# Quasi-coherent sheaves and concentrated scheme maps

*Written and mathematically self-checked by GPT-6.1 Sol (OpenAI), in Codex at Ultra, October 2026. The affine, small-submodule and Čech arguments adapt the Stacks project authors' proofs in the pinned AI Integrated Stacks Project edition; those adaptations retain GFDL-1.2-or-later. The unbounded comparisons and expanded explanations are independently written. See the [course notice](../LICENCE.md) for authorship and terms. No independent review or formalization is claimed.*

On a scheme, quasi-coherent cohomology permits stronger projection and base-change results than arbitrary sheaves on a ringed space. This lesson develops the extra algebra and cohomology before proving those results for unbounded complexes.

Use the [module-sheaf lesson](sheaves-of-modules-on-a-ringed-space.md), Lemma 1.3, Theorem 3.1 and Lemma 5.1, for stalks, exact filtered colimits and open extension by zero; the [common reading](complexes-cones-and-localization.md), Theorem 5.1 and Proposition 5.2, for cohomology and triangles; the [bounded-resolution lesson](injective-modules-and-bounded-below-derived-functors.md), Sections 2–4, for injectives and flasque acyclicity; and the [Grothendieck-category lesson](k-injective-resolutions-in-grothendieck-categories.md), Theorems 2.4, 4.1 and 5.1 and Proposition 5.5, for enough injectives, unbounded resolutions and acyclic models. For the tower in Section 4, first read Sections 1–2 of [Inverse limits and unbounded resolutions](inverse-limits-and-unbounded-resolutions.md), Lemmas 1.1 and 2.1 and Propositions 1.2 and 2.2. Its geometric Theorem 6.2 is read after Sections 1–2 of the present lesson; it is not a prerequisite for any proof here. The tensor and adjunction conventions are those of the existing derived-operation lessons.

Rings are commutative with identity. A scheme is a locally ringed space locally isomorphic to \(\operatorname{Spec}A\). We use its defining principal opens \(D(a)\), with rings \(A_a\), and the usual gluing definition of a morphism. A sheaf is quasi-coherent if locally it is an associated module sheaf; the associated sheaves and their gluing are constructed below. Write \(D_{\mathrm{qc}}(X)\) for complexes of all \(\mathcal O_X\)-modules whose cohomology sheaves are quasi-coherent. Lemma 1.4 states the geometric definitions and proves the affine-product, separation and cover facts used below.

## 1. Affine sheaves and localization

**Lemma 1.1 (localization and a finite cover).** Localization of modules is exact, \(A_a\otimes_A M=M_a\), and a finite principal cover \(D(a_1),\ldots,D(a_m)\) of \(\operatorname{Spec}A\) gives an exact augmented ordered Čech complex
\[
0\longrightarrow M\longrightarrow
\prod_i M_{a_i}\longrightarrow
\prod_{i<j}M_{a_i a_j}\longrightarrow\cdots
\longrightarrow M_{a_1\cdots a_m}\longrightarrow0.
\tag{1.1}
\]
The differential is the alternating sum of restriction maps, deleting one index. Every principal open is quasi-compact.

**Proof.** The fraction \(m/s\) is zero precisely when some allowed denominator annihilates \(m\). Thus localization preserves injections; if a fraction maps to zero in a localized quotient, multiplication by an additional denominator makes its numerator lie in the submodule, proving exactness in the middle. Lifting a numerator proves surjectivity. The balanced maps
\[
(b/s)\otimes m\mapsto bm/s,\qquad m/s\mapsto(1/s)\otimes m
\]
are inverse, proving the tensor assertion.

An ideal contained in no prime is the unit ideal: a proper ideal is contained in a maximal ideal by Zorn's lemma, and a maximal ideal is prime. Consequently a principal cover means \((a_1,\ldots,a_m)=A\). For an arbitrary principal cover the same argument gives \(1=\sum_{\text{finite }i} b_i a_i\), and these finitely many opens already cover. Applying this to \(A_a\) proves quasi-compactness of \(D(a)\). Principal opens form a basis, so this also reduces arbitrary covers to principal covers.

Localize (1.1) at \(a_i\). The \(i\)-th member of the cover is now the whole affine scheme. Regard an ordered cochain as an alternating cochain on all index tuples, zero on tuples with repetitions. Define
\[
(hc)_{j_0\ldots j_{p-1}}=c_{i,j_0,\ldots,j_{p-1}}
\]
after reordering with its sign; in degree zero take the \(i\)-th component. In \(\delta h+h\delta\), the terms deleting a \(j_r\) cancel in pairs, and the term deleting \(i\) is \(c\). Thus the localized augmented complex is contractible. Localization commutes with its finite products and with cohomology by exactness.

A module \(N\) with \(N_{a_i}=0\) for every \(i\) is zero. For \(n\in N\), choose a common exponent \(r\) with \(a_i^r n=0\) for all \(i\). The ideal generated by the \(a_i^r\) has radical \(A\), hence is \(A\); expressing \(1\) in that ideal gives \(n=0\). Applying this detection to each cohomology module proves (1.1). \(\square\)

**Theorem 1.2 (the affine equivalence).** On \(X=\operatorname{Spec}A\), the assignments
\[
M\longmapsto\widetilde M,\qquad
\mathcal F\longmapsto\Gamma(X,\mathcal F)
\tag{1.2}
\]
are inverse exact equivalences between \(A\)-modules and quasi-coherent \(\mathcal O_X\)-modules. They satisfy
\[
\Gamma(D(a),\widetilde M)=M_a,\qquad
\widetilde M_{\mathfrak p}=M_{\mathfrak p}.
\tag{1.3}
\]
For an arbitrary module sheaf \(\mathcal G\), not necessarily quasi-coherent,
\[
\operatorname{Hom}_{\mathcal O_X}(\widetilde M,\mathcal G)
=\operatorname{Hom}_A(M,\Gamma(X,\mathcal G)).
\tag{1.4}
\]

**Proof.** Define \(\widetilde M\) on principal opens by \(M_a\). Lemma 1.1 proves the sheaf equalizer condition on finite principal covers. Every principal open is quasi-compact, so this proves the condition on all covers by refinement and finite gluing; it uniquely extends to a sheaf on arbitrary opens by compatible sections on the basis. The principal neighbourhoods of \(\mathfrak p\) give the stalk \(M_{\mathfrak p}\). In particular the sections on \(X=D(1)\) are \(M\).

Given an \(A\)-linear map \(M\to\Gamma(X,\mathcal G)\), send \(m/a^r\) on \(D(a)\) to \(a^{-r}\) times the restriction of the image of \(m\). Fraction relations hold because \(a\) is invertible there. These maps glue. Conversely every sheaf morphism gives the map on \(X\), and its restriction is forced by multiplication by \(a^{-1}\). This proves (1.4).

Let \(\mathcal F\) be quasi-coherent. It has a finite principal trivializing cover \(D(a_i)\): inside any affine neighbourhood where it is associated to a module, restrict to principal opens of \(X\) lying in that neighbourhood, and use quasi-compactness. To justify this restriction before using the equivalence, if \(\operatorname{Spec}B\subset\operatorname{Spec}C\) is an affine open, the natural map from the associated sheaf of \(B\otimes_C N\) to the restriction of \(\widetilde N\) is an isomorphism on stalks: the two local rings at each point agree, and both module stalks are \(N_{\mathfrak p}\). This uses only the defining local rings and the already constructed associated sheaf. Write \(N_i=\Gamma(D(a_i),\mathcal F)\); restrictions to intersections are localizations of \(N_i\). The sheaf condition identifies \(M=\Gamma(X,\mathcal F)\) with the kernel of
\[
\prod_i N_i\longrightarrow\prod_{i<j}\Gamma(D(a_i a_j),\mathcal F).
\]
Localize this finite equalizer at \(b\). Exact localization and finite products identify the result with the sheaf equalizer for the cover \(D(a_i b)\) of \(D(b)\). Hence \(M_b=\Gamma(D(b),\mathcal F)\). The morphism \(\widetilde M\to\mathcal F\) from (1.4) is therefore an isomorphism on the basis. Finally \(\widetilde{(-)}\) is exact on stalks by Lemma 1.1; its inverse on quasi-coherent sheaves is exact as well. This proves (1.2). \(\square\)

The definition by associated modules agrees with the definition by local presentations by free sheaves. A presentation of \(M\) gives a presentation of \(\widetilde M\), since association is exact and preserves sums. Conversely the cokernel of a matrix between free sheaves on an affine neighbourhood is the sheaf associated to the cokernel of that matrix of modules, by Theorem 1.2 and (1.4).

