§1. Cohomology of affine schemes

1.1. Review of the exterior algebra complex

1.1.1.

Let $A$ be a ring, $\mathbf{f}=(f_i{)}_{1\le i\le r}$ a system of $r$ elements of $A$. The exterior algebra complex $K_{\bullet }(\mathbf{f})$ corresponding to $\mathbf{f}$ is a chain complex (G, I, 2.2) defined as follows: the graded $A$-module $K_{\bullet }(\mathbf{f})$ is equal to the exterior algebra $\Lambda (A^r)$, graded in the usual way, and the boundary operator is the interior multiplication $i_{\mathbf{f}}$ by $\mathbf{f}$, viewed as an element of the dual $(A^r{)}^{\vee }$. Recall that $i_{\mathbf{f}}$ is an antiderivation of degree $-1$ on $\Lambda (A^r)$, and that if $({\mathbf{e}}_i{)}_{1\le i\le r}$ is the canonical basis of $A^r$ then $i_{\mathbf{f}}({\mathbf{e}}_i)=f_i$; the verification of $i_{\mathbf{f}}\circ i_{\mathbf{f}}=0$ is immediate.

An equivalent definition is the following. For each $i$, consider the chain complex $K_{\bullet }(f_i)$ defined by $K_0(f_i)=K_1(f_i)=A$, $K_n(f_i)=0$ for $n\ne 0,1$, and with boundary operator $d_1:A\to A$ equal to multiplication by $f_i$. Then $K_{\bullet }(\mathbf{f})$ is the tensor product

  K_•(𝐟) = K_•(f_1) ⊗ K_•(f_2) ⊗ ⋯ ⊗ K_•(f_r)                                (G, I, 2.7)

equipped with its total grading; the isomorphism between this complex and the one defined above is immediate.

Translator's note. EGA's $complexede{l}^{\prime }alg\grave{e}breext\acute{e}rieure$ is what is now universally called the Koszul complex. We retain EGA's name on first introduction and use it interchangeably below.

1.1.2.

For every $A$-module $M$, define the chain complex

  K_•(𝐟, M) = K_•(𝐟) ⊗_A M                                                   (1.1.2.1)

and the cochain complex (G, I, 2.2)

  K^•(𝐟, M) = Hom_A(K_•(𝐟), M).                                              (1.1.2.2)

If $g$ is a $k$-cochain of the latter complex, and if we set

  g(i_1, …, i_k) = g(𝐞_{i_1} ∧ ⋯ ∧ 𝐞_{i_k}),

then $g$ identifies with an alternating map from $[1,r{]}^k$ to $M$, and the above definitions give

  d^k g(i_1, i_2, …, i_{k+1})
        = Σ_{h=1}^{k+1} (−1)^{h−1} f_{i_h} g(i_1, …, î_h, …, i_{k+1}).        (1.1.2.3)

1.1.3.

From the preceding complexes one deduces, as usual, the homology and cohomology $A$-modules (G, I, 2.2)

  H_•(𝐟, M) = H_•(K_•(𝐟, M)),                                                (1.1.3.1)
  H^•(𝐟, M) = H^•(K^•(𝐟, M)).                                                (1.1.3.2)

One defines an $A$-isomorphism $K_{\bullet }(\mathbf{f},M)\stackrel{\sim }{\to }{K}^{\bullet }(\mathbf{f},M)$ by associating to each chain $z=\Sigma ({\mathbf{e}}_{i_1}\wedge \cdots \wedge {\mathbf{e}}_{i_k})\otimes z_{i_1,\cdots ,i_k}$ the cochain $g_z$ with

  g_z(j_1, …, j_{r−k}) = ε · z_{i_1, …, i_k},

where $(j_h{)}_{1\le h\le r-k}$ is the strictly increasing sequence complementary to the strictly increasing sequence $(i_h{)}_{1\le h\le k}$ in [1, r] and $ϵ=(-1{)}^{\nu }$, with $\nu$ the number of inversions of the permutation $i_1,\cdots ,i_k,j_1,\cdots ,j_{r-k}$ of [1, r]. One checks that $g_{dz}=d(g_z)$, yielding an isomorphism

  H^i(𝐟, M) ≅ H_{r−i}(𝐟, M)                  for 0 ≤ i ≤ r.                 (1.1.3.3)

In this chapter we shall mostly consider the cohomology modules $H^{\bullet }(\mathbf{f},M)$.

