Chapter 0_III

§12. Complements on the cohomology of sheaves

12.1. Cohomology of sheaves of modules on ringed spaces

12.1.1.

Let $(X,{\mathcal{O}}_X)$ be a ringed space. Recall that for every ${\mathcal{O}}_X$-module $\mathcal{F}$ one defines the cohomology $H^{\bullet }(X,\mathcal{F})$, which is a universal cohomological functor (T, 2.2) from the category $C(X)$ of ${\mathcal{O}}_X$-modules to the category of abelian groups; it is the right-derived functor of the left-exact functor $\mathcal{F}\mapsto \Gamma (X,\mathcal{F})$. The functor $H^{\bullet }(X,\mathcal{F})$ is isomorphic

to the restriction to the category $C(X)$ of the cohomological functor defined in the same way on the category of sheaves of abelian groups on $X$ (G, II, 7.2.1).

12.1.2.

Set $A=\Gamma (X,{\mathcal{O}}_X)$. Since every element of $A$ defines an endomorphism of the abelian group $\Gamma (X,\mathcal{F})$, it defines by functoriality an endomorphism of the $\partial$-functor $H^{\bullet }(X,\mathcal{F})$; these endomorphisms equip each $H^p(X,\mathcal{F})$ with a structure of $A$-module, and the operator $\partial$ is $A$-linear. Moreover, for any two non-negative integers $p$, $q$ and any two ${\mathcal{O}}_X$-modules $\mathcal{F}$, $\mathcal{G}$, one has a homomorphism of $A$-modules, called the cup product

  H^p(X, ℱ) ⊗_A H^q(X, 𝒢) → H^{p+q}(X, ℱ ⊗_{𝒪_X} 𝒢)                          (12.1.2.1)

(G, II, 6.6). These homomorphisms make the direct sum $S$ of the $H^p(X,{\mathcal{O}}_X)$ (for $p\ge 0$) into a graded anticommutative $A$-algebra, and the direct sum of the $H^p(X,\mathcal{F})$ into a graded $S$-module.

For every open cover $\mathfrak{U}$ of $X$, we shall always denote by $C^{\bullet }(\mathfrak{U},\mathcal{F})$ (contrary to (G, II, 5.1)) the complex of alternating cochains of the nerve of $\mathfrak{U}$ with values in the system of coefficients $\Gamma (U_{\sigma },\mathcal{F})$. It is clear that $C^{\bullet }(\mathfrak{U},\mathcal{F})$ is a graded $A$-module, so the cohomology groups $H^{\bullet }(\mathfrak{U},\mathcal{F})$ of this complex are endowed with a structure of $A$-module; moreover, the canonical maps $H^{\bullet }(\mathfrak{U},\mathcal{F})\to H^{\bullet }(X,\mathcal{F})$ (G, II, 5.4) are $A$-linear.

12.1.3.

Let $(X^{\prime },{\mathcal{O}}_{X^{\prime }})$ be a second ringed space, and let $f=(\psi ,\theta )$ be a morphism from $X^{\prime }$ to $X$.

Set $A^{\prime }=\Gamma (X^{\prime },{\mathcal{O}}_{X^{\prime }})$; the $\psi$-morphism $\theta$ defines canonically a ring homomorphism $A\to A^{\prime }$. Let $\mathcal{F}$ be an ${\mathcal{O}}_X$-module and ${\mathcal{F}}^{\prime }$ an ${\mathcal{O}}_{X^{\prime }}$-module; for every $f$-morphism $u:\mathcal{F}\to {\mathcal{F}}^{\prime }$ $(0_I,\mathrm{4.4.1})$ we shall see that one can define, for every $p\ge 0$, a di-homomorphism

  u_p : H^p(X, ℱ) → H^p(X', ℱ').                                             (12.1.3.1)

Indeed, since ${\psi }^{\ast }$ is exact in the category of sheaves of abelian groups on $X$, $\mathcal{F}\mapsto H^{\bullet }(X^{\prime },{\psi }^{\ast }(\mathcal{F}))$ is a $\partial$-functor in this category, and one knows that one has a canonical homomorphism of $\partial$-functors

  H^•(X, ℱ) → H^•(X', ψ^*(ℱ))                                                (12.1.3.2)

uniquely determined by the condition of reducing to the canonical homomorphism $\Gamma (X,\mathcal{F})\to \Gamma (X^{\prime },{\psi }^{\ast }(\mathcal{F}))$ in degree 0 (T, 3.2.2). Moreover, every element of $A$ determines an endomorphism $\mu$ of $\Gamma (X,\mathcal{F})$ and an endomorphism ${\mu }^{\prime }$ of $\Gamma (X^{\prime },{\psi }^{\ast }(\mathcal{F}))$ such that the diagram

                 Γ(X, ℱ)   →   Γ(X', ψ^*(ℱ))
                    ↓μ              ↓μ'                                       (12.1.3.3)
                 Γ(X, ℱ)   →   Γ(X', ψ^*(ℱ))

is commutative; by the uniqueness property of extension of morphisms for universal cohomological functors (T, 2.2), one deduces unique extensions of $\mu$ and ${\mu }^{\prime }$ to the cohomology making the diagrams analogous to (12.1.3.3) commutative, which means that (12.1.3.2) is a homomorphism of $A$-modules. Now note that one has $f^{\ast }(\mathcal{F})={\psi }^{\ast }(\mathcal{F}){\otimes }_{{\psi }^{\ast }({\mathcal{O}}_X)}{\mathcal{O}}_{X^{\prime }}$, so one has a canonical di-homomorphism ${\psi }^{\ast }(\mathcal{F})\to f^{\ast }(\mathcal{F})$ of the ${\psi }^{\ast }({\mathcal{O}}_X)$-module ${\psi }^{\ast }(\mathcal{F})$ into the ${\mathcal{O}}_{X^{\prime }}$-module $f^{\ast }(\mathcal{F})$. By functoriality, one therefore deduces a functorial di-homomorphism

  H^p(X', ψ^*(ℱ)) → H^p(X', f^*(ℱ))                                          (12.1.3.4)

with corresponding rings $A$ and $A^{\prime }$; composing this di-homomorphism with (12.1.3.2), one obtains a canonical di-homomorphism functorial in $\mathcal{F}$

  θ_p : H^p(X, ℱ) → H^p(X', f^*(ℱ)).                                         (12.1.3.5)

Finally, by functoriality, one deduces from the homomorphism $u^{♭}:f^{\ast }(\mathcal{F})\to {\mathcal{F}}^{\prime }$ a homomorphism of $A^{\prime }$-modules $H^p(X^{\prime },f^{\ast }(\mathcal{F}))\to H^p(X^{\prime },{\mathcal{F}}^{\prime })$, which, composed with (12.1.3.5), gives (12.1.3.1).

Let $f^{\prime }=({\psi }^{\prime },{\theta }^{\prime }):X^{\prime \prime }\to X^{\prime }$ be a second morphism of ringed spaces, $f^{\prime \prime }=f\circ f^{\prime }$ the composite morphism. Taking into account the commutativity of the functor ${\psi }^{\ast }$ with tensor product $(0_I,\mathrm{4.3.3})$, one verifies immediately that the composite of the di-homomorphism $H^p(X^{\prime },f^{\ast }(\mathcal{F}))\to H^p(X^{\prime \prime },f^{\prime \ast }(f^{\ast }(\mathcal{F})))$ with (12.1.3.5) is the corresponding di-homomorphism $H^p(X,\mathcal{F})\to H^p(X^{\prime \prime },f^{\prime \prime \ast }(\mathcal{F}))$.

12.1.4.

