Draft lesson.

Universal derived functors and compatible composition

Original text public domain (CC0).

Applying an additive functor to a complex always respects homotopies and signed cone triangles. It can nevertheless destroy acyclicity. A derived functor repairs this second problem by localizing. Its universal definition uses formal objects; a suitable resolution class makes those objects ordinary and computable. We keep these two existence criteria distinct and construct the specified comparison for a composition.

Fix an ambient universe containing all categories, denominator diagrams and Hom sets used below. The formal constructions take place there; no smaller-universe smallness or ordinary colimit in a derived category is presumed. Let \(F:\mathcal A\to\mathcal B\) be additive between abelian categories. Complexes are cochain complexes, with \(X[1]^n=X^{n+1}\) and \(d_{X[1]}^n=-d_X^{n+1}\). The positive cone is

The four homotopy and derived categories, their quasi-isomorphism calculi, full bounded embeddings and uniform-bound denominator refinements are proved in Cohomological bounds, denominators and truncation adjunctions, Sections 1–3. The formal universal properties are proved in Localizing functors through formal objects, Sections 1–3 and 5. We use the complete cone and cohomology arguments of Chain homotopies, mapping cones and triangles.

1. Additivity gives exactness on the homotopy category

Proposition 1.1. Every additive \(F\) induces an exact functor \(K^*(F):K^*(\mathcal A)\to K^*(\mathcal B)\), for \(*\) unbounded, bounded below, bounded above or bounded. No exactness on abelian rows is required.

Proof. Apply \(F\) to every differential and every component of a chain map. It preserves zero maps and compositions, so the complex and chain-map equations persist. It also preserves addition and negatives. Thus the homotopy equation \(f-g=dh+hd\) becomes the same equation after applying \(F\), and the construction descends to \(K^*\).

The maps of a biproduct exhibit \(F(U\oplus V)\) as the biproduct of \(FU,FV\): their two retraction equations, two zero equations and sum-of-projections identity are preserved. This gives a specified coordinate isomorphism, natural in both variables. Under it, \(F(C(f))\) has exactly the differential (0.1) for \(Ff\). Translation commutes with \(F\) under the corresponding degreewise identification, including its negative differential. The inclusion of \(Y\), the projection to \(X[1]\), and the first map \(f\) all agree under these comparisons. Hence the image of a cone triangle is a cone triangle. The distinguished triangles of \(K^*\) are the triangles isomorphic to these, which proves exactness. The shift comparisons iterate coherently because each is given by the same degreewise coordinate maps. \(\square\)

This assertion does not say that \(F\) preserves quasi-isomorphisms. Exercise 1 computes a failure.

2. Formal values and general triangle preservation

For \(*\in\{\text{unbounded},+,b\}\), write \(Q_{\mathcal A}:K^*(\mathcal A)\to D^*(\mathcal A)\), and put \(G=Q_{\mathcal B}K^*(F)\). For \(X\in K^*(\mathcal A)\), the outgoing denominator category \(O_X\) has objects quasi-isomorphisms \(s:X\to X_s\). An arrow from \(s\) to \(t\) is a homotopy-category map \(a:X_s\to X_t\) with \(as=t\). This category is filtered. Its formal value is

Here \(y\) is Yoneda, and the colimit is a presheaf colimit. We say that \(R^*F\) exists universally when every (2.1) is represented by an object of \(D^*(\mathcal B)\). This is equivalent to an ordinary localization \(R^*F\) with unit

which remains a localization after every further functor on \(D^*(\mathcal B)\). The complete equivalence, including transport of the unit, is Localizing functors through formal objects, Theorem 3.1. In particular, for any such further functor \(T\), the formal diagram \(T(GX_s)\) is represented by \(T(R^*F(X))\). This includes functors that do not preserve ordinary colimits. We sometimes write \(F(X_s)\) in (2.1); it still means its image under \(Q_{\mathcal B}\).

Lemma (triangle denominators). Fix a distinguished triangle \(\Delta=(X\to Y\to Z\to X[1])\) in \(K^*(\mathcal A)\). Let \(I_\Delta\) have as objects morphisms \(\Delta\to\Delta_i\) of distinguished triangles whose three vertex maps are quasi-isomorphisms. Its arrows are morphisms \(\Delta_i\to\Delta_j\) commuting with those specified maps from \(\Delta\). Then \(I_\Delta\) is filtered, and each vertex projection

is cofinal. The assertion holds in the unbounded, bounded-below, bounded-above and bounded homotopy categories, with a containing universe as above.

Proof. We use the outgoing squares, cancellation and equality tests proved in Derived morphisms through quasi-isomorphism roofs, Sections 1–4. In particular, each \(O_V\) is filtered; if two maps become equal under \(Q\), a quasi-isomorphism out of their common target makes them equal already in \(K^*(\mathcal A)\). Every arrow \(a:s\to t\) of \(O_V\) is itself a quasi-isomorphism: \(as=t\), and the cohomology maps of \(s,t\) are invertible.

