The Stacks project

35.9 Cohomology of quasi-coherent modules and topologies

In this section we prove that cohomology of quasi-coherent modules is independent of the choice of topology.

Lemma 35.9.1. Let $S$ be a scheme. Let

  1. $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $ and $\mathcal{C} = (\mathit{Sch}/S)_\tau $, or

  2. let $\tau = {\acute{e}tale}$ and $\mathcal{C} = S_{\acute{e}tale}$, or

  3. let $\tau = Zariski$ and $\mathcal{C} = S_{Zar}$.

Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be affine. Let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ be a standard affine $\tau $-covering in $\mathcal{C}$. Then

  1. $\mathcal{V} = \{ \coprod _{i = 1, \ldots , n} U_ i \to U\} $ is a $\tau $-covering of $U$,

  2. $\mathcal{U}$ is a refinement of $\mathcal{V}$, and

  3. the induced map on Čech complexes (Cohomology on Sites, Equation (

    \[ \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]

    is an isomorphism of complexes.

Proof. This follows because

\[ (\coprod \nolimits _{i_0 = 1, \ldots , n} U_{i_0}) \times _ U \ldots \times _ U (\coprod \nolimits _{i_ p = 1, \ldots , n} U_{i_ p}) = \coprod \nolimits _{i_0, \ldots , i_ p \in \{ 1, \ldots , n\} } U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \]

and the fact that $\mathcal{F}(\coprod _ a V_ a) = \prod _ a \mathcal{F}(V_ a)$ since disjoint unions are $\tau $-coverings. $\square$

Lemma 35.9.2. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau $, $\mathcal{C}$, $U$, $\mathcal{U}$ be as in Lemma 35.9.1. Then there is an isomorphism of complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^ a) \cong s((A/R)_\bullet \otimes _ R M) \]

(see Section 35.3) where $R = \Gamma (U, \mathcal{O}_ U)$, $M = \Gamma (U, \mathcal{F}^ a)$ and $R \to A$ is a faithfully flat ring map. In particular

\[ \check{H}^ p(\mathcal{U}, \mathcal{F}^ a) = 0 \]

for all $p \geq 1$.

Proof. By Lemma 35.9.1 we see that $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^ a)$ is isomorphic to $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}^ a)$ where $\mathcal{V} = \{ V \to U\} $ with $V = \coprod _{i = 1, \ldots n} U_ i$ affine also. Set $A = \Gamma (V, \mathcal{O}_ V)$. Since $\{ V \to U\} $ is a $\tau $-covering we see that $R \to A$ is faithfully flat. On the other hand, by definition of $\mathcal{F}^ a$ we have that the degree $p$ term $\check{\mathcal{C}}^ p(\mathcal{V}, \mathcal{F}^ a)$ is

\[ \Gamma (V \times _ U \ldots \times _ U V, \mathcal{F}^ a) = \Gamma (\mathop{\mathrm{Spec}}(A \otimes _ R \ldots \otimes _ R A), \mathcal{F}^ a) = A \otimes _ R \ldots \otimes _ R A \otimes _ R M \]

We omit the verification that the maps of the Čech complex agree with the maps in the complex $s((A/R)_\bullet \otimes _ R M)$. The vanishing of cohomology is Lemma 35.3.6. $\square$


Proposition 35.9.3. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $.

  1. There is a canonical isomorphism

    \[ H^ q(S, \mathcal{F}) = H^ q((\mathit{Sch}/S)_\tau , \mathcal{F}^ a). \]
  2. There are canonical isomorphisms

    \[ H^ q(S, \mathcal{F}) = H^ q(S_{Zar}, \mathcal{F}^ a) = H^ q(S_{\acute{e}tale}, \mathcal{F}^ a). \]

Proof. The result for $q = 0$ is clear from the definition of $\mathcal{F}^ a$. Let $\mathcal{C} = (\mathit{Sch}/S)_\tau $, or $\mathcal{C} = S_{\acute{e}tale}$, or $\mathcal{C} = S_{Zar}$.

