Draft lesson.

Nested Hom classes and bounded derived embeddings

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Level: advanced. Prerequisites: abelian categories, bounded derived categories, resolution classes, normalized derived Hom and truncation triangles.

A full exact inclusion of abelian categories whose image is closed under extensions identifies ordinary Hom and short-extension classes. Higher extensions can still change. Compatible resolution classes supply the extra control: the smaller category can resolve both inputs using objects that also compute the ambient derived Hom. We prove that this gives the actual comparison isomorphism, then reconstruct bounded ambient complexes from their cohomology and connecting maps. The degree-one comparison is proved in Thick abelian subcategories and derived comparison, Proposition 1.1. Its Exercise 1 gives a full exact inclusion of vector spaces into dual-number modules that is not closed under extensions: all intrinsic short extensions split, but an ambient short extension does not.

Conventions and prerequisites

Work in a fixed Grothendieck universe containing the integers, with locally small abelian categories and choices of the indicated resolutions; a larger ambient universe contains the localization diagrams when necessary. Complexes are cohomological:

For a chain map \(f:X\to Y\), the cone is

Its triangle has positive last projection to \(X[1]\); rotating a triangle \(U\xrightarrow{a}V\xrightarrow{b}W\xrightarrow{c}U[1]\) gives last arrow \(-a[1]\).

Derived Hom and bounded resolution pairs, Sections 1–7 proves the complete adapted-pair computation, its coherent unit and its interpretation as derived morphisms. It also proves the one-sided injective and projective specializations and free module covers. Thick abelian subcategories and derived comparison, Sections 1–4 proves the exact inclusion and the finite triangle comparison argument. Cohomological bounds, denominators and truncation adjunctions, Sections 1–5 proves the bounded embeddings, cohomology detection and the specified natural truncation triangles. These interfaces keep their actual maps and signs.

The freely accessible Stacks, Section 13.17, Tag 06UP, describes the ambient derived subcategory with cohomology in a weak Serre subcategory. Its introductory construction agrees with the subcategory used here. The resolution-pair hypothesis below is stated explicitly; membership in that ambient cohomology subcategory alone does not prove a derived equivalence.

1. Exact inclusion and the four resolution classes

Let \(\mathcal B\subset\mathcal A\) be a thick abelian subcategory, meaning a full replete subcategory containing zero, closed under ambient kernels and cokernels of arrows between its objects, and closed under extensions. This is the weak Serre condition, which does not require all ambient subobjects of its objects to belong to it.

The inclusion \(j:\mathcal B\to\mathcal A\) is exact. Its kernel and cokernel objects have the ambient universal properties restricted to \(\mathcal B\), and its coimage-to-image map is the ambient isomorphism. Extension closure supplies finite biproducts. Thus the abelian structure and cohomology calculations agree. Applied termwise, \(j\) preserves shifts, cones and quasi-isomorphisms; it induces exact functors \(F^*:D^*(\mathcal B)\to D^*(\mathcal A)\) for \(*=b,+,-\).

For complexes define the normalized Hom complex

When \(P\) is zero above \(u\) and \(I\) is zero below \(a\), its degree-\(n\) indices satisfy \(a-n\leq i\leq u\), so the product is finite and the complex is zero below \(a-u\).

Here is the precise meaning of a Hom-adapted pair \((\mathcal J,\mathcal P)\) in an abelian category \(\mathcal C\). Both classes are full and additive, contain zero and are closed under finite biproducts. They satisfy:

1. Every bounded-below complex has an outgoing quasi-isomorphism \(Y\to I\) with \(I\in K^+(\mathcal J)\).
2. Every bounded-above complex has an incoming quasi-isomorphism \(P\to X\) with \(P\in K^-(\mathcal P)\).
3. For such class complexes \(P,I\), the complex \(\mathcal H(P,I)\) is acyclic if either \(P\) or \(I\) is acyclic.

This is the Hom-injective pair condition for the target-first bifunctor \((Y,Z)\mapsto\operatorname{Hom}_{\mathcal C}(Z,Y)\) on \(\mathcal C\times\mathcal C^{\mathrm{op}}\), after reversing source degrees and using the normalization (1.1). It is a condition on two classes, not a claim that every member is an injective or projective object.