A direct definition of the homomorphism (12.1.3.2) can be obtained as follows: one considers an injective resolution ${\mathcal{L}}^{\bullet }=({\mathcal{L}}^i)$ of $\mathcal{F}$ formed of sheaves of abelian groups on $X$; since the functor ${\psi }^{\ast }$ is exact, ${\psi }^{\ast }({\mathcal{L}}^{\bullet })$ is a resolution of ${\psi }^{\ast }(\mathcal{F})$ formed of sheaves on $X^{\prime }$. If ${\mathcal{L}}^{\prime \bullet }=({\mathcal{L}}^{\prime i})$ is an injective resolution of ${\psi }^{\ast }(\mathcal{F})$ in the category of sheaves of abelian groups on $X^{\prime }$, there is therefore a morphism ${\psi }^{\ast }({\mathcal{L}}^{\bullet })\to {\mathcal{L}}^{\prime \bullet }$ of complexes of sheaves of abelian groups, compatible with the augmentations (M, V, 1.1 a)), well-determined up to homotopy. One thus deduces homomorphisms

  Γ(X, ℒ^•) → Γ(X', ψ^*(ℒ^•)) → Γ(X', ℒ'^•)

of complexes of abelian groups, whose composite, by passage to cohomology, gives a morphism of $\partial$-functors $H^{\bullet }(X,\mathcal{F})\to H^{\bullet }(X^{\prime },{\psi }^{\ast }(\mathcal{F}))$; since it coincides with (12.1.3.2) in degree 0, it is identical to it (T, 2.2).

Now consider an open cover $\mathfrak{U}=(U_{\alpha })$ of $X$, and let ${\mathfrak{U}}^{\prime }=({\psi }^{-1}(U_{\alpha }))$ be the open cover of $X^{\prime }$ obtained as the inverse image of $\mathfrak{U}$. The canonical homomorphisms $\Gamma (V,\mathcal{F})\to \Gamma ({\psi }^{-1}(V),f^{\ast }(\mathcal{F}))$ for every open $V$ of $X$ define at once (cf. G, II, 5.1) a homomorphism of complexes $C^{\bullet }(\mathfrak{U},\mathcal{F})\to C^{\bullet }({\mathfrak{U}}^{\prime },f^{\ast }(\mathcal{F}))$, whence the canonical homomorphisms

  θ_p : H^p(𝔘, ℱ) → H^p(𝔘', f^*(ℱ)).                                         (12.1.4.1)

Moreover, one has commutative diagrams

              H^p(𝔘, ℱ)    →^{θ_p}    H^p(𝔘', f^*(ℱ))
                  ↓                       ↓                                   (12.1.4.2)
              H^p(X, ℱ)    →^{θ_p}    H^p(X', f^*(ℱ))

where the vertical arrows are the canonical homomorphisms of (G, II, 5.2). To establish the commutativity of (12.1.4.2), consider the complex of sheaves of (alternating) cochains of $\mathcal{F}$ relative to $\mathfrak{U}$, ${\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F})$, such that $\Gamma (X,{\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F}))=C^{\bullet }(\mathfrak{U},\mathcal{F})$ (G, II, 5.2). The canonical homomorphisms $\Gamma (V,\mathcal{F})\to \Gamma ({\psi }^{-1}(V),{\psi }^{\ast }(\mathcal{F}))$ then define a $\psi$-morphism ${\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F})\to {\mathcal{C}}^{\bullet }({\mathfrak{U}}^{\prime },{\psi }^{\ast }(\mathcal{F}))$, and one has, with the notation above, a commutative diagram

   Γ(X, 𝒞^•(𝔘, ℱ))   →   Γ(X', 𝒞^•(𝔘', ψ^*(ℱ)))
         ↓                          ↓
   Γ(X, ℒ^•)         →   Γ(X', ψ^*(ℒ^•))   →   Γ(X', ℒ'^•)

which, on passing to cohomology, gives commutative diagrams

              H^p(𝔘, ℱ)    →    H^p(𝔘', ψ^*(ℱ))
                  ↓                  ↓
              H^p(X, ℱ)    →    H^p(X', ψ^*(ℱ))

where the vertical arrows are the canonical homomorphisms of (G, II, 5.2). It then suffices to combine these diagrams with the commutative diagrams

              H^p(𝔘', ψ^*(ℱ))   →   H^p(𝔘', f^*(ℱ))
                  ↓                       ↓
              H^p(X', ψ^*(ℱ))   →   H^p(X', f^*(ℱ))

which come from the homomorphism ${\psi }^{\ast }(\mathcal{F})\to f^{\ast }(\mathcal{F})$ and from the functorial character of the canonical homomorphisms of (G, II, 5.2), to obtain the commutativity of (12.1.4.2).

Note that if $A=\Gamma (X,{\mathcal{O}}_X)$, $A^{\prime }=\Gamma (X^{\prime },{\mathcal{O}}_{X^{\prime }})$, the homomorphism (12.1.4.1) is a di-homomorphism of modules corresponding to the rings $A$ and $A^{\prime }$. One has a transitivity property for (12.1.4.1) with respect to the composite of two morphisms, analogous to the transitivity of (12.1.3.5). Finally, note that in the preceding definitions, instead of an injective resolution ${\mathcal{L}}^{\bullet }$ of $\mathcal{F}$, one could equally well have started from a resolution such that $H^p(X,{\mathcal{L}}^i)=0$ for all $i$ and all $p>0$ (G, II, 4.7.1).

12.1.5.

Let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be three ${\mathcal{O}}_X$-modules, and consider an ${\mathcal{O}}_X$-homomorphism $u:\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}\to \mathcal{H}$, which gives, for cohomology, homomorphisms

  H^p(X, ℱ) ⊗_A H^q(X, 𝒢) → H^{p+q}(X, 𝓗)                                    (12.1.5.1)

deduced from the cup product (12.1.2.1). We show that, with the hypotheses and notation of (12.1.3), one has commutative diagrams

   H^p(X, ℱ) ⊗_A H^q(X, 𝒢)                  →   H^{p+q}(X, 𝓗)
            ↓                                          ↓                      (12.1.5.2)
   H^p(X', f^*(ℱ)) ⊗_{A'} H^q(X', f^*(𝒢))   →   H^{p+q}(X', f^*(𝓗))

where the vertical arrows come from the canonical homomorphisms (12.1.3.5). For this, recall that (12.1.5.1) can be obtained by starting from the canonical resolutions (G, II, 4.3) ${\mathcal{L}}^{\bullet }$, ${\mathcal{M}}^{\bullet }$, ${\mathcal{N}}^{\bullet }$ of $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ respectively (which are formed of ${\mathcal{O}}_X$-modules), from the linear map ${\mathcal{L}}^{\bullet }{\otimes }_{{\mathcal{O}}_X}{\mathcal{M}}^{\bullet }\to {\mathcal{N}}^{\bullet }$ of complexes of ${\mathcal{O}}_X$-modules corresponding to $u$, which yields a homomorphism of complexes of $A$-modules $\Gamma (X,{\mathcal{L}}^{\bullet }){\otimes }_A\Gamma (X,{\mathcal{M}}^{\bullet })\to \Gamma (X,{\mathcal{N}}^{\bullet })$, and by passing to cohomology, the homomorphisms $H^p(\Gamma (X,{\mathcal{L}}^{\bullet })){\otimes }_A{H}^q(\Gamma (X,{\mathcal{M}}^{\bullet }))\to H^{p+q}(\Gamma (X,{\mathcal{N}}^{\bullet }))$ (G, II, 6.6). Now, one clearly has a commutative diagram

   Γ(X, ℒ^•) ⊗_A Γ(X, ℳ^•)                                              →   Γ(X, 𝒩^•)
            ↓                                                                  ↓        (12.1.5.3)
   Γ(X', ψ^*(ℒ^•)) ⊗_{Γ(X', ψ^*(𝒪_X))} Γ(X', ψ^*(ℳ^•))                  →   Γ(X', ψ^*(𝒩^•))

which gives, on passing to cohomology, the commutative diagrams

   H^p(X, ℱ) ⊗_A H^q(X, 𝒢)                                                                                →   H^{p+q}(X, 𝓗)
            ↓                                                                                                       ↓                        (12.1.5.4)
   H^p(Γ(X', ψ^*(ℒ^•))) ⊗_{Γ(X', ψ^*(𝒪_X))} H^q(Γ(X', ψ^*(ℳ^•)))                                       →   H^{p+q}(Γ(X', ψ^*(𝒩^•))).