We are going to apply Cohomology on Sites, Lemma 21.10.9 with $\mathcal{F} = \mathcal{F}^ a$, $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the set of affine schemes in $\mathcal{C}$, and $\text{Cov} \subset \text{Cov}_\mathcal {C}$ the set of standard affine $\tau $-coverings. Assumption (3) of the lemma is satisfied by Lemma 35.9.2. Hence we conclude that $H^ p(U, \mathcal{F}^ a) = 0$ for every affine object $U$ of $\mathcal{C}$.

Next, let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be any separated object. Denote $f : U \to S$ the structure morphism. Let $U = \bigcup U_ i$ be an affine open covering. We may also think of this as a $\tau $-covering $\mathcal{U} = \{ U_ i \to U\} $ of $U$ in $\mathcal{C}$. Note that $U_{i_0} \times _ U \ldots \times _ U U_{i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$ is affine as we assumed $U$ separated. By Cohomology on Sites, Lemma 21.10.7 and the result above we see that

\[ H^ p(U, \mathcal{F}^ a) = \check{H}^ p(\mathcal{U}, \mathcal{F}^ a) = H^ p(U, f^*\mathcal{F}) \]

the last equality by Cohomology of Schemes, Lemma 30.2.6. In particular, if $S$ is separated we can take $U = S$ and $f = \text{id}_ S$ and the proposition is proved. We suggest the reader skip the rest of the proof (or rewrite it to give a clearer exposition).

Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ on $S$. Choose an injective resolution $\mathcal{F}^ a \to \mathcal{J}^\bullet $ on $\mathcal{C}$. Denote $\mathcal{J}^ n|_ S$ the restriction of $\mathcal{J}^ n$ to opens of $S$; this is a sheaf on the topological space $S$ as open coverings are $\tau $-coverings. We get a complex

\[ 0 \to \mathcal{F} \to \mathcal{J}^0|_ S \to \mathcal{J}^1|_ S \to \ldots \]

which is exact since its sections over any affine open $U \subset S$ is exact (by the vanishing of $H^ p(U, \mathcal{F}^ a)$, $p > 0$ seen above). Hence by Derived Categories, Lemma 13.18.6 there exists map of complexes $\mathcal{J}^\bullet |_ S \to \mathcal{I}^\bullet $ which in particular induces a map

\[ R\Gamma (\mathcal{C}, \mathcal{F}^ a) = \Gamma (S, \mathcal{J}^\bullet ) \longrightarrow \Gamma (S, \mathcal{I}^\bullet ) = R\Gamma (S, \mathcal{F}). \]

Taking cohomology gives the map $H^ n(\mathcal{C}, \mathcal{F}^ a) \to H^ n(S, \mathcal{F})$ which we have to prove is an isomorphism. Let $\mathcal{U} : S = \bigcup U_ i$ be an affine open covering which we may think of as a $\tau $-covering also. By the above we get a map of double complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{J}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{J}|_ S) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}). \]

This map induces a map of spectral sequences

\[ {}^\tau \! E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F}^ a)) \longrightarrow E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \]

The first spectral sequence converges to $H^{p + q}(\mathcal{C}, \mathcal{F})$ and the second to $H^{p + q}(S, \mathcal{F})$. On the other hand, we have seen that the induced maps ${}^\tau \! E_2^{p, q} \to E_2^{p, q}$ are bijections (as all the intersections are separated being opens in affines). Whence also the maps $H^ n(\mathcal{C}, \mathcal{F}^ a) \to H^ n(S, \mathcal{F})$ are isomorphisms, and we win. $\square$

Proposition 35.9.4. Let $f : T \to S$ be a morphism of schemes.

  1. The equivalences of categories of Proposition 35.8.9 are compatible with pullback. More precisely, we have $f^*(\mathcal{G}^ a) = (f^*\mathcal{G})^ a$ for any quasi-coherent sheaf $\mathcal{G}$ on $S$.