**Proposition 1.3 (operations and flatness).** Quasi-coherent sheaves form an abelian category, and their colimits are the ordinary sheaf colimits. For a morphism of affines associated to \(A\to B\),
\[
f^*\widetilde M=\widetilde{B\otimes_A M},\qquad
f_*\widetilde N=\widetilde{N\text{ regarded as an }A\text{-module}}.
\tag{1.5}
\]
Association commutes with tensor products. A flat ring map gives exact pullback. An affine open immersion gives a flat ring map. Flatness is preserved by base change.

**Proof.** Kernels, cokernels and colimits can be checked on an affine neighbourhood using the equivalence and the stalk test. Arbitrary sheaf colimits have the corresponding module colimits on stalks. Tensor compatibility follows from
\((M\otimes_A N)_{\mathfrak p}=M_{\mathfrak p}\otimes_{A_{\mathfrak p}}N_{\mathfrak p}\), with the balanced maps of Lemma 1.1.

For pullback in (1.5), stalkwise the natural balanced map is
\[
B_{\mathfrak q}\otimes_{A_{\mathfrak p}}M_{\mathfrak p}
=(B\otimes_A M)_{\mathfrak q}.
\]
For pushforward, the inverse image of \(D(a)\) is \(D(f(a))\), whose sections are \(N_{f(a)}\), precisely the restrictions of the underlying \(A\)-module. This proves (1.5). Flat \(B\) makes \(B\otimes_A-\) exact, including after localization, so pullback is exact. Conversely, a morphism flat on local rings has exact pullback by the stalk formula and exact localization. On affines, apply this to associated module sheaves and use the exact affine equivalences to conclude that \(B\otimes_A-\) is exact; hence \(B\) is flat over \(A\). If \(f\) is an affine open immersion, pullback is restriction and is exact, giving the same conclusion. For \(A\to B\) flat and any \(A\)-algebra \(C\), tensoring a \(C\)-module with \(C\otimes_A B\) over \(C\) is tensoring its underlying \(A\)-module with \(B\). Exactness proves base-change flatness; localizing proves the sheaf assertion. \(\square\)

**Lemma 1.4 (affine geometry and geometric cover criteria).** We use the following scheme definitions. A morphism \(h:S\to T\) is *quasi-compact* if \(h^{-1}V\) is quasi-compact for every affine open \(V\subset T\). It is *quasi-separated* if its diagonal \(S\to S\times_T S\) is quasi-compact, and *separated* if that diagonal is a closed immersion. A scheme has either separation property when its structure morphism to \(\operatorname{Spec}\mathbb Z\) does. A closed immersion is an embedding as the closed subscheme defined by a quasi-coherent ideal sheaf \(\mathcal J\): its underlying subset is \(V(\mathcal J)=\{t:\mathcal J_t\subset\mathfrak m_t\}\), and its structure sheaf is the restriction of \(\mathcal O_T/\mathcal J\) to that subset. Equivalently, the closed inclusion has a surjective map \(\mathcal O_T\to i_*\mathcal O_Z\) with quasi-coherent kernel; the affine descriptions below verify this equivalence. These definitions give the following facts.

1. For every scheme \(S\) and ring \(C\), there is a natural bijection
   \[
   \operatorname{Hom}_{\mathrm{Sch}}(S,\operatorname{Spec}C)
   =\operatorname{Hom}_{\mathrm{Ring}}(C,\Gamma(S,\mathcal O_S)).
   \]
   For \(A\)-algebras \(C,B\), the fibre product of their affine schemes over \(\operatorname{Spec}A\) is \(\operatorname{Spec}(C\otimes_A B)\), with its universal property valid for arbitrary testing schemes.
2. A closed subscheme of an affine scheme is affine. Closed immersions remain closed immersions under arbitrary base change. Affine schemes and their open subschemes are separated. Hence intersections of affine opens in a separated scheme are affine.
3. Quasi-compactness of a morphism can be checked on an affine open cover of its target. A morphism \(h:S\to T\) is quasi-separated exactly when \(U_1\cap U_2\) is quasi-compact for every affine open \(V\subset T\) and affine opens \(U_1,U_2\subset h^{-1}V\). Equivalently, every \(h^{-1}V\) is a quasi-separated scheme. In particular a scheme is quasi-separated exactly when intersections of its affine opens are quasi-compact; intersections of arbitrary quasi-compact opens are then quasi-compact too.
4. Quasi-compact and quasi-separated morphisms retain these properties after arbitrary base change. For an affine base change \(\operatorname{Spec}B\to\operatorname{Spec}A\), affine opens over \(\operatorname{Spec}A\) become affine, and pullback preserves their covers and intersections. Thus any chosen finite affine cover, and any chosen finite affine covers of its intersections, pull back to covers with the same number of members. If the intersections in a chosen cover are affine, they remain affine.

**Proof.** We give the constructions and the locality arguments.

For (1), let \(\varphi:C\to\Gamma(S,\mathcal O_S)\). At \(s\in S\), compose with the germ map and the residue map to the field \(\kappa(s)=\mathcal O_{S,s}/\mathfrak m_s\), and let \(\mathfrak p_s\) be its prime kernel. This defines \(f(s)=\mathfrak p_s\). The inverse image of \(D(c)\) is the locus where the section \(\varphi(c)\) has invertible germ. This locus is open: an inverse in a stalk is represented near that point, and its product with the section equals one after shrinking. These inverses are unique and glue, so \(\varphi(c)\) is invertible as a section on that locus. Consequently
\[
C_c\longrightarrow\Gamma(f^{-1}D(c),\mathcal O_S),
\qquad a/c^r\longmapsto\varphi(a)\varphi(c)^{-r},
\]
defines compatible maps on the principal-open basis. They give a structure-sheaf map. Its stalk map \(C_{\mathfrak p_s}\to\mathcal O_{S,s}\) is local: a fraction \(a/t\), with \(t\notin\mathfrak p_s\), maps into \(\mathfrak m_s\) precisely when \(a\in\mathfrak p_s\). Thus \(f\) is a scheme morphism. Conversely a scheme morphism supplies \(\varphi\) on global sections. Its local stalk maps force its point map to be \(s\mapsto\mathfrak p_s\), and localization forces the displayed maps on principal opens. The two constructions are inverse and natural.

For any testing scheme \(S\), two maps to \(\operatorname{Spec}C\) and \(\operatorname{Spec}B\) over \(\operatorname{Spec}A\) are therefore two compatible ring maps into \(\Gamma(S,\mathcal O_S)\). Their unique extension from \(C\otimes_A B\) sends \(c\otimes b\) to the product of the two images. This proves the claimed fibre-product universal property on arbitrary schemes, without assuming that \(S\) is affine. General scheme fibre products are obtained by gluing these affine products. More explicitly, choose affine opens in the two sources mapping into affine opens of the base. In an affine product, imposing that either coordinate land in a smaller open defines an open subscheme representing exactly that condition on test maps. Overlaps of two product charts represent the same compatible pairs with both coordinates in the source overlaps; their identifications are therefore unique and obey the cocycle identity. The usual gluing of schemes gives the product, and the chartwise universal property glues to its universal property. This argument also proves that base-changing an open immersion gives the corresponding open subscheme, and that pullback of open covers and intersections is ordinary inverse image.