But since ${\psi }^{\ast }({\mathcal{L}}^{\bullet })$, ${\psi }^{\ast }({\mathcal{M}}^{\bullet })$ and ${\psi }^{\ast }({\mathcal{N}}^{\bullet })$ are resolutions of ${\psi }^{\ast }(\mathcal{F})$, ${\psi }^{\ast }(\mathcal{G})$, ${\psi }^{\ast }(\mathcal{H})$ respectively, one has a commutative diagram (G, II, 6.6.1)

   H^p(Γ(X', ψ^*(ℒ^•))) ⊗_{Γ(X', ψ^*(𝒪_X))} H^q(Γ(X', ψ^*(ℳ^•)))   →   H^{p+q}(Γ(X', ψ^*(𝒩^•)))
            ↓                                                                  ↓                                 (12.1.5.5)
   H^p(X', ψ^*(ℱ)) ⊗_{Γ(X', ψ^*(𝒪_X))} H^q(X', ψ^*(𝒢))            →   H^{p+q}(X', ψ^*(𝓗)).

Finally, by functoriality, one has a commutative diagram

   H^p(X', ψ^*(ℱ)) ⊗_{Γ(X', ψ^*(𝒪_X))} H^q(X', ψ^*(𝒢))   →   H^{p+q}(X', ψ^*(𝓗))
            ↓                                                       ↓                       (12.1.5.6)
   H^p(X', f^*(ℱ)) ⊗_{A'} H^q(X', f^*(𝒢))                →   H^{p+q}(X', f^*(𝓗))

and by combining the three diagrams (12.1.5.4), (12.1.5.5) and (12.1.5.6), one obtains the desired commutative diagram (12.1.5.2).

Remark (12.1.6).

With the notation of (12.1.3), suppose one has a commutative diagram

   0  →  ℱ   →^r   𝒢   →^s   𝓗   →  0
              ↓u         ↓v         ↓w                                        (12.1.6.1)
   0  →  ℱ'  →^{r'}  𝒢'  →^{s'}  𝓗'  →  0

where $r$, $s$ are homomorphisms of ${\mathcal{O}}_X$-modules, $r^{\prime }$, $s^{\prime }$ are homomorphisms of ${\mathcal{O}}_{X^{\prime }}$-modules, $u$, $v$, $w$ are $f$-morphisms, and the rows are exact. One then deduces a commutative diagram

   ⋯ → H^p(X, ℱ)   → H^p(X, 𝒢)   → H^p(X, 𝓗)   →^∂  H^{p+1}(X, ℱ)   → ⋯
            ↓u_p          ↓v_p          ↓w_p           ↓u_{p+1}                (12.1.6.2)
   ⋯ → H^p(X', ℱ') → H^p(X', 𝒢') → H^p(X', 𝓗') →^∂ H^{p+1}(X', ℱ') → ⋯.

Indeed, (12.1.6.1) factors as

$$0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0\downarrow \downarrow \downarrow 0\to {\psi }^{\ast }(\mathcal{F})\to {\psi }^{\ast }(\mathcal{G})\to {\psi }^{\ast }(\mathcal{H})\to 0\downarrow \downarrow \downarrow 0\to {\mathcal{F}}^{\prime }\to {\mathcal{G}}^{\prime }\to {\mathcal{H}}^{\prime }\to 0$$

where the middle row is exact $(0_I,\mathrm{3.7.2})$, and it suffices to use the fact that (12.1.3.2) is a homomorphism of $\partial$-functors and that the $H^p(X^{\prime },{\mathcal{F}}^{\prime })$ form a $\partial$-functor in ${\mathcal{F}}^{\prime }$.

12.1.7.

The hypotheses and notation being those of (12.1.3), consider now the case where $\mathcal{F}=f_{\ast }({\mathcal{F}}^{\prime })={\psi }_{\ast }({\mathcal{F}}^{\prime })$; we shall see that the di-homomorphism defined in (12.1.3)

  H^p(X, f_*(ℱ')) → H^p(X', ℱ')                                              (12.1.7.1)

can be obtained (up to an automorphism of $H^p(X^{\prime },{\mathcal{F}}^{\prime })$) as an edge homomorphism of a spectral sequence of the composite functor ${\mathcal{F}}^{\prime }\mapsto \Gamma (X^{\prime },{\psi }_{\ast }({\mathcal{F}}^{\prime }))$ (T, 2.4). The description of the homomorphism (12.1.7.1) given in (12.1.4) shows here that one can obtain this homomorphism as follows: one considers injective resolutions ${\mathcal{L}}^{\bullet }$ and ${\mathcal{L}}^{\prime \bullet }$ of ${\psi }_{\ast }({\mathcal{F}}^{\prime })$ and of ${\mathcal{F}}^{\prime }$ respectively, then one takes a homomorphism of complexes $v:{\psi }^{\ast }({\mathcal{L}}^{\bullet })\to {\mathcal{L}}^{\prime \bullet }$ "lying above" the canonical homomorphism ${\psi }^{\ast }({\psi }_{\ast }({\mathcal{F}}^{\prime }))\to {\mathcal{F}}^{\prime }$;

one then notes that one has $\Gamma (X^{\prime },{\mathcal{L}}^{\prime \bullet })=\Gamma (X,{\psi }_{\ast }({\mathcal{L}}^{\prime \bullet }))$ and that the composite homomorphism

  Γ(X, ℒ^•) → Γ(X', ψ^*(ℒ^•)) →^{Γ(v)} Γ(X', ℒ'^•)

is none other than

  Γ(v^♭) : Γ(X, ℒ^•) → Γ(X, ψ_*(ℒ'^•))                                       (12.1.7.2)

$(0_I,\mathrm{3.7.1})$, and (12.1.7.1) is obtained by passage to cohomology in (12.1.7.2). On the other hand, the spectral sequences of the composite functor ${\mathcal{F}}^{\prime }\mapsto \Gamma (X,{\psi }_{\ast }({\mathcal{F}}^{\prime }))$ are obtained by considering an injective Cartan–Eilenberg resolution ${\mathcal{M}}^{\bullet \bullet }=({\mathcal{M}}^{ij})$ of the complex ${\psi }_{\ast }({\mathcal{L}}^{\prime \bullet })$ in the category of sheaves of abelian groups on $X$; the spectral sequences in question are those of the bicomplex $\Gamma (X,{\mathcal{M}}^{\bullet \bullet })$ (which are biregular since ${\mathcal{M}}^{ij}=0$ for $i<0$ or $j<0$). Now, the first spectral sequence of this bicomplex degenerates, for the sheaves ${\psi }_{\ast }({\mathcal{L}}^{\prime i})$ are flasque (G, II, 3.1.1), hence $H_{II}^q(\Gamma (X,{\mathcal{M}}^{i\bullet }))=H^q({\psi }_{\ast }({\mathcal{L}}^{\prime i}))=0$ for $q>0$ (G, II, 4.4.3); one therefore has bijective edge homomorphisms (11.1.6)

  'E_2^{i,0} = H^i(H_{II}^0(Γ(X, ℳ^{••}))) → H^i(Γ(X, ℳ^{••}))               (12.1.7.3)

and one knows (11.3.4) that this homomorphism comes, by passage to cohomology, from the augmentation

  Γ(X, ψ_*(ℒ'^•)) → Γ(X, ℳ^{••})                                             (12.1.7.4)

which itself comes from the augmentation $\eta :{\psi }_{\ast }({\mathcal{L}}^{\prime \bullet })\to {\mathcal{M}}^{\bullet 0}$. On the other hand, for the second spectral sequence one has edge homomorphisms