Choose an ambient pair \((\mathcal J_A,\mathcal P_A)\) in \(\mathcal A\) and an intrinsic pair \((\mathcal J_B,\mathcal P_B)\) in \(\mathcal B\), with the two nesting conditions

Membership up to isomorphism suffices after transporting the replacement maps. These four classes and both adapted-pair conditions are the hypotheses of the comparison theorem. No additional global supply of actual injectives or projectives is imposed.

2. One Hom complex computes both derived values

Write \(R_C(X,Y)=R\operatorname{Hom}_{\mathcal C}(X,Y)\). The adapted-pair theorem in the prerequisite gives

Its target is the category of complexes of abelian groups, even for a category of modules over a field. In examples the Hom groups also carry their usual linear structure.

Theorem 2.1. Under (1.2) and the two adapted-pair hypotheses, there is a canonical natural isomorphism

for \(X\in D^-(\mathcal B)\), \(Y\in D^+(\mathcal B)\). It commutes with the two shift comparisons, their negative exchange and the separately triangulated structures.

Proof. Choose intrinsic representatives and intrinsic class replacements

Both maps remain quasi-isomorphisms under the exact inclusion. By (1.2), their images are also ambient class replacements. Fullness of \(\mathcal B\) identifies every group in (1.1), and exactness retains its actual differentials. Thus there is an identity of complexes

Let \(\eta_C\) be the derived-Hom pair unit. It is invertible on a class pair. The exact computation is the isomorphism

For \(C=A\), interpret every argument in (2.5) after inclusion. The first input is contravariant: \(p:P\to X\) induces the arrow from the value at \(X\) to the value at \(P\). The second input is covariant: \(i:Y\to I\) induces the arrow from the value at \(Y\) to the value at \(I\). Thus both factors in (2.5) have the stated domains. Define

using the identity (2.4) between them.

We check that this defines a canonical natural comparison, rather than only an isomorphism for one set of choices. A quasi-isomorphism between source class complexes or between target class complexes has an acyclic cone inside its respective full class homotopy category. Condition 3 and the Hom cone comparisons in the prerequisite imply that its image under the fixed-variable Hom total is a quasi-isomorphism. Both intrinsic and ambient computations therefore invert all these same class denominators.

The replacement-density part of the pair theorem identifies the source-class localization with \(D^-(\mathcal B)\) and the target-class localization with \(D^+(\mathcal B)\). Both bifunctors in (2.2), restricted through these two localizations, are identified by their pair units with the identical functor \(Q\mathcal H(P,I)\). The identity of this functor is natural for all original class arrows. It is also natural for their inverse denominators: multiply the original naturality square by the invertible denominator images. Each localized arrow is a fraction of these arrows, so naturality holds for every derived arrow in either variable, and the two-variable squares commute as well.

For two replacement choices, the intrinsic denominator equality tests in the pair theorem supply common incoming source refinements and common outgoing target refinements, then replacements in the corresponding intrinsic classes. All homotopy equalities remain the same after inclusion. The two computation maps (2.5) commute with these common refinement maps by unit naturality, and the Hom maps on the refinements are inverted on both sides. Hence (2.6) agrees for the two choices. For three choices the same argument on a common refinement gives the cocycle equation. This proves representative independence and compatibility with identity and composition. It is equivalently descent of the identity natural transformation through the two class localizations.