For (2), let a closed subscheme of \(\operatorname{Spec}C\) have ideal sheaf \(\mathcal J\). Theorem 1.2 identifies it with \(\widetilde I\) for the ideal \(I=\Gamma(\operatorname{Spec}C,\mathcal J)\subset C\). Exact association and localization give quotient rings \(C_c/I_c=(C/I)_c\) on principal opens. The primes containing \(I\) correspond to the primes of \(C/I\), and these local descriptions identify the closed subscheme with \(\operatorname{Spec}(C/I)\). The ambient quotient sheaf has zero stalk at \(\mathfrak p\notin V(I)\), since \(I_{\mathfrak p}=C_{\mathfrak p}\), and stalk \(C_{\mathfrak p}/I_{\mathfrak p}\) at \(\mathfrak p\in V(I)\). These are the stalks of the direct image of the closed subscheme, so the canonical map is an isomorphism by the earlier stalk test. Conversely, for a closed inclusion with a surjective structure-sheaf map and quasi-coherent kernel, the kernel stalk is the whole ambient ring off the subset and is proper inside its maximal ideal on the subset, because the stalk map is local. Thus the kernel has exactly that subset as its zero locus, and the induced quotient gives its structure sheaf. This proves the equivalence of the two closed-immersion descriptions. After a map \(\operatorname{Spec}D\to\operatorname{Spec}C\), part (1) identifies its base change with
\[
\operatorname{Spec}\bigl(D\otimes_C(C/I)\bigr)
=\operatorname{Spec}(D/ID).
\]
The ring isomorphism sends \(d\otimes\bar c\) to \(d\varphi(c)\bmod ID\), with inverse \(d\bmod ID\mapsto d\otimes1\). It requires no flatness. The kernel ideal \(ID\) gives a quasi-coherent ideal sheaf by Theorem 1.2. These descriptions cover any base-changed target and prove the assertion for arbitrary base changes. Closedness, the quotient sheaf and its quasi-coherent kernel are all local on the target: the local closed images have open complements, and their quotient descriptions agree on overlaps.

The affine diagonal is the closed immersion induced by the surjective multiplication homomorphism \(C\otimes_{\mathbb Z}C\to C\). If \(W\) is open in an affine scheme \(S\), its diagonal is the base change of the diagonal of \(S\) along \(W\times W\to S\times S\), so \(W\) is separated. Finally, for affine opens \(U,V\) in a separated scheme \(S\), their intersection is the base change of the closed diagonal along \(U\times V\to S\times S\). It is a closed subscheme of the affine \(U\times V\), and hence affine. These identifications also follow directly from the universal property: a pair of maps to \(U,V\) with equal composite to \(S\) is exactly a map to \(U\cap V\).

Here are the locality details for (3). Every quasi-compact scheme has a finite affine cover. The nonvanishing locus of a global section on such a scheme is quasi-compact: on each member of that finite affine cover it is a principal open, which is quasi-compact by Lemma 1.1. Also every quasi-compact open of an affine has a finite principal cover.

Suppose an affine cover \((V_a)\) of \(T\) has all \(h^{-1}V_a\) quasi-compact. If \(W\subset T\) is affine, choose a finite principal cover of \(W\) subordinate to \((W\cap V_a)_a\). Each of its members is a quasi-compact open of some \(V_a\), so it has a finite principal cover inside that \(V_a\). The inverse image of each such principal open is the nonvanishing locus of the pulled-back section on the quasi-compact scheme \(h^{-1}V_a\), and is quasi-compact by the preceding paragraph. Their finite union is \(h^{-1}W\). This proves the target-local criterion without any separation hypothesis.

The affine products \(U_1\times_V U_2\), for the affine triples in (3), form an affine open cover of \(S\times_T S\). The inverse image of this product under the diagonal is \(U_1\cap U_2\): its universal property is again equality of the two maps to \(S\). The target-local criterion just proved therefore gives the stated criterion for quasi-separatedness. With \(T=\operatorname{Spec}\mathbb Z\), it gives the scheme criterion. Applied to each \(h^{-1}V\), it also proves the asserted equivalent form. Finally, each quasi-compact open of a scheme has a finite affine cover. In a quasi-separated scheme, the intersection of two such opens is the finite union of the intersections of their affine cover members, each quasi-compact by this criterion. This proves the last assertion in (3).

For (4), first let \(h\) be quasi-compact and let \(T'\to T\) be arbitrary. Fix an affine open \(W'\subset T'\). By continuity and the principal-open basis, it has a finite principal cover \(W'_r\) such that each \(W'_r\) maps into an affine open \(V_r\subset T\). The scheme \(h^{-1}V_r\) has a finite affine cover \(U_{ri}\). Part (1) makes each \(U_{ri}\times_{V_r}W'_r\) affine. These finitely many affine products cover the inverse image of \(W'_r\) under the base-changed morphism. That inverse image is quasi-compact; taking the finite union over \(r\) proves quasi-compactness over \(W'\), and thus over every affine open of \(T'\).

The diagonal of the base-changed morphism is a base change of the original diagonal. To check this identity, a test map into \(S'\times_{T'}S'\) factors through the new diagonal exactly when its two maps to \(S'\) are equal. Since the maps to \(T'\) are already equal, this is exactly equality of their maps to \(S\), the condition represented by the pullback of the original diagonal. The universal property therefore identifies the two schemes and their maps. Applying the quasi-compact base-change assertion just proved to the original diagonal proves preservation of quasi-separatedness.

For the final cover assertions, an affine open \(U=\operatorname{Spec}C\) over \(\operatorname{Spec}A\) pulls back to \(\operatorname{Spec}(C\otimes_A B)\). The open-subscheme part of (1) identifies pulled-back intersections with intersections of the pulled-back opens. Pulling back the same indexed covers thus proves all the statements about finite covers and affine intersections; a member which becomes empty contributes the empty scheme. No flatness, separation or noetherian assumption is needed for these cover identities. \(\square\)

## 2. Small submodules and enough injectives

**Theorem 2.1 (Gabber's construction).** For every scheme \(X\), \(\operatorname{QCoh}(X)\) is a Grothendieck abelian category. It has all limits and enough injectives.

**Proof.** Choose an affine cover \(U_i=\operatorname{Spec}A_i\), and for every pair choose an affine cover \(U_{ijk}\) of \(U_i\cap U_j\). These are sets of opens. Fix an infinite cardinal \(\kappa\) at least as large as their index sets. For \(\mathcal F\in\operatorname{QCoh}(X)\), put \(M_i=\Gamma(U_i,\mathcal F)\). On \(U_{ijk}\), the identifications
\[
M_i\otimes_{A_i}A_{ijk}
=\Gamma(U_{ijk},\mathcal F)
=M_j\otimes_{A_j}A_{ijk}
\tag{2.1}
\]
have flat coefficient maps by Proposition 1.3.

Start with sets \(S_i\subset M_i\), each of cardinal at most \(\kappa\), containing any prescribed set of at most \(\kappa\) local elements. For every \(m\in S_i\) and every \(j,k\), express its image in the right side of (2.1) as a finite sum of tensors. Add the finitely many \(M_j\) entries to \(S_j\). Do this for both directions on every overlap; repeat countably many times, always retaining earlier entries. The number of added elements is at most \(\kappa\) at each step, so the unions \(S_i^\infty\) still have that bound. Let \(N_i\) be the \(A_i\)-submodule they generate.

Flatness embeds \(N_i\otimes_{A_i}A_{ijk}\) in \(M_i\otimes_{A_i}A_{ijk}\). Each generator from \(N_i\) has its overlap image generated by entries from \(N_j\), and conversely. Thus these submodules have the same image in (2.1). Their associated sheaves agree on every overlap cover, with the same transition identifications inherited from \(\mathcal F\). They glue to a quasi-coherent subsheaf \(\mathcal N\subset\mathcal F\), generated by at most \(\kappa\) elements on each \(U_i\).