For a given $\mathbf{f}$, it is immediate (G, I, 2.1) that $M\mapsto H^{\bullet }(\mathbf{f},M)$ is a cohomological functor (T, II, 2.1) from the category of $A$-modules to the category of graded $A$-modules, vanishing in degrees < 0 and $>r$. In addition

  H^0(𝐟, M) = Hom_A(A/(𝐟), M),                                               (1.1.3.4)

writing $(\mathbf{f})$ for the ideal of $A$ generated by $f_1,\cdots ,f_r$; this follows immediately from (1.1.2.3), and $H^0(\mathbf{f},M)$ is clearly identified with the submodule of $M$ annihilated by $(\mathbf{f})$. Similarly, (1.1.2.3) gives

  H^r(𝐟, M) = M / (Σ_{i=1}^r f_i M) = (A/(𝐟)) ⊗_A M.                         (1.1.3.5)

We shall use the following well-known result, whose proof we recall for completeness.

Proposition (1.1.4).

Let $A$ be a ring, $\mathbf{f}=(f_i{)}_{1\le i\le r}$ a finite family of elements of $A$, and $M$ an $A$-module. If, for $1\le i\le r$, the multiplication $z\mapsto f_{i}z$ on $M_{i-1}=M/(f_{1}M+\cdots +f_{i-1}M)$ is injective, then $H^p(\mathbf{f},M)=0$ for $p\ne r$.

Proof. It suffices to prove that $H_i(\mathbf{f},M)=0$ for every $i>0$ by virtue of (1.1.3.3). We argue by induction on $r$, the case $r=0$ being trivial. Set ${\mathbf{f}}^{\prime }=(f_i{)}_{1\le i\le r-1}$; this family satisfies the hypotheses of the statement, so on setting $L_{\bullet }=K_{\bullet }({\mathbf{f}}^{\prime },M)$ we have $H_i(L_{\bullet })=0$ for $i>0$ by induction and $H_0(L_{\bullet })=M_{r-1}$ by (1.1.3.3) and (1.1.3.5). To abbreviate, set $K_{\bullet }=K_{\bullet }(f_r)=K_0\oplus K_1$, with $K_0=K_1=A$ and $d_1:K_1\to K_0$ equal to multiplication by $f_r$; by definition (1.1.1) we have $K_{\bullet }(\mathbf{f},M)=K_{\bullet }{\otimes }_A{L}_{\bullet }$. We invoke the following lemma.

Lemma (1.1.4.1).

Let $K_{\bullet }$ be a chain complex of free $A$-modules, zero except in dimensions 0 and 1. For every chain complex $L_{\bullet }$ of $A$-modules, there is an exact sequence

  0 → H_0(K_• ⊗ H_p(L_•)) → H_p(K_• ⊗ L_•) → H_1(K_• ⊗ H_{p−1}(L_•)) → 0

for every index $p$.

This is a particular case of an exact sequence of low-degree terms of the Künneth spectral sequence (M, XVII, 5.2 (a)) and (G, I, 5.5.2); one can also prove it directly, as follows. Regard K_0 and K_1 as chain complexes (zero in dimensions $\ne 0$ and $\ne 1$ respectively); then there is an exact sequence of complexes

  0 → K_0 ⊗ L_• → K_• ⊗ L_• → K_1 ⊗ L_• → 0,

to which the exact sequence in homology applies:

  ⋯ → H_{p+1}(K_1 ⊗ L_•) → H_p(K_0 ⊗ L_•) → H_p(K_• ⊗ L_•) → H_p(K_1 ⊗ L_•)
                       → H_{p−1}(K_0 ⊗ L_•) → ⋯.

But $H_p(K_0\otimes L_{\bullet })=K_0\otimes H_p(L_{\bullet })$ and $H_p(K_1\otimes L_{\bullet })=K_1\otimes H_{p-1}(L_{\bullet })$ for every $p$; one verifies immediately, in addition, that the boundary $\partial :K_1\otimes H_p(L_{\bullet })\to K_0\otimes H_p(L_{\bullet })$ coincides with $d_1\otimes 1$. The lemma follows from the exact sequence above and from the definitions of $H_0(K_{\bullet }\otimes H_p(L_{\bullet }))$ and $H_1(K_{\bullet }\otimes H_{p-1}(L_{\bullet }))$.