First establish a way to refine a triangle at any chosen vertex. Suppose its first arrow is \(f:X_0\to Y_0\), and \(u:X_0\to X_1\) is a quasi-isomorphism. The outgoing square gives \(v:Y_0\to Y_1\), also a quasi-isomorphism, and \(f_1:X_1\to Y_1\) with \(vf=f_1u\). Complete \(f_1\) to a distinguished triangle and extend this square by TR3. The third map is a quasi-isomorphism as well. Indeed, applying the exact \(Q\) gives a morphism of distinguished triangles with two invertible vertex maps. The two-vertex comparison of Translations and universal triangle comparisons, Theorem 6.2 makes the third image invertible. A map is inverted by \(Q\) exactly when it is a quasi-isomorphism, by cohomology detection in the localization prerequisite. Rotation gives the same refinement at the second or third vertex. Rotation includes its prescribed minus sign, and its inverse includes the translation comparisons; no strict inverse shift is assumed.

Call this the vertex refinement construction. It preserves the bounded variant: an actual completion may be taken to be the cone of \(f_1\), and shifts and finite biproducts preserve each of the four variants. Bounds may change during the finite construction. A category with one fixed cutoff is not being asserted triangulated.

The identity of \(\Delta\) makes \(I_\Delta\) nonempty. To find a common receiver for two of its objects \(\Delta\to\Delta_1,\Delta_2\), first use filteredness of \(O_X\) to obtain a common target for their first vertex maps. Apply the vertex refinement construction to the resulting arrow out of the first vertex of \(\Delta_2\). After replacing \(\Delta_2\) by that refinement, there is a map \(a:X_1\to X_2\) over \(X\). Do the same at the second vertex, carrying \(a\) forward through the new triangle map. Then do the same at the third vertex. We obtain maps

which commute with the maps from \(X,Y,Z\), respectively. They need not yet form a morphism of triangles.

They do form one after applying \(Q\): each component there is the corresponding map from \(\Delta\) to \(\Delta_2\), followed by the inverse of the map to \(\Delta_1\). Consequently the defect \(f_2a-bf_1\) becomes zero. The outgoing equality test gives a quasi-isomorphism out of \(Y_2\) killing this defect. Refine \(\Delta_2\) at that vertex and carry all three components forward. Their first square now commutes. Next kill \(g_2b-cg_1\) by a refinement at \(Z_2\). Finally kill \(h_2c-a[1]h_1\) by a refinement at \(X_2\): the equality test first gives a quasi-isomorphism out of \(X_2[1]\), which can be shifted back to \(X_2\). Its shifted map is identified with the chosen one by the inverse-translation comparisons. At each step the refinement is a whole triangle map, so previously established squares remain commutative. Thus the final \((a,b,c)\) is a triangle map over \(\Delta\). Together with the accumulated refinement out of the original \(\Delta_2\), it supplies a common receiver in \(I_\Delta\).

For two parallel arrows in \(I_\Delta\), their components become equal after \(Q\): each pair agrees after precomposition with the invertible image of its specified map from \(\Delta\). Kill the difference of their first components by an outgoing quasi-isomorphism and extend it by vertex refinement. Repeat at the second and third vertices. Postcomposition preserves equalities already obtained. The accumulated triangle map equalizes the two arrows. This proves every filteredness condition.

For cofinality of the first projection, fix \(s:X\to U\) in \(O_X\). Vertex refinement applied directly to \(\Delta\) extends \(s\) to an object of \(I_\Delta\). Hence the comma category \((s\downarrow I_\Delta)\) for that projection is nonempty. Two objects of this comma consist of triangle replacements and maps from \(U\) to their first vertices over \(X\). Choose a common receiver of their triangle replacements, using the filteredness just proved. The resulting two maps \(U\to X_j\) agree after precomposition with \(s\), hence become equal under \(Q\). Kill their difference by an outgoing quasi-isomorphism from \(X_j\), and extend it by vertex refinement. This gives a common receiver in the comma category itself. Any parallel arrows there are equalized by the equalizer construction in \(I_\Delta\), and remain comma arrows by postcomposition. Thus the comma is filtered, in particular connected. Its nonemptiness and connectedness prove cofinality. Rotating \(\Delta\) proves the other two projections, transporting the corresponding denominator categories along the translation equivalence. All index categories and finite refinements lie in the stated containing universe. \(\square\)

Theorem 2.1. If these formal values exist everywhere, \(R^*F\) is exact with its canonical shift comparison. Thus \(R^kF=H^kR^*F\) is cohomological for every integer \(k\).

Proof. The source quasi-isomorphisms form a saturated multiplicative system compatible with the triangulated structure, by the localization prerequisite. For a distinguished triangle \(X\to Y\to Z\to X[1]\), consider its category \(I\) of maps to distinguished triangles \(X_i\to Y_i\to Z_i\to X_i[1]\) whose three component maps are quasi-isomorphisms. The triangle-denominator lemma just proved makes \(I\) filtered and each vertex functor to \(O_X,O_Y,O_Z\) cofinal, in each of the bounded variants as well. We use all three endpoint comparisons, including their equality refinements.