Such small subsheaves are directed: apply the construction to the union of the generators of two of them. Their union is \(\mathcal F\), because every local element can be included initially. There is only a set of isomorphism classes of sheaves with this size bound on this fixed cover. Indeed each \(N_i\) is a quotient of \(A_i^{(\kappa)}\); its possible kernels form a set. The overlap isomorphisms between their localizations form sets, and the cocycle conditions select a subset of their product. Choose a set of representatives \(\mathcal N_\alpha\), and let \(G=\bigoplus_\alpha\mathcal N_\alpha\). For each \(\mathcal F\), the maps from these representatives onto its small subsheaves yield an epimorphism from a sum of copies of \(G\) onto \(\mathcal F\). Hence \(G\) is a generator.

Proposition 1.3 makes the category abelian with all colimits. Filtered colimits are exact, since on every affine chart they are filtered module colimits, whose elementwise exactness is proved in the module-sheaf lesson. It is locally small, since a sheaf morphism is determined by its maps on a fixed set of affine charts. We have proved AB5 and a generator. The complete Grothendieck-category construction in the earlier lesson, Theorem 2.4, therefore supplies enough injectives. All limits follow from the following explicit right-adjoint construction. \(\square\)

**Lemma 2.2 (the quasi-coherent right adjoint).** Inclusion of quasi-coherent sheaves into all module sheaves has a right adjoint \(Q\). In particular a diagram of quasi-coherent sheaves has limit \(Q(\lim_{\operatorname{Mod}}\mathcal F_i)\).

**Proof.** Fix the small representatives from Theorem 2.1. For a module sheaf \(\mathcal P\), form the sum \(\mathcal H\) of one copy of \(\mathcal N_\alpha\) for every sheaf morphism \(\mathcal N_\alpha\to\mathcal P\), and let \(h:\mathcal H\to\mathcal P\) be their sum. This is a set-indexed sum. Let \(\mathcal K\) be the sum of all quasi-coherent subobjects of \(\mathcal H\) killed by \(h\). Subobjects form a set by the Grothendieck-category lesson's Lemma 2.1. Their sum is a quasi-coherent subobject by Proposition 1.3, and is still killed by \(h\). Put \(\mathcal L=\mathcal H/\mathcal K\), with its induced map \(\ell:\mathcal L\to\mathcal P\).

Any map \(a:\mathcal M\to\mathcal L\) from a quasi-coherent sheaf with \(\ell a=0\) is zero. Indeed the pullback \(\mathcal H'=\mathcal H\times_{\mathcal L}\mathcal M\) is quasi-coherent and maps epimorphically to \(\mathcal M\). Its image in \(\mathcal H\) is a quasi-coherent subobject killed by \(h\), so it lies in \(\mathcal K\). The composite \(\mathcal H'\to\mathcal L\) is zero; its factor \(a\) is zero by that epimorphism.

Given \(b:\mathcal M\to\mathcal P\) from a quasi-coherent sheaf, express \(\mathcal M\) as the directed union of its small subsheaves. On each small subsheaf its map \(b\) is represented by a summand of \(\mathcal H\), and hence gives a lift to \(\mathcal L\). Two such lifts agree on an inclusion or intersection: their difference is killed by \(\ell\), and the preceding paragraph applies. They therefore define a lift on the directed union, unique by the same paragraph. Thus
\[
\operatorname{Hom}_{\mathrm{QCoh}}(\mathcal M,\mathcal L)
=\operatorname{Hom}_{\mathrm{Mod}}(\mathcal M,\mathcal P)
\]
naturally. Define \(Q\mathcal P=\mathcal L\); uniqueness supplies its functoriality. All module-sheaf limits exist by the first lesson's sectionwise construction. The displayed adjunction identifies cones into \(Q(\lim_{\operatorname{Mod}}\mathcal F_i)\) with compatible maps into the \(\mathcal F_i\), proving the limit formula. \(\square\)

Limits here are limits in \(\operatorname{QCoh}(X)\). In particular this theorem does not identify their products with products in the category of all module sheaves. The distinction matters in the inverse-limit lesson.

## 3. Affine cohomology and finite bounds

For a finite cover \(\mathcal U=(U_1,\ldots,U_m)\), let \(C^p(\mathcal U,\mathcal F)\) be the product of \(\Gamma(U_{i_0}\cap\cdots\cap U_{i_p},\mathcal F)\) for strictly increasing indices, with the alternating restriction differential. It is zero for \(p\ge m\).

**Lemma 3.1 (a basis criterion for acyclicity).** Suppose a basis \(\mathcal B\) consists of quasi-compact opens and is closed under finite intersections. If, on every \(U\in\mathcal B\), every finite covering by members of \(\mathcal B\) has \(H^p(C^\bullet(\mathcal U,\mathcal F))=0\) for \(p>0\), then \(H^p(U,\mathcal F)=0\) for all such \(U\) and \(p>0\).

**Proof.** First, an injective module sheaf \(\mathcal I\) has acyclic positive Čech groups for every finite cover. The augmented complex of sheaves
\[
\cdots\longrightarrow
\bigoplus_{i<j}j_{ij!}\mathcal O_{U_i\cap U_j}
\longrightarrow\bigoplus_i j_{i!}\mathcal O_{U_i}
\longrightarrow\mathcal O_U\longrightarrow0
\tag{3.1}
\]
is exact: at a point its nonzero summands are the simplex on the cover indices containing that point, contracted by insertion of one such index as in Lemma 1.1. Apply the exact contravariant functor \(\operatorname{Hom}(-,\mathcal I|_U)\). Restriction of an injective to an open is injective because its left adjoint \(j_!\) is exact. Lemma 5.1 of the module-sheaf lesson identifies the resulting Hom complex with the augmented section Čech complex.

Embed \(\mathcal F\) in an injective \(\mathcal I\) and put \(\mathcal Q=\mathcal I/\mathcal F\). For \(s\in\Gamma(U,\mathcal Q)\), choose local lifts to \(\mathcal I\). Refine their domains to \(\mathcal B\), and choose a finite subcover. Differences of lifts give a Čech one-cocycle in \(\mathcal F\). Its vanishing cohomology class means that subtracting an appropriate zero-cochain in \(\mathcal F\) makes the lifts agree. They glue, proving \(\Gamma(U,\mathcal I)\to\Gamma(U,\mathcal Q)\) surjective.

This holds on each intersection in any tested cover, so its Čech complexes form a short exact sequence. Their long exact cohomology sequence, the hypothesis for \(\mathcal F\), and the injective calculation show that \(\mathcal Q\) satisfies the same positive-Čech-vanishing hypothesis. The sheaf-cohomology long exact sequence gives \(H^1(U,\mathcal F)=0\), and \(H^{p+1}(U,\mathcal F)=H^p(U,\mathcal Q)\) for \(p\ge1\). Repeat the construction for \(\mathcal Q\). After \(p\) steps it proves the asserted vanishing in degree \(p\). \(\square\)

**Theorem 3.2 (affine acyclicity and Čech comparison).** If \(U\) is affine and \(\mathcal F\) is quasi-coherent, then \(H^p(U,\mathcal F)=0\) for \(p>0\). If \(X\) has a finite affine cover with affine multiple intersections, then
\[
H^p(X,\mathcal F)=H^p(C^\bullet(\mathcal U,\mathcal F)).
\tag{3.2}
\]
In particular a cover by \(m\) such affines gives vanishing in degrees \(p\ge m\).

**Proof.** On an affine, take the principal-open basis. Theorem 1.2 and Lemma 1.1 prove exactly the finite-cover hypothesis of Lemma 3.1, including covers of each principal open. This proves affine acyclicity.