  ″E_2^{i,0} = H^i(H_I^0(Γ(X, ℳ^{••}))) → H^i(Γ(X, ℳ^{••}))                  (12.1.7.5)

coming (11.3.4), by passage to cohomology, from the homomorphism of complexes $Z_I^0(\Gamma (X,{\mathcal{M}}^{\bullet \bullet }))\to \Gamma (X,{\mathcal{M}}^{\bullet \bullet })$. Now, since ${\psi }_{\ast }$ is left-exact, the sequence

$$0\to {\psi }_{\ast }({\mathcal{F}}^{\prime })\to {\psi }_{\ast }({\mathcal{L}}^{\prime 0})\to {\psi }_{\ast }({\mathcal{L}}^{\prime 1})$$

is exact; by the definition of a Cartan–Eilenberg resolution (11.4.2), one can therefore take $B_I^0({\mathcal{M}}^{\bullet \bullet })=0$, $Z_I^0({\mathcal{M}}^{\bullet \bullet })={\mathcal{L}}^{\bullet }$; since the diagram

                    ℒ^0   →^{i^0}   ℳ^{00}
                  ε ↓         ↗^{ε''}     ↑η^0
                    ψ_*(ℱ')   →^{ε'}    ψ_*(ℒ'^0)

is commutative, the injection of complexes $i:{\mathcal{L}}^{\bullet }\to {\mathcal{M}}^{\bullet 0}$ is compatible with the augmentations $ϵ$ and ${ϵ}^{\prime \prime }$. One thus has two homomorphisms of complexes from ${\mathcal{L}}^{\bullet }$ into ${\mathcal{M}}^{\bullet 0}$

                    ℒ^•   →^i      ℳ^{•0}
                  v^♭ ↘    ↗ η
                    ψ_*(ℒ'^•)

compatible with the augmentations $ϵ$ and ${ϵ}^{\prime \prime }$; since ${\mathcal{L}}^{\bullet }$ is an injective resolution and ${\mathcal{M}}^{\bullet 0}$ is formed of injective sheaves, it follows from (M, V, 1.1 a)) that these two homomorphisms are homotopic; the same is therefore true of the two corresponding homomorphisms $\Gamma (X,{\mathcal{L}}^{\bullet })\to \Gamma (X,{\mathcal{M}}^{\bullet 0})$, and on passing to cohomology one obtains the same homomorphism; in other words, one has shown that the edge homomorphism (12.1.7.5), which is written $H^p(X,{\psi }_{\ast }({\mathcal{F}}^{\prime }))\to H^p(\Gamma (X,{\mathcal{M}}^{\bullet \bullet }))$, is composed of (12.1.7.1) and of (12.1.7.3), which is written $H^p(X^{\prime },{\mathcal{F}}^{\prime })\to H^p(\Gamma (X,{\mathcal{M}}^{\bullet \bullet }))$ and which we have seen to be an isomorphism; whence our assertion.

12.2. Higher direct images

12.2.1.

Let $(X,{\mathcal{O}}_X)$, $(Y,{\mathcal{O}}_Y)$ be two ringed spaces, $f=(\psi ,\theta )$ a morphism from $X$ to $Y$, which defines the direct image functor $f_{\ast }:C(X)\to C(Y)$, identical moreover with the restriction to $C(X)$ of the functor ${\psi }_{\ast }$ defined in the category of sheaves of abelian groups on $X$. This last functor is additive and left-exact, and since every sheaf of abelian groups on $X$ is isomorphic to a subsheaf of an injective sheaf of abelian groups, one defines the right-derived functors $\mathcal{F}\mapsto R^p{\psi }_{\ast }(\mathcal{F})$ of the functor ${\psi }_{\ast }$; the $R^p{\psi }_{\ast }(\mathcal{F})$ are sheaves of abelian groups on $Y$, and the $R^p{\psi }_{\ast }$ form a universal cohomological functor (T, 2.3).

Moreover, the sheaf $R^p{\psi }_{\ast }(\mathcal{F})$ is the sheaf associated to the presheaf $V\mapsto H^p({\psi }^{-1}(V),\mathcal{F})$ (T, 3.7.2). Suppose now that $\mathcal{F}$ is an ${\mathcal{O}}_X$-module. Then $H^p({\psi }^{-1}(V),\mathcal{F})$ is naturally endowed with a structure of $\Gamma ({\psi }^{-1}(V),{\mathcal{O}}_X)$-module, hence of $\Gamma (V,{\psi }_{\ast }({\mathcal{O}}_X))$-module, and the data of the homomorphism $\theta :{\mathcal{O}}_Y\to {\psi }_{\ast }({\mathcal{O}}_X)$ allows one to deduce a structure of $\Gamma (V,{\mathcal{O}}_Y)$-module. For the structures thus defined, it is clear that restriction from an open set $V$ to an open set $V^{\prime }\subseteq V$ defines a di-homomorphism, and this permits us to define on each of the $R^p{\psi }_{\ast }(\mathcal{F})$ a structure of ${\mathcal{O}}_Y$-module; this is the ${\mathcal{O}}_Y$-module that we shall denote by $R^p{f}_{\ast }(\mathcal{F})$, with $R^p{f}_{\ast }$ thus defined as an additive functor from $C(X)$ to $C(Y)$. Moreover, the $R^p{f}_{\ast }$ form a $\partial$-functor, for if $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ is an exact sequence of ${\mathcal{O}}_X$-modules, the description of the $R^p{\psi }_{\ast }$ and of the ${\mathcal{O}}_Y$-module structure on $R^p{f}_{\ast }(\mathcal{F})$ given above shows at once that the homomorphism $\partial :R^p{f}_{\ast }({\mathcal{F}}^{\prime \prime })\to R^{p+1}{f}_{\ast }({\mathcal{F}}^{\prime })$ is in this case a homomorphism of ${\mathcal{O}}_Y$-modules. Finally, the $R^p{f}_{\ast }$ identify with the right-derived functors of $f_{\ast }$: indeed, every ${\mathcal{O}}_X$-module admits an injective resolution formed of ${\mathcal{O}}_X$-modules, and since such a resolution is formed of flasque sheaves of abelian groups (G, II, 7.1), it can serve to compute the $R^p{\psi }_{\ast }(\mathcal{F})$, since $R^n{\psi }_{\ast }(\mathcal{G})=0$ for $n\ge 1$ and every flasque sheaf $\mathcal{G}$ (T, 2.4.1, Remark 3, and cor. of Prop. 3.3.2). One thus concludes that the $R^p{f}_{\ast }$ form a universal cohomological functor from $C(X)$ to $C(Y)$ (T, 2.3).

12.2.2.

Let $\mathcal{F}$ and $\mathcal{G}$ be two ${\mathcal{O}}_X$-modules. With the notation of (12.2.1), for every open $V$ of $Y$ one has the cup-product homomorphism (12.1.2.1)

  H^p(ψ^{-1}(V), ℱ) ⊗_{Γ(ψ^{-1}(V), 𝒪_X)} H^q(ψ^{-1}(V), 𝒢)
      → H^{p+q}(ψ^{-1}(V), ℱ ⊗_{𝒪_X} 𝒢)

and it follows at once from the definition of the cup product (G, II, 6.6) that these homomorphisms commute with passage from $V$ to an open subspace $V^{\prime }$ of $V$. On the other hand, one has a homomorphism of rings

  Γ(V, 𝒪_Y) → Γ(V, ψ_*(𝒪_X)) = Γ(ψ^{-1}(V), 𝒪_X)

coming from $\theta$, whence a canonical homomorphism of tensor products

  H^p(ψ^{-1}(V), ℱ) ⊗_{Γ(V, 𝒪_Y)} H^q(ψ^{-1}(V), 𝒢)
      → H^p(ψ^{-1}(V), ℱ) ⊗_{Γ(ψ^{-1}(V), 𝒪_X)} H^q(ψ^{-1}(V), 𝒢)

which is also compatible with the restriction from $V$ to $V^{\prime }$. By composition, one therefore obtains a homomorphism of $\Gamma (V,{\mathcal{O}}_Y)$-modules, which defines a canonical functorial-in-$\mathcal{F}$-and-$\mathcal{G}$ homomorphism for the sheaves associated to the presheaves considered:

  R^p f_*(ℱ) ⊗_{𝒪_Y} R^q f_*(𝒢) → R^{p+q} f_*(ℱ ⊗_{𝒪_X} 𝒢).                  (12.2.2.1)

Note that for $p=q=0$, this homomorphism reduces to $(0_I,\mathrm{4.2.2.1})$.