The identity (2.4) also retains the Hom shift formulas. In particular the target unit-shift comparison is componentwise identity, while the source comparison from \(\mathcal H(P[1],I)\) to \(\mathcal H(P,I)[-1]\) sends a degree-\(n\) family to \(((-1)^n f^{k-1})_k\). Both intrinsic and ambient normalizations use those same multipliers, including the negative exchange of the two shifts. Their cone comparison maps are the same coordinate maps, too. Descent therefore preserves the entire separately triangulated structure specified by the pair theorem. This completes (2.2). \(\square\)

3. The comparison is the actual map on derived morphisms

The distinction between isomorphic complexes and the intended comparison map matters for full faithfulness. For every integer \(n\), the derived-Hom prerequisite gives a natural identification

Proposition 3.1. Under these identifications, \(H^n\Theta\) is precisely the map induced by the inclusion:

Proof. Compute both sides with the shared class pair (2.3). A degree-\(n\) cycle of (1.1) is a family \(f^k:P^k\to I^{k+n}\) with \(d_If=(-1)^n f d_P\). These same components form a chain map \(P\to I[n]\), because its target differential is \((-1)^n d_I\). If the family is a boundary \(\partial h\), the corresponding chain null-homotopy has components \((-1)^n h^k\): its differential expression is

Thus the cohomology class has exactly the homotopy-class interpretation used in (3.1). Its intrinsic derived arrow is

The full numerator and denominator formula in the prerequisite proves this for the unit-normalized computation (2.5). Under inclusion the same expression becomes

Equation (2.6) identifies the class through the identity (2.4), so takes (3.4) to (3.5). This is exactly (3.2), for every integer \(n\). It preserves addition and compositions because it is the actual functor on arrows. \(\square\)

Corollary 3.2. The functor \(F^b:D^b(\mathcal B)\to D^b(\mathcal A)\) is fully faithful.

Proof. Both bounded inputs belong to the domains of Theorem 2.1. Take \(n=0\) in Proposition 3.1. Its map is an isomorphism because \(\Theta\) is. The bounded-denominator theorem identifies the Hom groups of the bounded categories with the Hom groups in their full unbounded derived categories. Therefore the actual bounded inclusion map is bijective on each Hom set. The same argument with any \(n\) compares all shifted Hom groups, including positive Ext, degree zero and negative degrees. \(\square\)

4. Reconstructing bounded objects by their connecting maps

Let \(D^b_{\mathcal B}(\mathcal A)\) be the full subcategory of bounded ambient derived objects whose cohomology belongs to \(\mathcal B\) in every degree. It is triangulated. Indeed the long cohomology sequence of a triangle contains a five-term exact window with the missing vertex's cohomology in the middle. Its two neighboring kernel and cokernel objects belong to \(\mathcal B\), and their short extension is the middle cohomology object. This proves closure under cones; shifts and rotations cover all placements. Existing direct summands stay inside because a summand on cohomology is a kernel of an endomorphism of an object of \(\mathcal B\).

Theorem 4.1. Under the hypotheses of Theorem 2.1, the exact functor

is an equivalence. It preserves the natural truncation maps. Every object on the right with cohomology in a finite interval \([a,b]\) has an intrinsic representative with terms in that same interval.

Proof. The image is in the stated subcategory, since inclusion computes the same cohomology. Full faithfulness is Corollary 3.2. We prove essential surjectivity by a finite induction on the number of nonzero cohomology objects of an ambient \(Z\).

If there are none, cohomology detection gives \(Z\simeq0\). If there is one, in degree \(b\), smart truncation identifies \(Z\) with \(H^bZ[-b]\). Its cohomology object lies in \(\mathcal B\), so it is already an intrinsic object. Otherwise choose the largest nonzero cohomology degree \(b\) and use the natural top truncation triangle

The object \(L\) has fewer nonzero cohomology objects, all in \(\mathcal B\). By induction choose \(U\in D^b(\mathcal B)\) and \(\alpha:F^bU\xrightarrow{\sim}L\). Regard \(V\) as an intrinsic object too. Fullness lifts the actual connecting map \(\alpha[1]^{-1}\delta\) to an intrinsic map

Complete \(-\widetilde\delta[-1]:V[-1]\to U\) to an intrinsic distinguished triangle and rotate once. The rotation sign makes its last arrow \(-(-\widetilde\delta[-1])[1]=\widetilde\delta\), so the result is

The bounded derived category is triangulated, so \(W\) remains a bounded intrinsic object. Applying \(F^b\), using \(\alpha\), and rotating (4.4) and (4.2) twice puts their identical connecting arrow first. The triangle morphism axiom extends the identity square on that arrow to a triangle comparison. Its third component is an isomorphism: apply the long exact cohomology sequences and the five-term comparison lemma from the thick-subcategory prerequisite. Rotate back to obtain \(F^bW\simeq Z\).