Apply the exact functor \(G\) of Proposition 1.1 to this diagram. Its first two formal presheaf colimits are represented by \(A=R^*F(X)\) and \(B=R^*F(Y)\). The induced formal arrow is the represented map \(u=R^*F(X\to Y)\). Complete it to an actual distinguished triangle \(A\xrightarrow uB\to C\to A[1]\).

The full finite-stage representation proof in Representing filtered triangles by a finite stage, Lemma 2.1 and Theorem 3.1 applies to precisely this filtered diagram. To recall its mechanism: lift \(1_A\) and \(1_B\) to one stage, refine until their first square commutes, and use TR3 to obtain a map from the actual completion to that stage triangle. It induces \(yC\to\operatorname{colim}_I y(GZ_i)\). At every test object, the five-term Hom rows are exact; filtered colimits of abelian groups preserve exactness, and the four outer comparison maps are isomorphisms. The proved five-term argument makes this middle map an isomorphism. The same triangle comparison identifies its other two arrows with those of the actual completion. Thus the third formal value is represented by \(C\), with all three canonical maps forming a distinguished triangle.

Cofinality identifies these three formal colimits with (2.1). Shift of the denominator diagrams, together with the actual inverse of translation, gives the canonical representing shift comparison; its compatibility with the unit is also proved in Localizing triangulated functors, Lemma 2.1. The formal localization inverts denominators and descends to \(D^*(\mathcal A)\), as proved in the formal-localization prerequisite. Every distinguished triangle of that localization is isomorphic to the image of an actual source triangle. The preceding comparison therefore proves exactness there. Finally \(H^k\) takes distinguished target triangles to their long exact cohomology sequences. Composing with \(R^*F\) proves the last assertion, with every shift and connecting map retained. \(\square\)

This argument starts with an actual distinguished completion. Exactness of Hom rows alone is not used to characterize distinguished triangles. For the general derived-functor statement compare Stacks, Lemma 13.14.6; the finite-stage theorem supplies the complete representation argument used here.

3. Restricting an unbounded derived functor

Proposition 3.1. If the universal unbounded \(RF\) exists, its restriction to \(D^+(\mathcal A)\) takes values in \(D^+(\mathcal B)\) and is the universal \(R^+F\), with the same unit.

Proof. Represent an object by a complex \(X\) zero below \(m\). The uniform-bound denominator theorem in the prerequisite makes the full outgoing subcategory with \(X_s\) zero below this same \(m\) cofinal in the unbounded \(O_X\). It preserves common refinements and equality witnesses. Every \(F(X_s)\) is also zero below \(m\).

Apply the absolute representation property in Section 2 to \(H^k:D(\mathcal B)\to\mathcal B\), for \(k<m\). The resulting formal diagram has all values zero, so is the constant formal zero object. Its representing object is \(H^k(RF(X))\); hence this object is zero. This proves the target bound, which cofinality by itself would not establish.

The bounded-representative theorem transports \(RF(X)\) to an actual bounded-below representative. The full embedding \(D^+(\mathcal B)\hookrightarrow D(\mathcal B)\) now lets the representation of (2.1) restrict to tests in \(D^+(\mathcal B)\). Its stages and representing object both belong to that full subcategory. The bounded-below formal localization is therefore represented by exactly these values. Transporting its structural maps gives \(R^+F\), and the full representation theorem gives its universal property, even after arbitrary further functors defined only on \(D^+(\mathcal B)\). Both units come from the identity denominator. Their equality specifies the canonical restriction isomorphism. Naturality and shift compatibility follow from the representing formal transformations. \(\square\)

4. Resolution classes and a finite-prefix exactness test

A full additive class \(\mathcal J\subset\mathcal A\) is F-injective if every bounded-below complex admits an outgoing quasi-isomorphism into a bounded-below complex of class objects, and \(F(E)\) is acyclic for every acyclic bounded-below complex \(E\) of class objects. Isomorphic choices of class objects are transported with their maps. Finite biproducts are the ambient ones. These are exactly the adaptation conditions for \(K^+(\mathcal J)\subset K^+(\mathcal A)\): this full homotopy subcategory is closed under shifts and cones by (0.1), and its null objects are the acyclic class complexes.

Proposition 4.1. An F-injective class makes \(R^+F\) universally exist. Every outgoing replacement \(s:X\to I\), with terms of \(I\) in the class, computes

Proof. Use Localizing triangulated functors, Theorem 3.1, with source and target null systems the acyclic complexes. Class density is the first definition requirement; annihilation of the class null objects is the second. Its complete proof descends \(G|_{K^+(\mathcal J)}\) through the class localization and an exact inverse equivalence with \(D^+(\mathcal A)\). It supplies an invertible \(\eta_I:G(I)\to R^+F(I)\) and the normalization

This is the specified computation (4.1). The theorem compares two choices by common denominator refinement and a further class replacement, and proves the absolute universal property, so no functorial choice of resolutions is required. \(\square\)

Call \(\mathcal J\) cogenerating if every object embeds into one of its objects.