Proposition (12.2.3).

For every ${\mathcal{O}}_X$-module $\mathcal{F}$ and every ${\mathcal{O}}_Y$-module locally free of finite rank $\mathcal{L}$, one has canonical functorial isomorphisms

  R^p f_*(ℱ) ⊗_{𝒪_Y} ℒ ⥲ R^p f_*(ℱ ⊗_{𝒪_X} f^*(ℒ)).                          (12.2.3.1)

Proof. The homomorphism (12.2.3.1) is obtained by composing the homomorphism, a particular case of (12.2.2.1),

  R^p f_*(ℱ) ⊗_{𝒪_Y} f_*(f^*(ℒ)) → R^p f_*(ℱ ⊗_{𝒪_X} f^*(ℒ))                 (12.2.3.2)

with the homomorphism from the first member of (12.2.3.1) to that of (12.2.3.2) coming from the canonical homomorphism $(0_I,\mathrm{4.4.3.2})$. To verify that (12.2.3.1) is an isomorphism when $\mathcal{L}$ is locally free, one can immediately reduce to the case $\mathcal{L}={\mathcal{O}}_Y$, the question being local on $Y$, and the functors considered being additive in $\mathcal{L}$. But then, the proposition reduces, in view of the definition of (12.2.2.1), to the verification that the corresponding homomorphism of presheaves is bijective, which is immediate by virtue of the relation $f^{\ast }({\mathcal{O}}_Y)={\mathcal{O}}_X$.

12.2.4.

Let $(Z,{\mathcal{O}}_Z)$ be a third ringed space, $g:Y\to Z$ a morphism of ringed spaces. One knows (G, II, 7.1 and 3.1.1) that for every injective ${\mathcal{O}}_Y$-module $\mathcal{G}$, $f_{\ast }(\mathcal{G})$ is a flasque sheaf of abelian groups, and consequently (12.2.1) one has $R^p{g}_{\ast }(f_{\ast }(\mathcal{G}))=0$ for every $p>0$. It follows (T, 2.4.1) that the Leray spectral sequence of the composed functors is applicable to the composite functor $g_{\ast }{f}_{\ast }$: there is a biregular spectral sequence whose abutment is the functor $R^{\bullet }{h}_{\ast }$ where $h=g\circ f$, and whose E_2 term is given by

  E_2^{p,q} = R^p g_*(R^q f_*(ℱ)).                                           (12.2.4.1)

12.2.5.

Under the conditions of (12.2.4), we shall define directly canonical homomorphisms of ${\mathcal{O}}_Z$-modules

  R^n g_*(f_*(ℱ)) → R^n h_*(ℱ)                                               (12.2.5.1)
  R^n h_*(ℱ) → g_*(R^n f_*(ℱ))                                               (12.2.5.2)

which could be identified with the "edge homomorphisms" of the Leray spectral sequence (cf. (12.1.7)). It suffices to operate on the presheaves to which the higher direct image sheaves (12.2.1) are associated. For this, consider any open set $W$ of $Z$ and its inverse image $g^{-1}(W)$ in $Y$; one has a canonical di-homomorphism

  H^n(g^{-1}(W), f_*(ℱ)) → H^n(f^{-1}(g^{-1}(W)), f^*(f_*(ℱ)))                (12.2.5.3)

with corresponding rings $\Gamma (g^{-1}(W),{\mathcal{O}}_Y)$ and $\Gamma (h^{-1}(W),{\mathcal{O}}_X)$; on the other hand, the canonical homomorphism $(0_I,\mathrm{4.4.3.3})$ yields by functoriality canonical homomorphisms

  H^n(h^{-1}(W), f^*(f_*(ℱ))) → H^n(h^{-1}(W), ℱ)                            (12.2.5.4)

which are homomorphisms of $\Gamma (h^{-1}(W),{\mathcal{O}}_X)$-modules. Taking into account the ring homomorphism $\Gamma (W,{\mathcal{O}}_Z)\to \Gamma (h^{-1}(W),{\mathcal{O}}_X)$, one sees that by composing (12.2.5.4) and (12.2.5.3) one obtains a homomorphism of presheaves, which yields the homomorphism of sheaves (12.2.5.1).

The definition of (12.2.5.2) is even simpler; by definition, $R^n{h}_{\ast }(\mathcal{F})$ is associated to the presheaf $W\mapsto H^n(f^{-1}(g^{-1}(W)),\mathcal{F})$ and $R^n{f}_{\ast }(\mathcal{F})$ to the presheaf $V\mapsto H^n(f^{-1}(V),\mathcal{F})$; one therefore has a canonical homomorphism

  H^n(f^{-1}(g^{-1}(W)), ℱ) → Γ(g^{-1}(W), R^n f_*(ℱ)),

and it is immediate that these homomorphisms define a homomorphism of presheaves, which in turn defines (12.2.5.2).

12.2.6.

Under the hypotheses of (12.2.4), let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be three ${\mathcal{O}}_X$-modules and $u:\mathcal{F}\otimes \mathcal{G}\to \mathcal{H}$ an ${\mathcal{O}}_X$-homomorphism. One then has commutative diagrams

   R^p g_*(f_*(ℱ)) ⊗_{𝒪_Z} R^q g_*(f_*(𝒢))   →   R^{p+q} g_*(f_*(𝓗))
            ↓                                          ↓                       (12.2.6.1)
   R^p h_*(ℱ) ⊗_{𝒪_Z} R^q h_*(𝒢)             →   R^{p+q} h_*(𝓗)

and

   R^p h_*(ℱ) ⊗_{𝒪_Z} R^q h_*(𝒢)             →   R^{p+q} h_*(𝓗)
            ↓                                          ↓                       (12.2.6.2)
   g_*(R^p f_*(ℱ)) ⊗_{𝒪_Z} g_*(R^q f_*(𝒢))   →   g_*(R^{p+q} f_*(𝓗))

where the horizontal arrows come from (12.2.2.1) (the last combined with $(0_I,\mathrm{4.2.2.1})$) and the vertical arrows from the homomorphisms (12.2.5.1) and (12.2.5.2) respectively.

It indeed suffices to verify this for the corresponding homomorphisms of presheaves; returning to the definitions given in (12.2.2) and (12.2.5) for these homomorphisms, one is immediately reduced, for (12.2.6.1), to the commutative diagrams (12.1.5.2); the verification is even simpler for (12.2.6.2).

12.3. Complements on the Ext functors of sheaves

12.3.1.

Consider a ringed space $(X,{\mathcal{O}}_X)$; we shall not return to the definition and principal properties of the bifunctors $Ex{t}_{{\mathcal{O}}_X}^p(X;\mathcal{F},\mathcal{G})$ from the category of ${\mathcal{O}}_X$-modules to that of $\Gamma (X,{\mathcal{O}}_X)$-modules, and $\mathcal{E}x{t}_{{\mathcal{O}}_X}^p(\mathcal{F},\mathcal{G})$ from the category of ${\mathcal{O}}_X$-modules to itself, nor to the biregular spectral sequence $E(\mathcal{F},\mathcal{G})$ relating them (T, 4.2 and G, II, 7.3).

12.3.2.