The number of nonzero cohomology objects decreases at each step, so this construction terminates after finitely many steps even if there are gaps between their degrees. It retains the connecting maps; it does not replace every object by the direct sum of its cohomology objects. Exercise 3 gives a nonzero connecting map that prevents that replacement.

For the interval assertion, inclusion preserves the cohomology of \(W\), so its intrinsic cohomology also vanishes outside \([a,b]\). The bounded-representative theorem in the prerequisite applies inside \(\mathcal B\), using its own kernels and quotients, and gives a complex of \(\mathcal B\)-objects with terms only in that interval.

Finally inclusion commutes with smart truncations on intrinsic complexes, since their kernels, images and quotients agree ambiently. It takes the natural maps and short-row boundary triangles to the corresponding ambient maps and triangles. This remains true on derived arrows, since quasi-isomorphisms are inverted on both sides. Thus (4.1) is compatible with the truncation structure and its maps. A fully faithful essentially surjective functor is an equivalence: choose an intrinsic preimage and an isomorphism for each target object, and define its inverse on arrows using the unique lifted conjugate arrow. Uniqueness preserves identity and composition and gives the unit and counit isomorphisms. \(\square\)

5. One-sided injective and projective criteria

Two useful ways to obtain (1.2) use genuine injective or projective objects.

Suppose both categories have enough injectives and every intrinsic injective is ambient injective. Take \(\mathcal J_A,\mathcal J_B\) to be their injective classes and take \(\mathcal P_A=\mathcal A\), \(\mathcal P_B=\mathcal B\). The source replacements are identities. Proposition 6.1 of the derived-Hom prerequisite proves the two adapted-pair assertions: bounded-below injectives supply the target replacements and the required Hom acyclicity. The stated inclusion of injectives is exactly the first nesting condition; fullness supplies the second. Shared intrinsic injective replacements therefore prove Theorems 2.1 and 4.1.

Dually, suppose both categories have enough projectives and every intrinsic projective is ambient projective. Set \(\mathcal P_A,\mathcal P_B\) to those classes and take the target classes to be the entire respective categories. The target replacements are identities, and the same prerequisite's projective computation supplies the adapted pairs. The projective inclusion is the second nesting condition. This gives the same comparison and bounded equivalence without assuming enough injectives.

An intrinsic injective or projective tests only intrinsic extension or lifting problems. Fullness and exactness of the inclusion do not automatically enlarge its tests to every ambient object. The nesting hypothesis is therefore substantive; Exercise 2 exhibits a Serre subcategory for which it cannot give the claimed comparison.

6. Exercises with complete solutions

Exercise 1 — a central summand of a module category (18 points)

Let \(R=R_1\times R_2\) be a product of unital rings, let \(\mathcal A=\operatorname{Mod}(R)\), and let \(\mathcal B\) consist of modules killed by \(e_2=(0,1)\).

1. Identify \(\mathcal B\), prove that it is a Serre subcategory, and verify a nested projective pair (7 points).
2. Compute the intrinsic-to-ambient derived-Hom comparison for \(R_1\)-modules, including its map on Ext (6 points).
3. Give an explicit intrinsic preimage for a bounded ambient complex whose cohomology is killed by \(e_2\) (5 points).

Solution. The central idempotents \(e_1=(1,0)\) and \(e_2\) satisfy \(e_1+e_2=1\) and \(e_1e_2=0\). Hence every module decomposes canonically as

The projections are exact: an exact module sequence stays exact after either idempotent projection, as follows by applying the projection to a lift or to a kernel equation. Thus \(\mathcal B\) is equivalent to \(\operatorname{Mod}(R_1)\) by \(M_1\mapsto(M_1,0)\). Any subobject or quotient of such a module has zero second coordinate. In a short exact sequence with zero second coordinate at both ends, exactness gives zero second coordinate in the middle. This proves the Serre property.