Resolve \(\mathcal F\to\mathcal I^\bullet\) by a bounded-below injective resolution, starting in degree zero. Form the finite-width double complex \(C^p(\mathcal U,\mathcal I^q)\) with total differential \(\delta+(-1)^p d\). Its horizontally augmented rows are exact by (3.1); its vertical cohomology is \(C^p(\mathcal U,\mathcal F)\) in degree zero and zero in positive degrees, by affine acyclicity and the open restriction of injectives. The two comparisons therefore identify its total cohomology with both sides of (3.2). For completeness, the filtrations used here are finite in each total degree: \(0\le p<m\) and \(q\ge0\). Successively quotienting by a column produces the vertical calculation; successively eliminating the exact augmented rows produces the horizontal calculation. The long exact sequences for these finite filtrations justify both identifications without an infinite-convergence assumption. Finally \(C^p=0\) for \(p\ge m\). \(\square\)

A scheme is *semi-separated* here when intersections of any two affine opens are affine. Multiple intersections are then affine by induction. Lemma 1.4(2) proves that a separated scheme is semi-separated and that every open subscheme of an affine is separated. Its proof includes the affine-product and closed-subscheme identifications used in these assertions.

**Proposition 3.3 (quasi-compact, quasi-separated cohomology).** Let \(W\) be quasi-compact and quasi-separated. There is a finite integer \(d_W\) such that
\[
H^p(W,\mathcal F)=0\quad(p>d_W)
\tag{3.3}
\]
for all quasi-coherent \(\mathcal F\). For \(W\to\operatorname{Spec}A\), formation of these groups commutes with localization in \(A\). Consequently, for a quasi-compact, quasi-separated morphism \(f:X\to Y\), all \(R^p f_*\mathcal F\) are quasi-coherent. On every affine open of \(Y\) there is a uniform finite bound for \(p\).

**Proof.** Choose a finite affine cover. If its intersections are affine, Theorem 3.2 gives \(d_W=m-1\). More generally, induct on the number of affine members. Write \(W=U\cup V\), with \(U\) affine and \(V\) the union of the remaining members. If \(U_2,\ldots,U_m\) are the affine cover members forming \(V\), then \(U\cap V=\bigcup_{i=2}^m(U\cap U_i)\). Each intersection is quasi-compact by Lemma 1.4(3), so their finite union is quasi-compact. It is separated because it is open in the affine \(U\), by Lemma 1.4(2). It therefore has a finite affine cover with affine intersections, and Theorem 3.2 gives its bound. The Mayer–Vietoris sequence gives a bound
\[
d_W=\max(0,d_V,d_{U\cap V}+1).
\tag{3.4}
\]
The empty scheme contributes zero groups and may simply be omitted.

Here the sequence itself follows by applying sections to an injective resolution: a flasque sheaf \(\mathcal I\) has
\[
0\to\Gamma(W,\mathcal I)\to
\Gamma(U,\mathcal I)\oplus\Gamma(V,\mathcal I)
\xrightarrow{s-t}\Gamma(U\cap V,\mathcal I)\to0.
\tag{3.5}
\]
The left part is gluing, and the last map is onto by flasque restriction. Injectives are flasque by the bounded-resolution lesson. The short exact sequence of complexes gives Mayer–Vietoris.

On a semi-separated finite cover, replacing \(W\) by \(W_a=W\times_{\operatorname{Spec}A}D(a)\) localizes each affine term of its Čech complex. Exactness of localization gives
\[
H^p(W,\mathcal F)_a=H^p(W_a,\mathcal F|_{W_a}).
\tag{3.6}
\]
For a general \(W\), apply the same induction using (3.5). Localization preserves its long exact sequence, and the already established isomorphisms for \(U,V,U\cap V\) imply the isomorphism for \(W\) by the Five Lemma (or by the kernels and cokernels of two adjacent maps). All restriction maps used are the natural ones, so (3.6) is canonical.

The sheaf \(R^p f_*\mathcal F\) is the sheafification of \(T\mapsto H^p(f^{-1}T,\mathcal F)\). Indeed an injective resolution gives this presheaf cohomology, and exact sheafification identifies it with cohomology of the pushforward complex. On an affine \(T=\operatorname{Spec}A\), formula (3.6) identifies this sheaf with the associated module \(H^p(f^{-1}T,\mathcal F)\). This proves quasi-coherence and the local uniform bound. \(\square\)

By Lemma 1.4(4), the chosen affine pieces and the chosen finite covers of their intersections pull back under every affine base change \(A\to B\) with the same indices, and all previously affine intersections remain affine. The base-changed scheme is again quasi-compact and quasi-separated. Thus the same finite-cover induction, using the pulled-back covers at each step, gives the same recursive bound (3.4). No flatness is needed for this statement about the bound.

**Proposition 3.4 (continuity on quasi-compact opens).** On a quasi-compact, quasi-separated scheme \(W\), all \(H^p(W,-)\) commute with filtered colimits of arbitrary module sheaves. Hence, for a quasi-compact, quasi-separated morphism, all \(R^p f_*\) commute with filtered colimits and arbitrary direct sums of module sheaves.

**Proof.** Sections on a quasi-compact open commute with filtered colimits when intersections of quasi-compact opens are quasi-compact. To prove surjectivity, represent a section of the colimit locally at some indices, choose a finite covering, and move the representatives to a common index. Agreement in the colimit is local on each overlap; a finite subcover of each quasi-compact overlap and a further common index make the representatives agree there. They glue at that index. For injectivity, equality in the colimit is again local; quasi-compactness gives a finite cover and a common index where the sections agree. This proves the section assertion, including on all finite intersections.

We may embed a diagram \((\mathcal F_i)\) in a diagram \((\mathcal I_i)\) whose terms are injective. Here is an existence argument that does not require arbitrarily chosen embeddings to be compatible. For the small indexing category \(J\), the category of \(J\)-diagrams in module sheaves has componentwise exactness and exact filtered colimits. If \(G\) is a generator for module sheaves, the left adjoints \(L_jG\) to evaluation at \(j\) give a set of generators, since
\[
(L_j M)_k=\bigoplus_{\alpha:j\to k}M.
\]
This left adjoint is exact. The diagram category is therefore Grothendieck by the same earlier theorem, and has enough injectives. Evaluation sends its injectives to injective module sheaves, since its left adjoint is exact. Embed the diagram and take its componentwise quotient \((\mathcal Q_i)\).

Let \(\mathcal I=\operatorname{colim}_i\mathcal I_i\). On every finite cover by quasi-compact opens, its Čech complex is the filtered colimit of the injectives' Čech complexes: sections commute with this colimit, and finite products commute with filtered module colimits. Filtered colimits are exact. The injective calculation (3.1) thus makes \(\mathcal I\) Čech-acyclic. The basis of all quasi-compact opens is closed under intersections by Lemma 1.4(3) and covers \(W\); Lemma 3.1 shows \(\mathcal I\) is acyclic on each of them.

Compare the long exact sequences for
\(0\to\mathcal F_i\to\mathcal I_i\to\mathcal Q_i\to0\)
with the sequence for their filtered colimit. Sections commute, and the middle sheaves are acyclic both before and after the colimit. Thus \(H^1\) commutes by the cokernel of the section map, and higher degrees commute by dimension shifting, applying the same argument to the quotient diagram. This proves the assertion for every \(p\).

For \(f\), work on the affine-open basis of \(Y\), whose inverse images are quasi-compact and quasi-separated. Sheafify the preceding cohomology comparison; filtered colimits commute with sheafification. A direct sum is the filtered colimit of its finite partial sums, and cohomology commutes with finite sums. This gives the last assertion. \(\square\)

## 4. Controlling unbounded complexes

Call a scheme morphism *concentrated* if it is quasi-compact and quasi-separated. This term concerns the scheme morphism; it does not assert a finite cohomological dimension on all module sheaves.

**Lemma 4.1 (stable split towers).** Let \(C_n\) be a tower of complexes of abelian groups with degreewise surjective transitions. If, in degrees \(q-1\) and \(q\), its cohomology transitions are eventually isomorphisms, then
\[
H^q(\lim C_n)=\lim H^q(C_n)
\tag{4.1}
\]
and the latter is the stable value.