Theorem 4.2. Suppose \(F\) is left exact. The class \(\mathcal J\) is F-injective if and only if it is cogenerating and the following finite-prefix test holds: whenever

is exact and there is a finite exact sequence

its image under \(F\) is a short exact row.

Proof. First assume the class is F-injective. Replace \(A[0]\) by \(I\). The map \(A\to I^0\) factors through \(Z^0I\), and its composite to \(H^0I\) is an isomorphism. It is therefore monic, proving cogeneration, even if \(I\) initially has negative degrees.

The checked construction in Relative replacements, Theorem 2.1 now gives an augmented class coresolution \(0\to A''\to T^0\to T^1\to\cdots\). Splice it with (4.4) and (4.3), to form

The splice maps are \(J_r\to A'\to A\) and \(A\to A''\to T^0\). Exactness follows at each splice from the corresponding image and kernel in the two original exact rows. All terms are class objects, so (4.5) remains acyclic after \(F\). At \(F(T^0)\), its kernel towards \(F(T^1)\) is precisely \(F(A'')\), by left exactness. Thus \(F(A)\to F(A'')\) is epic. Left exactness supplies the other two positions of the required row.

Conversely, cogeneration supplies all bounded-below class replacements by that same full replacement theorem. Let an acyclic class complex \(E\) be zero below \(m\). For each \(n>m\), its initial finite exact segment

has precisely form (4.4). Apply the test to \(0\to Z^nE\to E^n\to Z^{n+1}E\to0\). At \(n=m\), its left term is zero, whose finite class prefix may be taken to be the single zero class object. Therefore every cycle row becomes exact under \(F\). The differential of \(F(E)\) factors through \(F(Z^{n+1}E)\); those exact rows identify its image with the next differential's kernel. Hence \(F(E)\) is acyclic in every degree. These are both class requirements. We have never assumed that the original cycles of \(E\) belong to \(\mathcal J\). \(\square\)

5. Heart values, two acyclicity conventions and injectives

Proposition 5.1. Whenever the universal \(R^+F\) exists, \(R^kF(A)=0\) for \(k<0\) and \(A\in\mathcal A\). If \(F\) is left exact, the unit identifies \(R^0F(A)\) naturally with \(F(A)\).

Proof. Restrict the outgoing denominator diagram of \(A[0]\) to targets \(I\) zero below zero. The bound theorem again makes this cofinal. Their images under \(F\) are zero in negative degrees, and absolute representation after \(H^k\) gives the first assertion. At degree zero, the quasi-isomorphism \(A[0]\to I\) gives an exact sequence \(0\to A\to I^0\to I^1\). Left exactness identifies \(F(A)\) with \(\ker(FI^0\to FI^1)=H^0(FI)\). These identifications commute with every denominator map and carry the unit to the identity on \(F(A)\). Thus the entire formal \(H^0\)-diagram is constant at \(F(A)\), which proves the second assertion by absolute representation. \(\square\)

For an arbitrary additive functor with a derived functor, call \(A\) cohomologically right F-acyclic if \(R^kF(A)=0\) for \(k\ne0\). Separately, say that \(A\) computes the derived value if its unit \(F(A)[0]\to R^+F(A)\) is an isomorphism. Computing the value implies cohomological acyclicity. For left exact \(F\), Proposition 5.1 and cohomology detection show that the two conditions are equivalent, and both say that \(R^kF(A)=0\) for \(k>0\). Without left exactness the unit condition is stronger; Exercise 1 exhibits the difference. Every object of an F-injective class computes the value, by (4.2) with its identity replacement.

For left derivation the cohomological definition is \(L^kF(A)=0\) for \(k\ne0\). One may again distinguish this from invertibility of the counit. Positive-degree values vanish on the heart; if \(F\) is right exact, \(L^0F(A)=F(A)\) with its counit, by the incoming, upper-bound version of the preceding proof. Then the two left acyclicity conditions coincide.

Proposition 5.2. If \(\mathcal A\) has enough injectives, its full class of injective objects is F-injective for every additive \(F\).

Proof. Enough injectives means exactly that this class is cogenerating, so the general class replacement theorem supplies its outgoing replacements. An acyclic bounded-below complex \(I\) of injectives is contractible: Relative replacements, Lemma 4.1, applied to \(1_I\), supplies \(1_I=dh+hd\). Additivity preserves this equation, making \(F(I)\) contractible and thus acyclic. No abelian exactness of \(F\) was used. \(\square\)

6. The largest acyclic class retains a supply hypothesis

Theorem 6.1. Let \(F\) be left exact with a universal \(R^+F\). Let \(\mathcal J_F\) be the full class of its cohomologically right acyclic objects. It is additive, closed under cokernels of monomorphisms between its objects, and \(F\) preserves every short exact row whose first two objects belong to it. If a full additive cogenerating subclass of \(\mathcal J_F\) exists, then \(\mathcal J_F\) is itself F-injective. Moreover, a full additive \(\mathcal J\) is F-injective exactly when it is cogenerating and consists of acyclic objects.