Both module categories have enough projectives: a free module on the underlying set of a module maps onto it. If \(P\) is projective over \(R_1\), then \((P,0)\) is projective over \(R\). Indeed an ambient epimorphism is a pair of epimorphisms, and a map from \((P,0)\) lifts by the projective property of \(P\) in its first coordinate; the second map is zero. Consequently the classes

give the nested pairs of Section 5.

Choose a bounded-above projective resolution \(P\to M\) over \(R_1\), with \(M\) in degree zero. The same resolution \((P,0)\to(M,0)\) is ambient projective. Fullness identifies every term and differential of the normalized Hom complexes, giving the canonical comparison

In particular \(\operatorname{Ext}_{R_1}^n(M,N)\to \operatorname{Ext}_{R}^n((M,0),(N,0))\) is an isomorphism for every \(n\geq0\). Its map is the actual inclusion on derived arrows, by Proposition 3.1, rather than an isomorphism obtained by an unrelated choice of resolutions. The same argument computes bounded source complexes and bounded-below targets.

Finally write an ambient bounded complex as \(Z=(Z_1,Z_2)\), term by term. Its second cohomology coordinate is \(H^i(Z_2)\). Under the stated hypothesis \(Z_2\) is acyclic, so the canonical inclusion \((Z_1,0)\to Z\) is a quasi-isomorphism. The bounded \(R_1\)-complex \(Z_1\) is the required intrinsic preimage. Its cohomology has exactly the same degree support as that of \(Z\); smart truncation supplies the same finite cohomology interval if desired. \(\square\)

Exercise 2 — a Serre subcategory with different higher Ext (22 points)

Fix a field \(k\). Let \(\mathcal A\) be the category of representations of

in arbitrary \(k\)-vector spaces. Let \(S_i\) have \(k\) at vertex \(i\) and zero elsewhere, and let \(\mathcal B\) consist of representations supported at vertices \(0\) and \(3\).

1. Prove that \(\mathcal A\) is abelian and \(\mathcal B\) is Serre and semisimple (6 points).
2. For \(i=0,1,2\), let \(P_i\) have \(k\) at \(i,i+1\) and identity on their arrow, and set \(P_3=S_3\). Prove projectivity and compute \(\operatorname{Ext}_{\mathcal A}^3(S_0,S_3)\) (10 points).
3. Explain exactly why the conclusion of Corollary 3.2 fails and why Section 5 does not apply (6 points).

Solution. Kernels and cokernels of a morphism of representations are formed vertex by vertex; the arrow maps restrict to kernels and descend to quotients. The two zero-composite relations remain true. Images and coimages likewise agree vertexwise, since they do in vector spaces, so the category is abelian and exactness is vertexwise.

The subcategory \(\mathcal B\) is equivalent to \(\operatorname{Vect}_k\times\operatorname{Vect}_k\), with its two endpoint spaces as coordinates. Vanishing at either interior vertex passes to subobjects, quotients and extensions by vertexwise exactness. Thus it is Serre. Every surjection of vector spaces splits by choosing lifts of a basis, so every short exact sequence in \(\mathcal B\) splits. Every object is projective and injective and

A map \(P_i\to V\) is determined by a vector \(v\in V_i\); its next component is \(a_i(v)\). The next arrow equation is precisely the zero-composite relation. For \(i=3\) only the terminal vector is needed. Hence

Evaluation at a vertex is exact, so each \(P_i\) is projective. The category has enough projectives: take a basis of each \(V_i\), one copy of \(P_i\) for each basis vector, and the maps specified by (6.6). Their direct sum maps onto \(V\) at every vertex. Such a sum is projective, since a map from it lifts across an epimorphism by lifting separately the defining vector of each summand.