**Proof.** Recursively lift coordinates to show that \(1-t:\prod C_n\to\prod C_n\) is degreewise onto; its kernel is \(\lim C_n\). Products of abelian groups are exact, so the long exact cohomology sequence has the segment
\[
\prod H^{q-1}(C_n)\xrightarrow{1-t}\prod H^{q-1}(C_n)
\to H^q(\lim C_n)\to
\prod H^q(C_n)\xrightarrow{1-t}\prod H^q(C_n).
\]
For an eventually isomorphic tower, \(1-t\) is onto: solve the equations recursively on the tail using the inverse transition maps, and then solve the finitely many earlier coordinates backwards. Its kernel is the stable inverse limit. This proves (4.1). \(\square\)

**Theorem 4.2 (unbounded affine comparison and pushforward).** For an affine \(U=\operatorname{Spec}A\),
\[
D(A)\ \simeq\ D_{\mathrm{qc}}(U),\qquad
M^\bullet\longmapsto\widetilde{M^\bullet},\quad
E\longmapsto R\Gamma(U,E).
\tag{4.2}
\]
For a concentrated \(f:X\to Y\), \(Rf_*\) preserves \(D_{\mathrm{qc}}\). On an affine open of \(Y\) with bound \(d\) from Proposition 3.3, it has cohomological amplitude \([0,d]\) on this category, and it commutes with arbitrary direct sums of its objects. Derived pullback along any scheme morphism preserves \(D_{\mathrm{qc}}\).

**Proof.** Start with \(E\in D_{\mathrm{qc}}(X)\), represented by a complex of all module sheaves. By the inverse-limit lesson, choose compatible bounded-below injective resolutions
\[
\tau^{\ge -n}E\hookrightarrow I_n,\qquad
I_{n+1}\twoheadrightarrow I_n
\]
with degreewise split transitions. The complex \(I=\lim I_n\) is K-injective by that lesson's Proposition 2.2. This assertion uses exact products of abelian groups in Hom complexes and does not require exact products of module sheaves.

On an affine open \(U\), the bounded-below cohomology calculation gives
\[
H^q\Gamma(U,I_n)=\Gamma(U,H^q(\tau^{\ge -n}E)).
\tag{4.3}
\]
To justify this calculation explicitly, for a fixed \(q\) replace the bounded-below input by its good upper truncation \(\tau^{\le q}\). The omitted tail has cohomology in degrees at least \(q+1\), and its bounded-below injective resolution starts there; its derived sections have no cohomology below \(q+1\). The truncation comparison is therefore an isomorphism in degree \(q\). The remaining input has cohomology in a finite interval. Its successive good-truncation triangles have factors \(H^r(E)[-r]\). Theorem 3.2 sends each such factor to its sections in degree \(r\) alone. Induction on this finite interval, using the long exact sequences, gives (4.3), with its natural cohomology identification.

For a fixed \(q\), (4.3) and its degree \(q-1\) version stabilize when \(n\) is large. Sections commute with limits and preserve the split transitions. Lemma 4.1 gives
\[
H^q\Gamma(U,I)=\Gamma(U,H^q(E)).
\tag{4.4}
\]
The map \(E\to I\) is a quasi-isomorphism. To check this without assuming that sections of the original \(E\) are exact, sheafify the presheaves \(U\mapsto H^q\Gamma(U,I)\) on the affine basis. This sheafification is \(H^q(I)\), since filtered stalk colimits are exact. Formula (4.4) identifies it with \(H^q(E)\). The identification of the map follows by projecting to \(I_n\), where \(E\to\tau^{\ge -n}E\to I_n\) is the identity on degree-\(q\) cohomology for large \(n\). Thus \(I\) is an actual K-injective model for \(E\).

Now let \(W\) be quasi-compact and quasi-separated, with bound \(d\). For an input with cohomology in a finite interval \([a,b]\), each factor \(H^r(E)[-r]\) has derived sections in \([r,r+d]\), by Proposition 3.3. Finite induction on its truncation triangles therefore puts the result in \([a,b+d]\). For a bounded-below input without an upper bound, the same good upper truncation as in the preceding paragraph computes any fixed degree; bounded-below injective resolutions also prove the lower bound directly. For \(m>n\), put \(T_r=\tau^{\ge-r}E\), and let \(F_{m,n}\) be the homotopy fibre of \(T_m\to T_n\). The truncation map is the identity on cohomology in degrees at least \(-n\), and its lower surviving cohomology is in \([-m,-n-1]\). The long exact cohomology sequence of
\[
F_{m,n}\longrightarrow T_m\longrightarrow T_n
\longrightarrow F_{m,n}[1]
\]
therefore puts \(F_{m,n}\) in \(D^{[-m,-n-1]}\). With the cohomological shift convention \(H^r(F[1])=H^{r+1}(F)\), its cone is
\[
C_{m,n}=\operatorname{Cone}(T_m\to T_n)\simeq F_{m,n}[1]
\ \in D^{[-m-1,-n-2]}.
\]
The finite-interval bound already proved gives \(H^rR\Gamma(W,C_{m,n})=0\) for \(r>-n-2+d\). Applying \(R\Gamma(W,-)\) to the cone triangle, its long exact sequence shows that \(H^rR\Gamma(W,T_m)\to H^rR\Gamma(W,T_n)\) is an isomorphism when both adjacent cone groups in degrees \(r-1\) and \(r\) vanish. For a fixed \(q\), choose \(n\ge0\) with \(n>d-q\). Those vanishings hold for both \(r=q-1\) and \(r=q\), uniformly for every \(m>n\). The compatible injective resolutions induce these same cohomology maps. Hence the towers \(H^q\Gamma(W,I_n)\) and \(H^{q-1}\Gamma(W,I_n)\) are eventually constant. Lemma 4.1 computes their limit and proves the same amplitude statement for arbitrary unbounded \(E\). More precisely,
\[
H^qR\Gamma(W,E)
\quad\hbox{depends only on}\quad
\tau^{\ge q-d}\tau^{\le q}E.
\tag{4.5}
\]
The two truncation comparisons are isomorphisms in degree \(q\): the lower omitted tail maps to degrees at most \(q-1\), and the upper omitted tail maps to degrees at least \(q+1\). The long exact sequences prove the assertion.

When \(W\) maps to an affine \(T=\operatorname{Spec}A\), the bounded-input comparisons commute with localization in \(A\), by Proposition 3.3 and a finite induction on the good-truncation triangles. Formula (4.5) makes that induction finite for every unbounded input and every \(q\). Sheafifying on \(T\) proves that \(H^q(Rf_*E)\) is associated to \(H^qR\Gamma(f^{-1}T,E)\), and is quasi-coherent. Restriction of an injective to an open is injective, so restricting \(Rf_*\) to \(T\) agrees with this computation. This proves preservation and the local amplitude claim.

For affine \(U\), association is an exact left adjoint to sections by (1.4), also on complexes. Thus it is left adjoint on derived categories to \(R\Gamma\): the exact left adjoint preserves acyclic complexes, and the right adjoint sends K-injectives to K-injectives by the Hom test. Formula (4.4) says that the derived unit on every module complex is a quasi-isomorphism. The counit on \(E\in D_{\mathrm{qc}}(U)\) is also a quasi-isomorphism, since its cohomology is the affine counit \(\widetilde{\Gamma(U,H^qE)}\to H^qE\) of Theorem 1.2. This proves (4.2).

For direct sums, first consider complexes with cohomology in a fixed finite interval. Proposition 3.4 proves the comparison on objects in degree zero, and finite induction on their good-truncation triangles proves it on that interval. For a general family, compute degree \(q\) after the single window in (4.5). Sums are exact and commute with good truncations and cohomology in module sheaves. All windows are now in the same finite interval, so the already proved comparison applies. This works locally on every affine of \(Y\), proving preservation of arbitrary sums.