Proof. Exactness of \(R^+F\) gives its additivity, so zero and finite biproducts belong to \(\mathcal J_F\). A short exact row \(0\to A'\xrightarrow u A\to A''\to0\) gives a derived distinguished triangle: the map \(C(u)\to A''[0]\) is a quasi-isomorphism, since the cone has cohomology \(\ker u=0\) in degree \(-1\), \(\operatorname{coker}u=A''\) in degree zero, and zero otherwise. Theorem 2.1 and Proposition 5.1 therefore give the natural long exact sequence beginning

If \(A',A\in\mathcal J_F\), the segments \(R^kF(A)\to R^kF(A'')\to R^{k+1}F(A')\) for \(k>0\) make every higher group of \(A''\) zero. The negative groups are already zero. Also \(R^1F(A')=0\), making the displayed F-row short exact. This proves the closure and exactness assertions.

When a cogenerating acyclic subclass exists, \(\mathcal J_F\) is cogenerating. Apply the fully proved quotient-closed criterion Compatible injective classes, Corollary 6.1 to the closure and exactness just proved. It makes \(\mathcal J_F\) F-injective.

Any F-injective \(\mathcal J\) is cogenerating by Theorem 4.2 and consists of computing, hence acyclic, objects by Proposition 4.1. Conversely, if \(\mathcal J\) is cogenerating and acyclic, it supplies the cogenerating subclass needed in the preceding paragraph. Every acyclic bounded-below complex of \(\mathcal J\)-objects is also a complex of \(\mathcal J_F\)-objects, so its F-image is acyclic. Cogeneration supplies its own class replacements. Thus \(\mathcal J\) is F-injective. It need not itself be cokernel-closed. \(\square\)

The supply premise is explicit. Existence of the formal derived values alone is not the assertion that the heart contains enough acyclic objects. The theorem requires cogeneration before constructing class replacements; it cannot infer that premise from mere derivability.

7. Incoming formal values and projective classes

For \(*\in\{\text{unbounded},-,b\}\), the incoming category \(P_X\) has objects quasi-isomorphisms \(s:X_s\to X\) and arrows \(a:X_s\to X_t\) satisfying \(ta=s\). It is cofiltered. The formal left value is the pro-object

For a constant test object \(V\), its maps to \(V\) are \(\operatorname{colim}_{P_X^{op}}\operatorname{Hom}(Q_{\mathcal B}F(X_s),V)\). Universally existing \(L^*F\) means that (7.1) is constant at an object of \(D^*(\mathcal B)\) for every \(X\). It has counit \(\epsilon:L^*FQ_{\mathcal A}\to Q_{\mathcal B}K^*(F)\) and universal property

The complete opposite-category proof in Localizing functors through formal objects, Section 5 reverses both category arguments and the denominator calculus; it proves the absolute property with precisely this variance. The triangulated statement is its dual applied with the opposite triangulated structure, so existing \(L^*F\) is exact. Reversal also gives the upper-bound and heart assertions stated in Section 5.

A full additive F-projective class supplies incoming bounded-above replacements and has acyclic F-images on acyclic bounded-above class complexes. These are exactly the adapted \(K^-(\mathcal J)\) requirements. The left version of the full adapted localization theorem gives universal \(L^-F\), with value \(F(P)\) for every incoming class replacement \(P\to X\), normalized by its counit. This is the reversed proof of Proposition 4.1, not an outgoing formula with its arrows left unchanged.

8. Flat left modules work over any unital ring

Fix an arbitrary unital ring \(R\) and a right \(R\)-module \(N\). Let \(F=N\otimes_R-\) from left \(R\)-modules to abelian groups. Tensor is right exact in both variables: its balanced-map universal property takes cokernels to cokernels, as fully proved in Hom, tensor and the exactness of module limits, Section 1, with \(k=\mathbb Z\). A left module \(P\) is flat if \(-\otimes_R P\) preserves monomorphisms of right modules. With right exactness, this means it preserves short exact rows.

Lemma 8.1. If \(0\to K\to P\to Q\to0\) is exact and \(Q\) is flat, its tensor with every right module \(V\) is short exact. If also \(P\) is flat, then \(K\) is flat.

Proof. Choose a free right-module presentation \(0\to U_1\to U_0\to V\to0\). Free modules tensor a monomorphism injectively: their tensor is the direct sum of copies of that monomorphism, and every vector has finite support. Right exactness already proves the two rightmost positions of the proposed V-row. Suppose \(x\in V\otimes K\) maps to zero in \(V\otimes P\). Lift it to \(\tilde x\in U_0\otimes K\). Its image in \(U_0\otimes P\) comes from some \(\ell\in U_1\otimes P\). The image of \(\ell\) in \(U_1\otimes Q\) maps to zero in \(U_0\otimes Q\). Flatness of \(Q\) makes the latter map injective, so that image is zero. Right exactness of the tensor of the original K-row with \(U_1\) lifts \(\ell\) from \(U_1\otimes K\). Its image in \(U_0\otimes K\) equals \(\tilde x\), because their images in \(U_0\otimes P\) agree and \(U_0\) is free. Thus \(x=0\).