  2. The equivalences of categories of Proposition 35.8.9 part (1) are not compatible with pushforward in general.

  3. If $f$ is quasi-compact and quasi-separated, and $\tau \in \{ Zariski, {\acute{e}tale}\} $ then $f_*$ and $f_{small, *}$ preserve quasi-coherent sheaves and the diagram

    \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^ a} & & \mathit{QCoh}(\mathcal{O}_ S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^ a} \\ \mathit{QCoh}(T_\tau , \mathcal{O}) \ar[rr]^{f_{small, *}} & & \mathit{QCoh}(S_\tau , \mathcal{O}) } \]

    is commutative, i.e., $f_{small, *}(\mathcal{F}^ a) = (f_*\mathcal{F})^ a$.

Proof. Part (1) follows from the discussion in Remark 35.8.6. Part (2) is just a warning, and can be explained in the following way: First the statement cannot be made precise since $f_*$ does not transform quasi-coherent sheaves into quasi-coherent sheaves in general. Even if this is the case for $f$ (and any base change of $f$), then the compatibility over the big sites would mean that formation of $f_*\mathcal{F}$ commutes with any base change, which does not hold in general. An explicit example is the quasi-compact open immersion $j : X = \mathbf{A}^2_ k \setminus \{ 0\} \to \mathbf{A}^2_ k = Y$ where $k$ is a field. We have $j_*\mathcal{O}_ X = \mathcal{O}_ Y$ but after base change to $\mathop{\mathrm{Spec}}(k)$ by the $0$ map we see that the pushforward is zero.

Let us prove (3) in case $\tau = {\acute{e}tale}$. Note that $f$, and any base change of $f$, transforms quasi-coherent sheaves into quasi-coherent sheaves, see Schemes, Lemma 26.24.1. The equality $f_{small, *}(\mathcal{F}^ a) = (f_*\mathcal{F})^ a$ means that for any étale morphism $g : U \to S$ we have $\Gamma (U, g^*f_*\mathcal{F}) = \Gamma (U \times _ S T, (g')^*\mathcal{F})$ where $g' : U \times _ S T \to T$ is the projection. This is true by Cohomology of Schemes, Lemma 30.5.2. $\square$

Lemma 35.9.5. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the étale or Zariski topology, there are canonical isomorphisms $R^ if_{small, *}(\mathcal{F}^ a) = (R^ if_*\mathcal{F})^ a$.

Proof. We prove this for the étale topology; we omit the proof in the case of the Zariski topology. By Cohomology of Schemes, Lemma 30.4.5 the sheaves $R^ if_*\mathcal{F}$ are quasi-coherent so that the assertion makes sense. The sheaf $R^ if_{small, *}\mathcal{F}^ a$ is the sheaf associated to the presheaf

\[ U \longmapsto H^ i(U \times _ S T, \mathcal{F}^ a) \]

where $g : U \to S$ is an object of $S_{\acute{e}tale}$, see Cohomology on Sites, Lemma 21.7.4. By our conventions the right hand side is the étale cohomology of the restriction of $\mathcal{F}^ a$ to the localization $T_{\acute{e}tale}/U \times _ S T$ which equals $(U \times _ S T)_{\acute{e}tale}$. By Proposition 35.9.3 this is presheaf the same as the presheaf

\[ U \longmapsto H^ i(U \times _ S T, (g')^*\mathcal{F}), \]

where $g' : U \times _ S T \to T$ is the projection. If $U$ is affine then this is the same as $H^0(U, R^ if'_*(g')^*\mathcal{F})$, see Cohomology of Schemes, Lemma 30.4.6. By Cohomology of Schemes, Lemma 30.5.2 this is equal to $H^0(U, g^*R^ if_*\mathcal{F})$ which is the value of $(R^ if_*\mathcal{F})^ a$ on $U$. Thus the values of the sheaves of modules $R^ if_{small, *}(\mathcal{F}^ a)$ and $(R^ if_*\mathcal{F})^ a$ on every affine object of $S_{\acute{e}tale}$ are canonically isomorphic which implies they are canonically isomorphic. $\square$

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