Finally, on an affine chart an object of \(D_{\mathrm{qc}}\) is associated to a module complex by (4.2). Resolve that module complex by the semifree resolution of the K-flat lesson, Theorem 3.4. Its associated complex is K-flat: on a stalk it is a localization of a semifree complex, and tensor with any acyclic stalk complex is acyclic. For a morphism of affine charts \(A\to B\), its pullback is the associated complex of \(B\otimes_A P\), by Proposition 1.3. Its cohomology is quasi-coherent. These charts cover the source, so derived pullback preserves \(D_{\mathrm{qc}}\). \(\square\)

No global equivalence between \(D(\operatorname{QCoh}(X))\) and \(D_{\mathrm{qc}}(X)\) has been used. The tower in this proof is a tower of all module sheaves; its augmentation was checked on affine opens.

**Proposition 4.3 (finite dimension on all module sheaves).** Suppose \(f\) is concentrated and there is \(d\) with \(R^p f_*\mathcal M=0\) for every module sheaf \(\mathcal M\) and \(p>d\). Then \(Rf_*\), on the entire unbounded module-sheaf derived category, has amplitude \([0,d]\) and commutes with arbitrary direct sums.

**Proof.** There is a finite exact coresolution by exact functors whose terms are \(f_*\)-acyclic. Here is its construction. Put
\[
C^0(\mathcal M)=\prod_{x\in X}\operatorname{Sky}_x(\mathcal M_x).
\]
On an open \(U\) its sections are \(\prod_{x\in U}\mathcal M_x\), with scalar action through germs. The germ map \(\mathcal M\hookrightarrow C^0(\mathcal M)\) is monic; the section products show that \(C^0\) is exact and flasque. Its quotient functor \(Q\) is exact by the kernel-cokernel sequence applied to a short exact sequence and these monic maps. Iterate \(C^r=C^0Q^r\).

Flasque sheaves are \(f_*\)-acyclic: on every inverse-image open they are flasque and have no higher sections. Dimension shifting in \(0\to Q^r\mathcal M\to C^r\mathcal M\to Q^{r+1}\mathcal M\to0\) gives
\[
R^p f_*Q^d\mathcal M=R^{p+d}f_*\mathcal M=0\qquad(p>0).
\]
We therefore have an exact coresolution of width \(d\),
\[
0\to\mathcal M\to C^0\mathcal M\to\cdots
\to C^{d-1}\mathcal M\to Q^d\mathcal M\to0.
\tag{4.6}
\]
For \(d=0\), use \(\mathcal M\) itself, which is already acyclic.

Apply (4.6) to each term of a complex and take the finite-width total complex, with differential \(\delta+(-1)^p d\). The augmentation is a quasi-isomorphism: its row cohomology is zero, and the finite width permits the same finite filtration argument as in Theorem 3.2, even when the vertical complex is unbounded. Its terms are finite sums of \(f_*\)-acyclic objects. Proposition 5.5 of the Grothendieck-category lesson proves that any termwise \(f_*\)-acyclic complex computes \(Rf_*\) under this uniform finite-dimension hypothesis.

For \(E\in D^{\le b}\), replace \(E\) by its good upper truncation, a complex zero above \(b\). The total model (4.6) is zero above \(b+d\), proving the upper amplitude bound. For \(E\in D^{\ge a}\), the ordinary bounded-below resolution computes \(Rf_*\) in degrees at least \(a\). This proves amplitude \([0,d]\), and gives the finite-window comparison (4.5) for all inputs. Proposition 3.4 and finite induction on cohomology triangles prove preservation of sums on each finite window. Exactness of sums and the window comparison then prove it on all inputs, as in Theorem 4.2. \(\square\)

## 5. The unbounded projection formula

**Theorem 5.1 (concentrated projection formula).** Let \(f:X\to Y\) be concentrated and \(K\in D_{\mathrm{qc}}(Y)\), with no boundedness or perfection assumption on \(K\). The canonical projection morphism
\[
(Rf_*E)\otimes^L_{\mathcal O_Y}K
\longrightarrow
Rf_*(E\otimes^L_{\mathcal O_X}Lf^*K)
\tag{5.1}
\]
is an isomorphism in either of these cases:

- \(E\in D_{\mathrm{qc}}(X)\);
- \(E\) is any complex of module sheaves, and \(f_*\) has a uniform finite cohomological dimension on all module sheaves.

**Proof.** Use the canonical morphism constructed from the tensor adjunction and counit in [Projection formula and base change](projection-formula-and-base-change.md), Section 2. The construction commutes with restriction to opens and is natural in \(K\). We can therefore work with \(Y=\operatorname{Spec}A\). By Theorem 4.2, \(K\) is associated to an \(A\)-complex. The semifree theorem in the K-flat lesson supplies a quasi-isomorphic semifree \(P\), filtered by stages
\[
0=P_{-1}\subset P_0\subset P_1\subset\cdots,\qquad
P_s/P_{s-1}=\bigoplus_\alpha A[-r_\alpha],
\quad P=\bigcup_sP_s,
\tag{5.2}
\]
with the inclusions split as graded modules.

For fixed \(E\), both sides of (5.1) are triangulated functors of \(K\). They preserve arbitrary sums. For the left side this is the derived tensor construction. For the right side, tensor and pullback preserve sums; in the first case their outputs lie in \(D_{\mathrm{qc}}(X)\) by Theorem 4.2 and the affine K-flat tensor model, and \(Rf_*\) preserves their sums by that theorem. In the second case Proposition 4.3 gives the required preservation on all module complexes.

For \(K=\mathcal O_Y\), (5.1) is the identity on \(Rf_*E\), after the unit identifications: its transpose is the pushforward counit, which defines that identity under adjunction. It is therefore an isomorphism for shifts and sums of the unit. The short exact sequences in (5.2) give triangles, so induction and their long exact cohomology sequences make it an isomorphism for every \(P_s\).

Finally,
\[
P\simeq\operatorname{hocolim}_s P_s
=\operatorname{Cone}\!\left(
\bigoplus_sP_s\xrightarrow{1-\mathrm{shift}}\bigoplus_sP_s
\right).
\tag{5.3}
\]
To check this assertion, the map \(1-\mathrm{shift}\) is degreewise injective: the first coordinate of an element in its kernel is zero, then the second, and so on. Its cokernel is the direct limit, with the relations identifying an element with its image at the next stage. Thus its cone maps quasi-isomorphically to \(\operatorname{colim}P_s=P\); exactness of filtered module colimits justifies the cohomology comparison. The two functors preserve the sums and the triangle in (5.3). Naturality of the canonical morphism then makes it an isomorphism at \(P\), hence at \(K\). This proves both cases. \(\square\)

## 6. Flat base change

**Lemma 6.1 (ordinary quasi-coherent flat base change).** Let \(W\) be quasi-compact and quasi-separated over \(A\), let \(A\to B\) be flat, and let \(\mathcal F\) be quasi-coherent. If \(W_B=W\times_{\operatorname{Spec}A}\operatorname{Spec}B\), then the natural maps
\[
B\otimes_A H^p(W,\mathcal F)
\longrightarrow H^p(W_B,\mathcal F_B)
\tag{6.1}
\]
are isomorphisms for all \(p\).

**Proof.** Lemma 1.4(4) makes \(W_B\) quasi-compact and quasi-separated and identifies the pulled-back affine covers and intersections. If \(W=\operatorname{Spec}C\) and \(\mathcal F=\widetilde M\), the degree-zero map is the balanced isomorphism
\[
(C\otimes_A B)\otimes_C M=B\otimes_A M,
\]
and higher groups vanish by Theorem 3.2. With a finite affine cover and affine intersections, (6.1) is computed by the map of Čech complexes sending \(b\otimes s\) to \(b\,v^*s\) on each intersection. The affine calculation identifies the complexes, and flatness identifies their cohomology.