If \(P\) is also flat, apply these proved tensor rows to a monomorphism \(V'\hookrightarrow V\). An element of \(V'\otimes K\) that dies in \(V\otimes K\) dies in \(V\otimes P\). Flatness of \(P\) makes it zero in \(V'\otimes P\); the already proved row makes it zero in \(V'\otimes K\). This proves flatness of \(K\). \(\square\)

Lemma 8.2. Every complex of left modules zero above \(b\) admits an incoming quasi-isomorphism from a complex of free left modules zero above the same \(b\).

Proof. A module \(M\) is a quotient of the free left module on its underlying set, by \(e_m\mapsto m\). Start with \(P^n=0\) for \(n>b\). Construct lower terms downwards. Suppose the terms above \(n\), their differential and map \(f:P\to X\), have been constructed, with \(Z^{n+1}P\to Z^{n+1}X\) surjective and cohomology already correct in degrees greater than \(n+1\). Form the pullback module

Take a free surjection \(P^n\to T_n\). Its two projections define \(f^n\) and \(d_P^n\), the latter followed by inclusion in \(P^{n+1}\). Equation (8.1) gives the chain-map identity, and the cycle condition gives \(d_P^{n+1}d_P^n=0\).

If \(x\in Z^nX\), lift the pair \((x,0)\); its preimage is a cycle of \(P^n\). Thus \(Z^nP\to Z^nX\) is surjective. If a cycle \(z\in Z^{n+1}P\) maps to a boundary \(d_X^nx\), lift \((x,z)\). Its differential is \(z\), so the kernel of the map on \(H^{n+1}\) is zero. Surjectivity on cycles gives surjectivity on this cohomology as well. Therefore the new \(H^{n+1}\)-map is an isomorphism. Earlier higher-degree assertions are unchanged. At \(n=b\) the required cycle surjection in degree \(b+1\) is \(0\to0\), so the recursion starts. Every fixed cohomology degree becomes correct after its next lower term has been added. The resulting complex and map are quasi-isomorphic in every degree. Every term and differential is fixed when constructed; no infinite categorical limit is being taken. \(\square\)

Theorem 8.3. The full additive class of flat left modules is \((N\otimes_R-)\)-projective, for every right module \(N\). Thus \(L^-(N\otimes_R-)\) exists universally and is computed by every bounded-above flat, in particular free, replacement.

Proof. Free left modules are flat: \(V\otimes_R R^{(S)}\simeq V^{(S)}\), naturally, and a finite-support coordinate check preserves injections. Finite biproducts of flat modules are flat for the same reason, and zero is flat. Lemma 8.2 therefore supplies the incoming flat replacements.

Let a bounded-above acyclic complex \(E\) of flats be zero above \(b\). Then \(Z^bE=E^b\) is flat. Downward induction in the exact cycle rows \(0\to Z^nE\to E^n\to Z^{n+1}E\to0\), using Lemma 8.1, makes every cycle flat and every tensor cycle row exact. Their image-kernel identities prove \(N\otimes_RE\) acyclic. These are exactly the two projective-class conditions in Section 7. No commutativity, finite generation, finite resolution length or flatness of the fixed right module \(N\) was assumed. \(\square\)

9. The normalized comparison for a composition

Let \(F:\mathcal A\to\mathcal B\), \(F':\mathcal B\to\mathcal C\) be additive. Assume \(R^*F,R^*F',R^*(F'F)\) universally exist, for the same \(*\in\{\text{unbounded},+,b\}\). Their units will be \(\eta,\eta',\theta\). The composite

and (2.2) give a unique natural transformation

Theorem 9.1. If \(\mathcal J\) is F-injective, \(\mathcal J'\) is F'-injective and \(F(\mathcal J)\subset\mathcal J'\), then \(\mathcal J\) is F'F-injective and (9.2) for \(*=+\) is an isomorphism of exact functors.

Proof. The class replacements are unchanged. For an acyclic bounded-below class complex \(E\), \(F(E)\) is acyclic with terms in \(\mathcal J'\), so \(F'F(E)\) is acyclic. This proves the composite adaptation and existence.

For a class complex \(I\), \(\eta_I\) and \(\theta_I\) are invertible by Proposition 4.1. Since \(F(I)\) is a complex of \(\mathcal J'\)-objects, \(\eta'_{F(I)}\) is invertible too. The defining equation gives the actual component

Each factor is invertible. If \(s:X\to I\) is any class replacement, naturality specifies

The outside factors are invertible as well. This proves invertibility of this particular universal comparison, with its unit normalization. Its natural inverse is obtained by multiplying the naturality equations by component inverses.