One defines, in the same way as in (M, XIV, 1), the notion of extension of an ${\mathcal{O}}_X$-module $\mathcal{F}$ by an ${\mathcal{O}}_X$-module $\mathcal{G}$ and the composition law between classes of equivalent extensions: the arguments made for modules adapt indeed in an obvious way to any abelian category. The second proof of (M, XIV, 1.1), which uses only the existence of embeddings in injective objects, is therefore still valid for the category of ${\mathcal{O}}_X$-modules, and thus shows that $Ex{t}_{{\mathcal{O}}_X}^1(X;\mathcal{F},\mathcal{G})$ is canonically identified with the abelian group of classes of extensions of $\mathcal{F}$ by $\mathcal{G}$.

Proposition (12.3.3).

Let $(X,{\mathcal{O}}_X)$ be a ringed space such that the sheaf of rings ${\mathcal{O}}_X$ is coherent. Then, for every pair of coherent ${\mathcal{O}}_X$-modules $\mathcal{F}$, $\mathcal{G}$ and every $p\ge 0$, $\mathcal{E}x{t}_{{\mathcal{O}}_X}^p(\mathcal{F},\mathcal{G})$ is a coherent ${\mathcal{O}}_X$-module.

Proof. Note that the $\mathcal{E}x{t}_{{\mathcal{O}}_X}^p(\mathcal{F},\mathcal{G})$ form a cohomological functor contravariant in $\mathcal{F}$. Since $\mathcal{F}$ is coherent, there exist, for every $p$ and every point $x\in X$, an open neighborhood $U$ of $x$ and an exact sequence of $({\mathcal{O}}_X\mid U)$-modules

  0 → ℛ → ℒ_{p−1} → ⋯ → ℒ_0 → ℱ ∣ U → 0

where each of the ${\mathcal{L}}_i$ ($0\le i\le p-1$) is isomorphic to an ${\mathcal{O}}_X^{n_i}\mid U$ and $\mathcal{R}$ is coherent: this follows by induction on $p$ from $(0_I,\mathrm{5.3.2})$ and $(0_I,\mathrm{5.3.4})$, in view of the hypothesis that ${\mathcal{O}}_X$ is coherent.

Now note that, for $p\ge 1$, one has $Ex{t}_{{\mathcal{O}}_X\mid U}^p(\mathcal{L}\mid U,\mathcal{G}\mid U)=0$ for every ${\mathcal{O}}_X$-module $\mathcal{L}$ such that $\mathcal{L}\mid U$ is isomorphic to an ${\mathcal{O}}_X^n\mid U$ (T, 4.2.3); the argument of (M, V, 7.2) therefore applies to the contravariant cohomological functor $\mathcal{F}\mapsto \mathcal{H}o{m}_{{\mathcal{O}}_X\mid U}(\mathcal{F}\mid U,\mathcal{G}\mid U)$, and gives an exact sequence

  ℋom_{𝒪_X ∣ U}(ℒ_{p−1}, 𝒢 ∣ U) → ℋom_{𝒪_X ∣ U}(ℛ, 𝒢 ∣ U) → ℰxt^p_{𝒪_X ∣ U}(ℱ ∣ U, 𝒢 ∣ U) → 0

and since the first two terms of this sequence are coherent $({\mathcal{O}}_X\mid U)$-modules $(0_I,\mathrm{5.3.5})$, so is the third $(0_I,\mathrm{5.3.4})$.

Proposition (12.3.4).

Let $f:X\to Y$ be a flat morphism of ringed spaces, and let $\mathcal{F}$, $\mathcal{G}$ be two ${\mathcal{O}}_Y$-modules.

(i) There exists a homomorphism of cohomological bifunctors

  f^*(ℰxt^p_{𝒪_Y}(ℱ, 𝒢)) ⥲ ℰxt^p_{𝒪_X}(f^*(ℱ), f^*(𝒢))                       (12.3.4.1)

reducing in degree 0 to the canonical homomorphism $(0_I,\mathrm{4.4.6})$.

(ii) There exists a canonical morphism of spectral sequences

  E(ℱ, 𝒢) → E(f^*(ℱ), f^*(𝒢))                                                (12.3.4.2)

which, for the E_2 terms, reduces to the homomorphisms

  H^p(Y, ℰxt^q_{𝒪_Y}(ℱ, 𝒢)) → H^p(X, ℰxt^q_{𝒪_X}(f^*(ℱ), f^*(𝒢)))             (12.3.4.3)

deduced from (12.3.4.1) and (12.1.3.1).

Proof.

(i) Since $f^{\ast }$ is an exact functor on the category of ${\mathcal{O}}_Y$-modules $(0_I,\mathrm{6.7.2})$, the functors $\mathcal{G}\mapsto f^{\ast }(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},\mathcal{G}))$ and $\mathcal{G}\mapsto \mathcal{H}o{m}_{{\mathcal{O}}_X}(f^{\ast }(\mathcal{F}),f^{\ast }(\mathcal{G}))$ are left-exact; one deduces canonically from $(0_I,\mathrm{4.4.6})$ a homomorphism of their derived functors. To compute the latter, one takes an injective resolution ${\mathcal{L}}^{\bullet }=({\mathcal{L}}^i)$ of $\mathcal{G}$, and one therefore has morphisms ${\mathcal{H}}^p(f^{\ast }(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},{\mathcal{L}}^{\bullet })))\to {\mathcal{H}}^p(\mathcal{H}o{m}_{{\mathcal{O}}_X}(f^{\ast }(\mathcal{F}),f^{\ast }({\mathcal{L}}^{\bullet })))$ of cohomologies of complexes of sheaves. Moreover, by the exactness of $f^{\ast }$, one has ${\mathcal{H}}^p(f^{\ast }(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},{\mathcal{L}}^{\bullet })))=f^{\ast }({\mathcal{H}}^p(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},{\mathcal{L}}^{\bullet })))=f^{\ast }(\mathcal{E}x{t}_{{\mathcal{O}}_Y}^p(\mathcal{F},\mathcal{G}))$ by definition. On the other hand, the exactness of $f^{\ast }$ entails that $f^{\ast }({\mathcal{L}}^{\bullet })$ is a resolution of $f^{\ast }(\mathcal{G})$; if ${\mathcal{L}}^{\prime \bullet }=({\mathcal{L}}^{\prime i})$ is an injective resolution of $f^{\ast }(\mathcal{G})$ in the category of ${\mathcal{O}}_X$-modules, there is therefore a homomorphism of complexes $f^{\ast }({\mathcal{L}}^{\bullet })\to {\mathcal{L}}^{\prime \bullet }$, determined up to homotopy, and which defines by consequence a well-determined homomorphism in cohomology; composing this homomorphism with the one defined above, one obtains (12.3.4.1).

(ii) With the preceding notation, one has a homomorphism of complexes of sheaves of ${\mathcal{O}}_X$-modules $f^{\ast }(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},{\mathcal{L}}^{\bullet }))\to \mathcal{H}o{m}_{{\mathcal{O}}_X}(f^{\ast }(\mathcal{F}),{\mathcal{L}}^{\prime \bullet })$. Let ${\mathcal{M}}^{\bullet \bullet }$ be a Cartan–Eilenberg injective resolution of the complex $\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},{\mathcal{L}}^{\bullet })$ in the category of ${\mathcal{O}}_Y$-modules; then, by the exactness of the functor $f^{\ast }$, $f^{\ast }({\mathcal{M}}^{\bullet \bullet })$ is a Cartan–Eilenberg resolution of the complex $f^{\ast }(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},{\mathcal{L}}^{\bullet }))$; if ${\mathcal{M}}^{\prime \bullet \bullet }$ is a Cartan–Eilenberg injective resolution of the complex $\mathcal{H}o{m}_{{\mathcal{O}}_X}(f^{\ast }(\mathcal{F}),{\mathcal{L}}^{\prime \bullet })$, there is therefore (11.4.2) a homomorphism (determined up to homotopy) $f^{\ast }({\mathcal{M}}^{\bullet \bullet })\to {\mathcal{M}}^{\prime \bullet \bullet }$ compatible with the homomorphism considered above, in other words an $f$-morphism ${\mathcal{M}}^{\bullet \bullet }\to {\mathcal{M}}^{\prime \bullet \bullet }$ of bicomplexes of sheaves. From this one deduces a di-homomorphism $\Gamma (Y,{\mathcal{M}}^{\bullet \bullet })\to \Gamma (X,{\mathcal{M}}^{\prime \bullet \bullet })$ of bicomplexes of modules, determined up to homotopy, and a well-determined morphism of spectral sequences (11.3.2), which is none other than the morphism (12.3.4.2) sought, the characterization of (12.3.4.3) following at once from the definitions.