For \(i=0,1,2\), define \(d_i:P_{i+1}\to P_i\) to be identity at their common vertex \(i+1\) and zero at other vertices. These are representation maps and adjacent composites vanish. The sequence

is exact. At vertex zero its nonzero map is \((P_0)_0\to(S_0)_0\), an identity. At vertices one, two and three the respective nonzero maps are \((d_0)_1\), \((d_1)_2\) and \((d_2)_3\), again identities. Thus every vertex sequence is exact. Apply \(\operatorname{Hom}_{\mathcal A}(-,S_3)\): by (6.6) the result has \(k\) only in cohomological degree three and zero elsewhere. Therefore

Equations (6.5) and (6.8) show that the actual map \(\operatorname{Hom}_{D^b(\mathcal B)}(S_0,S_3[3])\to \operatorname{Hom}_{D^b(\mathcal A)}(S_0,S_3[3])\) is the map \(0\to k\), so the bounded inclusion is not full. In particular the intrinsic projective \(S_0\) is not ambient projective. The epimorphism \(P_0\to S_0\) has no section: a section would be identity at vertex zero, but its arrow equation along \(0\to1\) would say that this identity followed by the identity arrow of \(P_0\) is zero.

The projective nesting required in Section 5 therefore fails. More generally there cannot be nested adapted pairs satisfying all the hypotheses of Theorem 2.1 for this inclusion, since those hypotheses would force the contradictory Ext isomorphism. Being Serre, and hence closed under extensions, alone does not settle higher derived morphisms. \(\square\)

Exercise 3 — retaining the truncation boundary (20 points)

Use the category \(\mathcal A\) and the objects of Exercise 2. This time put \(\mathcal C=\mathcal A\times\operatorname{Vect}_k\), and include \(\mathcal B'=\mathcal A\times\{0\}\) in \(\mathcal C\). Let

have terms in degrees \(0,1,2\).

1. Verify the nesting criterion for \(\mathcal B'\subset\mathcal C\) and compute the cohomology of \(W\) (6 points).
2. Prove that the top truncation boundary \(\delta:S_0[-2]\to S_3[1]\) is nonzero, and explain its relevance to Theorem 4.1 (9 points).
3. Whenever a connecting map \(\delta:V\to U[1]\) has an actual chain representative, write the cone used in the reconstruction and verify its differential and sign (5 points).

Solution. Exercise 2 gives enough projectives in \(\mathcal A\). Vector spaces have enough projectives as well. Products of the free covers give enough projectives in \(\mathcal C\). A projective \(P\) of \(\mathcal A\) becomes projective as \((P,0)\), since lifting across a product epimorphism reduces to its first component. The full target classes and the respective projective source classes are nested adapted pairs. Theorem 4.1 applies to this proper subcategory.

The vertexwise resolution (6.7) gives

For example the kernel in degree zero is the copy of \(S_3\) in \(P_2\), the middle kernel is the image of \(P_2\), and the final cokernel is \(S_0\). Thus its top truncation triangle is

Its last arrow belongs to the nonzero group \(\operatorname{Hom}_D(S_0[-2],S_3[1])= \operatorname{Ext}_{\mathcal A}^3(S_0,S_3)=k\). We prove that this particular arrow is nonzero, without identifying a scalar by an unchosen Yoneda sign convention.

The bounded projective complex \(W\) computes morphisms out of itself. By (6.6) all three groups \(\operatorname{Hom}_{\mathcal A}(P_i,S_3)\), \(i=0,1,2\), vanish. Its entire Hom complex into \(S_3\) is therefore zero, and

If \(\delta=0\), apply \(\operatorname{Hom}_D(S_0[-2],-)\) to (6.11). Exactness lifts the identity on \(S_0[-2]\) to a section of \(W\to S_0[-2]\). Together with the first map of (6.11), that section defines \(S_3\oplus S_0[-2]\to W\). It is an isomorphism on every cohomology object, by (6.10) and the section identity, so is a derived isomorphism. Projection to \(S_3\) then gives a nonzero arrow \(W\to S_3\): its composite with the first map from \(S_3\) is identity. This contradicts (6.12), since \(S_3\neq0\). Hence \(\delta\neq0\).

The object \((W,0)\) has an intrinsic preimage under the bounded equivalence, but it cannot be replaced by \((S_3\oplus S_0[-2],0)\). Lifting the connecting map in Theorem 4.1 retains the data that distinguishes them. Here that map is a derived arrow; there need not be an actual chain map between the displayed cohomology complexes representing it.