For a general \(W\), induct on a finite affine cover as in Proposition 3.3. For \(W=U\cup V\), the assertion is already known for the affine \(U\), the smaller union \(V\), and the quasi-compact separated \(U\cap V\). Tensor its Mayer–Vietoris long exact sequence with the flat \(B\) and compare it with the base-changed sequence. The natural cochain maps commute with restriction and the boundary maps, since they arise from the same short exact sequence (3.5). Two adjacent kernels and cokernels, or the Five Lemma, give the comparison for \(W\). This proves (6.1) in all degrees. \(\square\)

**Theorem 6.2 (unbounded concentrated flat base change).** In the cartesian square of schemes
\[
\begin{array}{ccc}
X'&\xrightarrow{v}&X\\
\downarrow g&&\downarrow f\\
Y'&\xrightarrow{u}&Y,
\end{array}
\tag{6.2}
\]
assume \(f\) concentrated and \(u\) flat. For every \(E\in D_{\mathrm{qc}}(X)\), the canonical base-change morphism
\[
Lu^*Rf_*E\longrightarrow Rg_*Lv^*E
\tag{6.3}
\]
is an isomorphism.

**Proof.** Lemma 1.4(4) shows that \(g\) is concentrated, because it is a base change of the concentrated morphism \(f\). Work locally on affine opens \(T'=\operatorname{Spec}B\) of \(Y'\) mapping into \(T=\operatorname{Spec}A\) of \(Y\); these opens cover \(Y'\) by the principal-open basis. Flatness gives flat \(A\to B\) by Proposition 1.3 and the affine equivalence. The affine-product formula of Lemma 1.4(1) and Proposition 1.3 show that the base change \(v\) is flat. Thus \(Lu^*=u^*\) and \(Lv^*=v^*\); both are exact and commute with good truncations.

First put \(E=\mathcal F[0]\). Proposition 3.3 identifies the higher direct images on \(T\) and \(T'\) with the modules in (6.1), so Lemma 6.1 proves the isomorphism on every cohomology sheaf. It also identifies the map with the canonical one. Indeed the base-change morphism is the transpose, under \(Lg^*\dashv Rg_*\), of
\[
Lg^*Lu^*Rf_*\mathcal F
=Lv^*Lf^*Rf_*\mathcal F
\xrightarrow{Lv^*(\text{counit})}Lv^*\mathcal F.
\tag{6.4}
\]
On affine pieces the counit is evaluation \(c\otimes s\mapsto cs\); transposing (6.4) sends \(b\otimes s\) to \(b\,v^*s\). These are precisely the Čech cochain maps of Lemma 6.1. To pass from pieces to the whole scheme, use the augmented cover complex or the Mayer–Vietoris restriction triangle (3.5). Its maps are restriction, restriction difference, and the boundary of that short exact sequence. The mate construction commutes with restriction and with composition of pushforwards: substitute (6.4), use the naturality of the counit for restriction, and cancel the inserted unit-counit pair by its triangle identity. This is also the pasting identity proved in Section 4 of the earlier base-change lesson. Consequently it gives a map of these cover complexes and restriction triangles. The affine formulas determine its Čech components; the same map of triangles determines its Mayer–Vietoris boundary components. Thus the cohomological isomorphism proved in Lemma 6.1 is induced by (6.3), not merely an abstract isomorphism of the two objects.

The cohomological bounds for \(f^{-1}T\) and its affine base change can be chosen to be the same finite \(d\), by the observation after Proposition 3.3. Theorem 4.2 gives amplitude \([0,d]\) for both derived pushforwards on \(D_{\mathrm{qc}}\). Finite induction on good-truncation triangles extends the canonical comparison from module sheaves to complexes with cohomology in a finite interval. For an arbitrary \(E\), its degree-\(q\) comparison depends only on the window \(\tau^{\ge q-d}\tau^{\le q}E\): the two exact pullbacks commute with that window, and both pushforwards have the stated amplitude. The finite-interval result therefore proves that (6.3) is an isomorphism on \(H^q\), for every \(q\). The stalk criterion proves it is an isomorphism in the derived category. \(\square\)

The cartesian, scheme, flatness and quasi-coherence hypotheses are part of the theorem. The arbitrary ringed-space comparison in the earlier lesson continues to exist without them, but its boundary-section and nonflat Tor examples show why existence alone does not give invertibility.

## 7. Exercises with solutions

**Exercise 1.** For \(A=k[\epsilon]/(\epsilon^2)\) and \(M=k=A/(\epsilon)\), determine \(R\Gamma(\operatorname{Spec}A,\widetilde M)\) and \(M\otimes_A^L M\). Explain why affine cohomological dimension zero does not make tensor exact.

**Solution.** Theorem 4.2 gives \(R\Gamma(\operatorname{Spec}A,\widetilde M)=M[0]\). The free resolution
\[
\cdots\xrightarrow{\epsilon}A\xrightarrow{\epsilon}A
\xrightarrow{\epsilon}A\to M\to0
\]
is exact: both the kernel and the image of multiplication by \(\epsilon\) are \((\epsilon)\). Place its last \(A\) in degree zero. Tensoring with \(M\) makes every differential zero, giving \(H^{-r}(M\otimes_A^L M)=k\) for every \(r\ge0\). Affine acyclicity concerns the sections functor on quasi-coherent sheaves; flatness concerns the tensor functor. The two calculations explicitly distinguish them.

**Exercise 2.** Cover \(\mathbf P^1_k\) by \(U_0=\operatorname{Spec}k[t]\) and \(U_1=\operatorname{Spec}k[t^{-1}]\). Use a transition trivialization for \(\mathcal O(-2)\) in which the degree-zero Čech image in \(k[t,t^{-1}]\) is \(k[t]+t^{-2}k[t^{-1}]\). Compute its cohomology and the affine-cover bound.

**Solution.** The section Čech complex has these two chart modules in degree zero and \(k[t,t^{-1}]\) in degree one, with differential \(a-t^{-2}b\). The intersection of \(k[t]\) and \(t^{-2}k[t^{-1}]\) is zero, so \(H^0=0\). Their sum contains all Laurent monomials except \(t^{-1}\), so \(H^1=k\cdot[t^{-1}]\). Higher cohomology is zero by Theorem 3.2, since the two opens and their intersection are affine. The bound \(d=1\) is attained.

**Exercise 3.** In the affine square \(A=\mathbb Z\), \(C=B=\mathbb Z/2\), \(X=\operatorname{Spec}C\), \(Y'=\operatorname{Spec}B\), take \(E=\mathcal O_X\). Compute the comparison after this nonflat base change.

**Solution.** The ordinary fibre product is \(\operatorname{Spec}(C\otimes_A B)=\operatorname{Spec}(\mathbb Z/2)\), so \(v\) is the identity. The source of base change corresponds to \(B\otimes_A^L C\), represented by \(B\xrightarrow0B\) in degrees \(-1,0\), using the free resolution \(\mathbb Z\xrightarrow2\mathbb Z\to C\). The target is \(C[0]\). The canonical map is multiplication in degree zero and zero in degree \(-1\); it has nonzero kernel on \(H^{-1}=B\), and is not an isomorphism. Theorem 6.2 does not apply because \(A\to B\) is not flat.

## Further reading

The [AI Integrated Stacks Project](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/tree/565b10e987aba5969b21145a0833f42d69f96790) chapters *Schemes*, *Properties of Schemes*, *Cohomology of Sheaves* and *Cohomology of Schemes* supply the affine, Gabber and Čech methods developed here. The upstream human authors are the Stacks project authors. The small-submodule theorem is attributed to Ofer Gabber; the classical affine and finite-cover cohomology method also appears in Jean-Pierre Serre's [*Faisceaux algébriques cohérents*, freely available English translation](https://www.jmilne.org/math/Documents/fac.pdf).

Joseph Lipman's freely readable [Notes on Derived Functors and Grothendieck Duality](https://www.math.purdue.edu/~jlipman/Duality.pdf) develops the concentrated projection and flat base-change theorems in a broader duality framework. The proofs needed for the statements of this lesson are given above.