^0p^0=p^1d_C^0\), and \(d_C^0i^0=i^1d_W^0\). Also \(pi=1\). The only nonzero homotopy component is \(h^0(y,x)=y\). In degree minus one, \(h^0d_C^{-1}(z)=z\). In degree zero, \(d_C^{-1}h^0(y,x)=(y,-y)=(y,x)-i^0p^0(y,x)\). In degree one, both \(1-ip\) and \(dh+hd\) are zero. These are every nonzero component of (4.5). The shift in the cone is the reason two tails starting in degree one have a cone starting in degree zero.

Exercise 4 — Cogeneration and closure do not replace F-exactness (advanced)

Take \(\mathcal A=\mathcal B=\mathsf{Ab}\), \(\mathcal J=\mathcal A\), and \(F=\operatorname{Hom}(\mathbb Z/2,-)\). Show that \(F\) is additive and left exact, that the class is cogenerating and cokernel closed, but that it is not F-injective. Locate the failed hypothesis of Corollary 6.1.

Solution. Additivity is addition of homomorphisms. A map from \(\mathbb Z/2\) into the kernel of an arrow is exactly a map into its source killed by that arrow, proving preservation of kernels; zero is preserved, so \(F\) is left exact. The whole abelian category is cogenerating by identity embeddings and contains every cokernel. Consider the exact complex \(E=(\mathbb Z\xrightarrow2\mathbb Z\to\mathbb Z/2)\) in degrees zero, one and two, with the last map reduction modulo two. Its first map is monic, the kernel of reduction is \(2\mathbb Z\), and reduction is epic. A homomorphism \(\mathbb Z/2\to\mathbb Z\) is zero, whereas \(\operatorname{Hom}(\mathbb Z/2,\mathbb Z/2)\simeq\mathbb Z/2\). Thus \(FE\) has zero terms in degrees zero and one and \(\mathbb Z/2\) in degree two. It has \(H^2(FE)\simeq\mathbb Z/2\), so acyclic preservation fails. The same short exact row has image \(0\to0\to0\to\mathbb Z/2\to0\), which is not exact at its last nonzero object. Precisely the F-exactness hypothesis is missing. The compatible extension criterion cannot hold either, since the proved theorem would otherwise force \(FE\) acyclic.

Freely accessible references

• The Stacks Project Authors, Derived functors on derived categories, especially Lemmas 13.15.5–13.15.6, supplies the replacement and quotient-closed class comparison. The stronger compatible extension criterion and its explicit tail-cone homotopy are proved in Sections 2–4 above.
• The Stacks Project Authors, Right derived functors and injective resolutions, gives the enough-injectives case. Its injective supply hypothesis is retained when applying that comparison; the general theorem here uses the displayed class hypotheses instead.
• The Stacks Project Authors, Abelian categories, pullbacks of epimorphisms, gives the finite abelian operation used in Lemma 2.1. The actual square and its comparison map are specified in that proof.