Proposition (12.3.5).

Under the hypotheses of (12.3.4), suppose in addition the sheaf of rings ${\mathcal{O}}_Y$ coherent; then, for every coherent ${\mathcal{O}}_Y$-module $\mathcal{F}$, the canonical homomorphisms (12.3.4.1) are bijective.

Proof. The question being local on $Y$, one may suppose there exists an exact sequence $0\to \mathcal{R}\to {\mathcal{O}}_Y^n\to \mathcal{F}\to 0$, and $\mathcal{R}$ is then also a coherent ${\mathcal{O}}_Y$-module $(0_I,\mathrm{5.3.4})$. To prove that the homomorphisms

  f^*(ℰxt^p_{𝒪_Y}(ℱ, 𝒢)) → ℰxt^p_{𝒪_X}(f^*(ℱ), f^*(𝒢))

are bijective, we argue by induction on $p$, the proposition resulting from $(0_I,\mathrm{6.7.6.1})$ when $p=0$. Now, one has the commutative diagram

   f^*(ℰxt^{p−1}_{𝒪_Y}(𝒪_Y^n, 𝒢)) → f^*(ℰxt^{p−1}_{𝒪_Y}(ℛ, 𝒢)) →^∂ f^*(ℰxt^p_{𝒪_Y}(ℱ, 𝒢)) → f^*(ℰxt^p_{𝒪_Y}(𝒪_Y^n, 𝒢))
            ↓                              ↓                              ↓                                ↓
   ℰxt^{p−1}_{𝒪_X}(𝒪_X^n, f^*(𝒢)) → ℰxt^{p−1}_{𝒪_X}(f^*(ℛ), f^*(𝒢)) →^∂ ℰxt^p_{𝒪_X}(f^*(ℱ), f^*(𝒢)) → ℰxt^p_{𝒪_X}(𝒪_X^n, f^*(𝒢))

since $f^{\ast }({\mathcal{O}}_Y)={\mathcal{O}}_X$; as $f^{\ast }$ is exact, the two rows are exact. Moreover, one has $\mathcal{E}x{t}_{{\mathcal{O}}_Y}^p({\mathcal{O}}_Y^n,\mathcal{G})=0$ for every $p>0$ and likewise $\mathcal{E}x{t}_{{\mathcal{O}}_X}^p({\mathcal{O}}_X^n,f^{\ast }(\mathcal{G}))=0$ for every $p>0$ (T, 4.2.3). In view of the induction hypothesis, the first two vertical arrows of the preceding diagram are isomorphisms, and the terms on the right are 0, hence $f^{\ast }(\mathcal{E}x{t}_{{\mathcal{O}}_Y}^p(\mathcal{F},\mathcal{G}))\to \mathcal{E}x{t}_{{\mathcal{O}}_X}^p(f^{\ast }(\mathcal{F}),f^{\ast }(\mathcal{G}))$ is an isomorphism.

12.4. Hypercohomology of the direct image functor

12.4.1.

Let $(X,{\mathcal{O}}_X)$, $(Y,{\mathcal{O}}_Y)$ be two ringed spaces, $f:X\to Y$ a morphism of ringed spaces. One can take the hypercohomology of $f_{\ast }$ with respect to any complex of ${\mathcal{O}}_X$-modules ${\mathcal{K}}^{\bullet }=({\mathcal{K}}^i{)}_{i\in \mathbb{Z}}$ (11.4.4), for in the abelian category of ${\mathcal{O}}_Y$-modules, filtered inductive limits exist and are exact (T, 3.1.1). The ${\mathcal{O}}_Y$-modules of hypercohomology $R^p{f}_{\ast }({\mathcal{K}}^{\bullet })$ will also be denoted ${\mathcal{H}}^p(f,{\mathcal{K}}^{\bullet })$ or ${\mathcal{H}}^p({\mathcal{K}}^{\bullet })$. Recall that ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is the cohomology of the bicomplex of ${\mathcal{O}}_Y$-modules $f_{\ast }({\mathcal{L}}^{\bullet \bullet })$, where ${\mathcal{L}}^{\bullet \bullet }$ is an injective Cartan–Eilenberg resolution of ${\mathcal{K}}^{\bullet }$ in the category of ${\mathcal{O}}_X$-modules; ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is the abutment of two spectral sequences ${}^{\prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ and ${}^{\prime \prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ whose E_2 terms are given by

  'E_2^{pq} = ℋ^p(ℋ^q(f, 𝒦^•))                                               (12.4.1.1)
  ″E_2^{pq} = ℋ^p(f, ℋ^q(𝒦^•))     (= R^p f_*(ℋ^q(𝒦^•)))                     (12.4.1.2)

In these formulas, we have adopted the general notation $T(A^{\bullet })$ for the transform of a complex by a functor (11.2.1), and one writes ${\mathcal{H}}^p(f,\mathcal{F})$ instead of $R^p{f}_{\ast }(\mathcal{F})$ for an ${\mathcal{O}}_X$-module $\mathcal{F}$. Recall further that the sequence ${}^{\prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ is always regular; the two spectral sequences ${}^{\prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ and ${}^{\prime \prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ are biregular when ${\mathcal{K}}^{\bullet }$ is bounded below,

or when there exists an integer $m$ such that every ${\mathcal{O}}_X$-module admits a flasque resolution of length $\le m$ (11.4.4).

12.4.2.

We shall likewise denote by $H^{\bullet }(X,{\mathcal{K}}^{\bullet })$ the hypercohomology of the functor $\Gamma$ with respect to a complex ${\mathcal{K}}^{\bullet }$ of ${\mathcal{O}}_X$-modules; the $H^p(X,{\mathcal{K}}^{\bullet })$ are therefore $\Gamma (X,{\mathcal{O}}_X)$-modules. One can moreover consider $H^{\bullet }(X,{\mathcal{K}}^{\bullet })$ as a particular case of ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$, where $f$ is a morphism of $(X,{\mathcal{O}}_X)$ to a ringed space reduced to a point endowed with the ring $\Gamma (X,{\mathcal{O}}_X)$.

For every open set $V$ of $X$, we shall write $H^{\bullet }(V,{\mathcal{K}}^{\bullet })$ instead of $H^{\bullet }(V,{\mathcal{K}}^{\bullet }\mid V)$.

Proposition (12.4.3).

For every integer $p\in \mathbb{Z}$, the ${\mathcal{O}}_Y$-module ${\mathcal{H}}^p(f,{\mathcal{K}}^{\bullet })$ is canonically isomorphic to the sheaf associated to the presheaf $U\mapsto H^p(f^{-1}(U),{\mathcal{K}}^{\bullet })$ on $Y$.

Proof. Indeed, with the notation of (12.4.1), the cohomology sheaf ${\mathcal{H}}^p(f_{\ast }({\mathcal{L}}^{\bullet \bullet }))$ is associated to the presheaf $U\mapsto H^p(\Gamma (U,f_{\ast }({\mathcal{L}}^{\bullet \bullet })))=H^p(\Gamma (f^{-1}(U),{\mathcal{L}}^{\bullet \bullet }))$. But it is clear that ${\mathcal{L}}^{\bullet \bullet }\mid f^{-1}(U)$ is an injective Cartan–Eilenberg resolution of ${\mathcal{K}}^{\bullet }\mid f^{-1}(U)$ (T, 3.1.3), so $H^p(\Gamma (f^{-1}(U),{\mathcal{L}}^{\bullet \bullet }))=H^p(f^{-1}(U),{\mathcal{K}}^{\bullet })$ by definition.