For the last part assume separately that \(\delta:V\to U[1]\) is an actual chain map, with components \(\delta^i:V^i\to U^{i+1}\). Its chain equation is

The source-shifted map \(-\delta[-1]:V[-1]\to U\) has degree-\(i\) component \(-\delta^{i-1}\). The target-first cone therefore has terms \(U^i\oplus V^i\) and differential

Its square is zero: the first off-diagonal entry is \(-d_U\delta-\delta d_V=0\), and both diagonal squares are zero. The unrotated triangle ends in the positive projection to \(V\). Rotation produces the triangle \(U\to\operatorname{Cone}(-\delta[-1])\to V\to U[1]\), whose last map is \(-(-\delta[-1])[1]=\delta\), as required. This calculation verifies the sign when chain representatives are available; the abstract reconstruction uses distinguished triangles and also covers maps represented by roofs. \(\square\)

Exercise 4 — a shared Hom computation with its actual generator (20 points)

Let \(\mathcal A=\operatorname{Vect}_k\times\operatorname{Vect}_k\) and \(\mathcal B=\operatorname{Vect}_k\times\{0\}\). Work in its first coordinate and define

with all other terms zero. Set \(X=k[1]\), \(Y=k[-1]\). Use \(p:P\to X\), projection to the second coordinate in degree \(-1\), and \(i:Y\to I\), inclusion into the first coordinate in degree \(1\).

1. Check the replacements and write the full normalized Hom complex (8 points).
2. Compute its cohomology over every field, including characteristic two (7 points).
3. Identify its surviving generator with the actual derived arrow and with the ambient comparison (5 points).

Solution. Vector spaces are projective and injective: a basis lifts across any surjection, and a linear map extends from a subspace by extending a basis. Thus the full source and target classes are nested adapted pairs. We have \(H^{-1}P=\{(0,y)\}\cong k\) and \(H^0P=0\), so \(p\) is a quasi-isomorphism. Likewise \(H^0I=0\) and \(H^1I=k^2/\{(0,z)\}\cong k\), so \(i\) is a quasi-isomorphism.

The Hom complex has nonzero terms only in degrees \(0,1,2\). Write a degree-zero element as \(\lambda:P^0\to I^0\). In degree one use a row \((a,b):P^{-1}\to I^0\) and a column \((c,d):P^0\to I^1\). Degree two is a matrix \(P^{-1}\to I^1\). Formula (1.1) gives

Indeed in degree zero the source contribution is \(-\lambda d_P\), giving \((-\lambda,0)\), and the target contribution is \(d_I\lambda=(0,\lambda)\). In degree one the sign reverses: \(d_I(a,b)+(c,d)d_P\) gives the matrix in (6.17). In particular \(D_1D_0=0\).

The map \(D_0\) is injective. The kernel of \(D_1\) consists of \(c=b=0\), \(d=-a\), exactly the image of \(D_0\). Its image consists of every matrix whose upper-right entry is zero: prescribe independently \(c,b,a+d\). Thus

where the upper-right entry is the quotient coordinate. These equations remain valid in characteristic two, since \(d=-a\) and \(a+d=0\) are equivalent there too. There are no further degrees. Hence the derived Hom object is isomorphic to \(k[-2]\); its nonzero cohomology lies in degree two.

A generator is the matrix \(E_{12}\), which takes \((x,y)\) to \((y,0)\). It is a degree-two cycle and represents a chain map \(f:P\to I[2]\). On the surviving source cohomology \((0,y)\), it takes the value \((y,0)\), which is the target quotient's standard generator. Therefore

is identity. More explicitly \(f=i[2]\,1_{k[1]}\,p\) as chain maps, which gives (6.19) directly after inverting the two quasi-isomorphisms.

The intrinsic and ambient Hom complexes have identical entries and differentials. Their canonical comparison is identity on (6.16)–(6.17), hence takes this same generator to the identity in the first ambient coordinate. This verifies the actual-arrow conclusion of Proposition 3.1 along with the cohomology computation; an abstract comparison of dimensions would not identify that arrow. \(\square\)