For shift compatibility, the units and (9.1) are compatible with their proved shift comparisons. The two candidate transformations in the compatibility equation for \(c\) agree after the shifted unit \(\theta\); injectivity of the shifted universal bijection in (2.2) makes them equal. This is also the complete argument in Localizing triangulated functors, Section 4. Thus (9.2) is a comparison of exact functors. \(\square\)

For three successive functors, assuming all the displayed derived composites exist, either route from the derived threefold composite to the threefold composition has, after its universal unit, the same composite of the three whiskered units. The universal bijection makes those two natural transformations equal. This proves associativity coherence. The same uniqueness makes changes of computing class preserve (9.2); no new choice of an objectwise isomorphism is made.

For left derivation, when the three left derived functors exist in a common bound, the counits form

Property (7.2) therefore gives the reversed comparison

If \(\mathcal P,\mathcal P'\) are projective computing classes and \(F(\mathcal P)\subset\mathcal P'\), the same acyclic-image argument makes \(\mathcal P\) composite-projective. For \(*=-\), its component at \(P\) is \(\epsilon_{F'F,P}^{-1}\epsilon'_{F(P)}L^-F'(\epsilon_P)\), with every factor invertible. Incoming replacements and naturality give invertibility everywhere. Shift and triple-composition coherence follow from counit universal injectivity, exactly as on the right. Existence of the three functors alone is not enough to make either comparison invertible.

10. Four graded exercises with complete solutions

Exercise 1 — Exact triangles versus acyclicity (introductory)

Take \(F=-\otimes_{\mathbb Z}\mathbb Z/2\) on abelian groups. Apply it to the acyclic complex \(E=(\mathbb Z\xrightarrow{2}\mathbb Z\to\mathbb Z/2)\), in degrees \(0,1,2\). Compute the cohomology of \(F(E)\). Then compute the universal right derived functor of \(F\) and compare cohomological acyclicity with invertibility of its unit on \(\mathbb Z\).

Solution. For every group \(M\), \(M\otimes\mathbb Z/2\simeq M/2M\): the balanced-map universal property identifies maps from either side with additive maps from \(M\) killed by two. Thus \(F(E)=(\mathbb Z/2\xrightarrow0\mathbb Z/2\xrightarrow1\mathbb Z/2)\). Its only nonzero cohomology is \(H^0=\mathbb Z/2\). The quasi-isomorphism \(0\to E\) is consequently not preserved. Proposition 1.1 still applies to every cone triangle.

Abelian groups have enough injectives, and injective groups are divisible. For completeness the ordinary supply is the evaluation embedding \(M\hookrightarrow(\mathbb Q/\mathbb Z)^{\operatorname{Hom}(M,\mathbb Q/\mathbb Z)}\): characters separate nonzero elements by first defining one on its cyclic subgroup and extending it into the divisible, hence injective, group \(\mathbb Q/\mathbb Z\). The full Zorn extension and small-product proof is Countable Mittag–Leffler systems, Section 4. Conversely an injective group is divisible: extend the map \(n\mathbb Z\to I\), \(n\mapsto x\), across \(n\mathbb Z\subset\mathbb Z\), obtaining an element whose n-fold multiple is \(x\).

Proposition 5.2 therefore computes \(R^+F\) on a complex of divisible groups. Its termwise F-image is zero, so \(R^+F=0\), universally and with its zero unit. Every heart object is cohomologically right F-acyclic. On \(\mathbb Z\), the unit is \(\mathbb Z/2[0]\to0\), which is not invertible. This demonstrates why left exactness was required for the equivalence of the two conditions in Section 5.

Exercise 2 — A left-derived cyclic tensor (intermediate)

For integers \(m,n\ge1\), compute \(L^-(\mathbb Z/n\otimes_{\mathbb Z}-)(\mathbb Z/m)\), including its degree-zero counit. State all nonzero cohomology groups and explicit generators.

Solution. The free incoming replacement of \(\mathbb Z/m[0]\) is \(P=(\mathbb Z\xrightarrow m\mathbb Z)\) in degrees \(-1,0\), with degree-zero quotient augmentation. Theorem 8.3 computes the value by \((\mathbb Z/n\xrightarrow m\mathbb Z/n)\). Put \(d=\gcd(m,n)\), and write \(m=dm_0,n=dn_0\), with \(m_0,n_0\) coprime. Its degree-minus-one kernel consists precisely of multiples of \(n_0\) modulo \(n\): \(n\mid mx\) is equivalent to \(n_0\mid x\). It is cyclic of order \(d\), with generator the class of \(n/d\). The degree-zero cokernel is \(\mathbb Z/(n,m)\simeq\mathbb Z/d\), generated by the image of one. Bezout gives \((n,m)=d\mathbb Z\), including when \(d=1\). All other groups vanish by the two nonzero degrees. In degree zero the counit is the quotient \(\mathbb Z/n\to(\mathbb Z/m)\otimes\mathbb Z/n\simeq\mathbb Z/d\); it induces the identity on the computed cokernel. If \(m=1\) or \(n=1\), \(d=1\) and the entire derived object is zero by cohomology detection. No flatness of the fixed \(\mathbb Z/n\) was used.