Proposition (12.4.4).

The hypercohomology ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is a cohomological functor in ${\mathcal{K}}^{\bullet }$ in each of the following cases:

a) ${\mathcal{K}}^{\bullet }$ varies in the category of complexes bounded below.

b) There exists an integer $m$ such that every ${\mathcal{O}}_X$-module admits a flasque resolution of length $\le m$.

c) $X$ is a Noetherian space.

Proof. Cases a) and b) are particular cases of (11.5.4). On the other hand, case c) follows from (11.5.2), for one knows that in this case the functor $f_{\ast }$ commutes with inductive limits (G, II, 3.10.1).

12.4.5.

Consider now an open cover $\mathfrak{U}=(U_{\alpha })$ of $X$, and for every complex of presheaves ${\mathcal{K}}^{\bullet }=({\mathcal{K}}^j)$ on $X$, the bicomplex $C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$, whose component of indices $(i,j)$ is $C^i(\mathfrak{U},{\mathcal{K}}^j)$, the group of $i$-cochains alternating in the nerve of $\mathfrak{U}$ with values in ${\mathcal{K}}^j$ (G, II, 5.1). We shall say that the cohomology of this bicomplex is the hypercohomology of the cover $\mathfrak{U}$ with coefficients in ${\mathcal{K}}^{\bullet }$, and we shall denote it $H^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })=H^{\bullet }(C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet }))$. The Leray spectral sequence of a cover (T, 3.8.1 and G, II, 5.9.1) generalizes as follows to hypercohomology:

Proposition (12.4.6).

Let ${\mathcal{K}}^{\bullet }=({\mathcal{K}}^j)$ be a complex of ${\mathcal{O}}_X$-modules. There exists a regular spectral functor in ${\mathcal{K}}^{\bullet }$ having as abutment the hypercohomology $H^{\bullet }(X,{\mathcal{K}}^{\bullet })$, and whose E_2 term is given by

$$E_2^{pq}=H^p(\mathfrak{U},h^q({\mathcal{K}}^{\bullet }))(\mathrm{12.4.6.1})$$

where $h^q({\mathcal{K}}^{\bullet })$ denotes the complex of presheaves $V\mapsto H^q(V,{\mathcal{K}}^{\bullet })$ on $X$. The preceding spectral sequence is biregular if ${\mathcal{K}}^{\bullet }$ is bounded below.

Proof. Consider an injective Cartan–Eilenberg resolution ${\mathcal{L}}^{\bullet \bullet }$ of ${\mathcal{K}}^{\bullet }$, and the tricomplex $C^{\bullet }(\mathfrak{U},{\mathcal{L}}^{\bullet \bullet })=(C^i(\mathfrak{U},{\mathcal{L}}^{jk}))$; consider this tricomplex first as a bicomplex for the degrees $i$ and $j+k$. Since $i$ takes only values $\ge 0$, the second spectral sequence of this bicomplex is regular (11.3.3) and degenerate, for one has $H^q(\mathfrak{U},{\mathcal{L}}^{jk})=0$ for every $q>0$, the ${\mathcal{O}}_X$-modules ${\mathcal{L}}^{jk}$ being flasque sheaves

(G, II, 5.2.3). One therefore has (11.1.6) a canonical isomorphism $H^n(C^{\bullet }(\mathfrak{U},{\mathcal{L}}^{\bullet \bullet }))\stackrel{\sim }{\to }{H}^n(\Gamma (X,{\mathcal{L}}^{\bullet \bullet }))$ (by virtue of (G, II, 5.2.2)), hence by definition (12.4.2) an isomorphism $H^n(C^{\bullet }(\mathfrak{U},{\mathcal{L}}^{\bullet \bullet }))\stackrel{\sim }{\to }{H}^n(X,{\mathcal{K}}^{\bullet })$. Consider on the other hand the tricomplex $C^{\bullet }(\mathfrak{U},{\mathcal{L}}^{\bullet \bullet })$ as a bicomplex for the degrees $i+j$ and $k$. Since $k$ takes only values $\ge 0$, the first spectral sequence of this bicomplex is always regular; it is biregular if ${\mathcal{L}}^{jk}=0$ for $j<j_0$, that is, when ${\mathcal{K}}^{\bullet }$ is bounded below (11.3.3). This spectral sequence is the one sought; indeed, for every $j$, ${\mathcal{L}}^{j\bullet }$ is an injective resolution of ${\mathcal{K}}^j$; consequently, $H^q(C^{\bullet }(\mathfrak{U},{\mathcal{L}}^{j\bullet }))$ is none other than the complex of cochains $C^{\bullet }(\mathfrak{U},h^q({\mathcal{K}}^j))$, which completes the proof.

Corollary (12.4.7).

If, for every simplex $\sigma$ of the nerve of $\mathfrak{U}$, and for every integer $i$, one has $H^q(U_{\sigma },{\mathcal{K}}^i)=0$ for $q>0$, then one has a canonical isomorphism

  H^•(𝔘, 𝒦^•) ⥲ H^•(X, 𝒦^•).                                                 (12.4.7.1)

Proof. Indeed, the hypothesis entails that $C^{\bullet }(\mathfrak{U},h^q({\mathcal{K}}^j))=0$ for $q>0$, hence $E_2^{pq}=0$ for $q>0$; the sequence (12.4.6.1) being degenerate and regular, the conclusion follows from the definition (12.4.5) of $H^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ and from (11.1.6).

12.4.8.

Let $(X^{\prime },{\mathcal{O}}_{X^{\prime }})$ be a second ringed space, and let $f=(\psi ,\theta )$ be a morphism from $X^{\prime }$ to $X$. By the same method as in (12.1.3) and (12.1.4), one defines a di-homomorphism for the hypercohomology of a complex ${\mathcal{K}}^{\bullet }$ of ${\mathcal{O}}_X$-modules

  H^p(X, 𝒦^•) → H^p(X', f^*(𝒦^•)).                                           (12.4.8.1)

One starts from a Cartan–Eilenberg injective resolution ${\mathcal{L}}^{\bullet \bullet }$ of ${\mathcal{K}}^{\bullet }$, and since ${\psi }^{\ast }$ is exact, ${\psi }^{\ast }({\mathcal{L}}^{\bullet \bullet })$ is a Cartan–Eilenberg resolution of ${\psi }^{\ast }({\mathcal{K}}^{\bullet })$ in the category of ${\psi }^{\ast }({\mathcal{O}}_X)$-modules; there is then a morphism ${\psi }^{\ast }({\mathcal{L}}^{\bullet \bullet })\to {\mathcal{L}}^{\prime \bullet \bullet }$, where ${\mathcal{L}}^{\prime \bullet \bullet }$ is a Cartan–Eilenberg injective resolution of ${\psi }^{\ast }({\mathcal{K}}^{\bullet })$, and from this one deduces a morphism for cohomology $H^{\bullet }(X,{\mathcal{K}}^{\bullet })\to H^{\bullet }(X^{\prime },{\psi }^{\ast }({\mathcal{K}}^{\bullet }))$; by composition with the morphism deduced by functoriality from ${\psi }^{\ast }({\mathcal{K}}^{\bullet })\to f^{\ast }({\mathcal{K}}^{\bullet })$, one obtains the morphism (12.4.8.1) sought.

Starting from (12.4.8.1) and (12.4.3), one can then, reasoning as in (12.2.5), define, for two morphisms $f:X\to Y$, $g:Y\to Z$ of ringed spaces, homomorphisms for the hypercohomology of a complex ${\mathcal{K}}^{\bullet }$ of ${\mathcal{O}}_X$-modules

  ℋ^n(g, f_*(𝒦^•)) → ℋ^n(h, 𝒦^•)                                             (12.4.8.2)
  ℋ^n(h, 𝒦^•) → g_*(ℋ^n(f, 𝒦^•)).                                            (12.4.8.3)

We leave the details of the definitions to the reader.