Exercise 3 — An acyclic class larger than injectives (advanced)

For \(F=\operatorname{Hom}_{\mathbb Z}(\mathbb Z/2,-)\), identify all right acyclic groups. Prove that they form an F-injective class, and exhibit one which is not injective. Verify directly that \(F\) preserves the short exact rows required by the quotient-closed criterion.

Solution. Enough injectives and Proposition 5.2 give \(R^+F\). For a resolution \(M\to I\), its kth cohomology is \(\operatorname{Hom}_K(\mathbb Z/2[0],I[k])\): chain maps into this shifted injective complex are cycles of its Hom complex, and the homotopy equation is precisely its boundary equation. The vanishing proved in Relative replacements, Lemma 4.1 applies to every acyclic source and every shift of \(I\). The exact Hom sequence of a quasi-isomorphism cone therefore makes precomposition into \(I[k]\) invertible. The incoming fraction formula consequently identifies the homotopy Hom group with \(\operatorname{Hom}_D(\mathbb Z/2,M[k])\).

The projective source \(P=(\mathbb Z\xrightarrow2\mathbb Z)\), in degrees \(-1,0\), can be used instead: the complete projective Hom test is Hereditary categories, Lemma 3.1. Chain maps and homotopies give \(R^0F(M)=M[2]=\{x:2x=0\}\), \(R^1F(M)=M/2M\), and zero in every other degree. Indeed the only map for \(k=1\) is an arbitrary element of \(M\), and a homotopy changes it by twice an element; for \(k=0\), its element must be killed by two. Hence the acyclic class is precisely the 2-divisible groups.

This class contains all divisible groups, so is cogenerating by the embedding in Exercise 1. It is closed under finite sums and under quotients: lift an element of a quotient to the middle group and take its half there. In a row \(0\to A'\to A\to A''\to0\) with \(A',A\) 2-divisible, a 2-torsion element of \(A''\) lifts to \(a\in A\) with \(2a\in A'\). Choose \(a'\in A'\) with \(2a'=2a\). Then \(a-a'\) is a 2-torsion lift. This proves surjectivity of \(F(A)\to F(A'')\); its other positions are the ordinary Hom kernel test. The quotient-closed criterion therefore makes the class F-injective, also as concluded by Theorem 6.1. The group \(\mathbb Z[1/2]\) is 2-divisible but not 3-divisible, since it contains no element whose triple is one. The preceding injective-implies-divisible proof shows that it is not injective.

Exercise 4 — Three derived functors without a composition isomorphism (advanced)

Fix a prime \(p\). On abelian groups take \(F=\operatorname{Hom}_{\mathbb Z}(\mathbb Z/p,-)\) and \(F'=-\otimes_{\mathbb Z}\mathbb Z/p\). Show that all three bounded-below right derived functors exist but their comparison (9.2) is not an isomorphism. Identify the failure of the ordinary injective-image hypothesis.

Solution. Proposition 5.2 supplies all three functors, since both given functors and their composite are additive and abelian groups have enough injectives. Every divisible group has \(F'(I)=I/pI=0\); thus \(R^+F'=0\) by the same class computation as Exercise 1. Every group \(F(M)\) is killed by \(p\). The natural quotient identification \(V\otimes\mathbb Z/p\simeq V/pV\) therefore gives \(F'F\simeq F\) on all groups, not just on injectives. Uniqueness of the universal localization gives \(R^+(F'F)\simeq R^+F\).

At \(M=\mathbb Z/p\), Proposition 5.1 gives \(H^0R^+F(M)=\operatorname{Hom}(\mathbb Z/p,\mathbb Z/p)\simeq\mathbb Z/p\), whereas \(R^+F'R^+F(M)=0\). The specified comparison is therefore a map with nonzero degree-zero source to zero and cannot be invertible. Take the injective group \(I=\mathbb Q/\mathbb Z\). Then \(F(I)\simeq\mathbb Z/p\), which is not divisible and hence not injective. Thus the ordinary injective classes do not satisfy \(F(\mathcal J)\subset\mathcal J'\). This concrete failure agrees with the additional premise of Theorem 9.1; existence of the three functors supplies the comparison, not its inverse.

Freely accessible references

• Stacks Project, cofinal systems of triangle denominators, Tag 05R9, gives another proof of the common triangle index constructed in Section 2.
• Stacks Project, derived functors, Tag 05S7, for general formal derivability and its triangulated statement. The complete finite-stage representation proof is included through the linked course prerequisite.
• Stacks Project, essentially constant systems, Tags 05PU and 05PY, for the finite-witness and representability formulations. The course formal-object proofs establish the absolute universal property used above.