With this lemma in hand, the rest of the proof of (1.1.4) is immediate: the induction hypothesis combined with (1.1.4.1) gives $H_p(K_{\bullet }\otimes L_{\bullet })=0$ for $p\ge 2$; moreover, if we show that $H_1(K_{\bullet },H_0(L_{\bullet }))=0$, then the same lemma also gives $H_1(K_{\bullet }\otimes L_{\bullet })=0$. But by definition $H_1(K_{\bullet },H_0(L_{\bullet }))$ is the kernel of the multiplication $z\mapsto f_{r}z$ on $M_{r-1}$, and that kernel is zero by hypothesis. The proof is complete.

1.1.5.

Let $\mathbf{g}=(g_i{)}_{1\le i\le r}$ be a second sequence of $r$ elements of $A$, and set $\mathbf{fg}=(f_i{g}_i{)}_{1\le i\le r}$. One can define a canonical homomorphism of complexes

$${\varphi }_{\mathbf{g}}:K_{\bullet }(\mathbf{fg})\to K_{\bullet }(\mathbf{f})(\mathrm{1.1.5.1})$$

as the canonical extension to the exterior algebra $\Lambda (A^r)$ of the $A$-linear map $(x_1,\cdots ,x_r)\mapsto (g_1{x}_1,\cdots ,g_r{x}_r)$ from $A^r$ to itself. To see that this is a homomorphism of complexes, note in general that if $u:E\to F$ is an $A$-linear map, $\mathbf{x}\in F^{\vee }$, and $\mathbf{y}={}^{t}u(\mathbf{x})\in E^{\vee }$, then

  (Λ u) ∘ i_𝐲 = i_𝐱 ∘ (Λ u);                                                 (1.1.5.2)

indeed, both sides are antiderivations of $\Lambda F$, and it suffices to check that they coincide on $F$, which follows immediately from the definitions.

When one identifies $K_{\bullet }(\mathbf{f})$ with the tensor product of the $K_{\bullet }(f_i)$ (1.1.1), ${\varphi }_{\mathbf{g}}$ is the tensor product of the ${\varphi }_{g_i}$, where ${\varphi }_{g_i}$ reduces to the identity in degree 0 and to multiplication by $g_i$ in degree 1.

1.1.6.

In particular, for every pair of integers $m$, $n$ with $0\le n\le m$, we have homomorphisms of complexes

$${\varphi }_{{\mathbf{f}}^{m-n}}:K_{\bullet }({\mathbf{f}}^m)\to K_{\bullet }({\mathbf{f}}^n)(\mathrm{1.1.6.1})$$

and hence homomorphisms

  φ_{𝐟^{m−n}} : K^•(𝐟^n, M) → K^•(𝐟^m, M),                                   (1.1.6.2)
  φ_{𝐟^{m−n}} : H^•(𝐟^n, M) → H^•(𝐟^m, M).                                   (1.1.6.3)

The latter satisfy the obvious transitivity relation ${\varphi }_{{\mathbf{f}}^{m-p}}={\varphi }_{{\mathbf{f}}^{m-n}}\circ {\varphi }_{{\mathbf{f}}^{n-p}}$ for $p\le n\le m$; they thus define two inductive systems of $A$-modules. We set

  C^•((𝐟), M) = lim_→ K^•(𝐟^n, M),                                           (1.1.6.4)
                  n
  H^•((𝐟), M) = H^•(C^•((𝐟), M)) = lim_→ H^•(𝐟^n, M),                        (1.1.6.5)
                                     n

the last equality coming from the fact that the inductive limit commutes with the functor $H^{\bullet }$ (G, I, 2.1). We shall see later (1.4.3) that $H^{\bullet }((\mathbf{f}),M)$ depends only on the ideal $(\mathbf{f})$ of $A$ (in fact only on the $(\mathbf{f})$-preadic topology on $A$), which justifies the notations.

It is clear that $M\mapsto C^{\bullet }((\mathbf{f}),M)$ is an exact $A$-linear functor and $M\mapsto H^{\bullet }((\mathbf{f}),M)$ a cohomological functor.

1.1.7.

Let $\mathbf{f}=(f_i)\in A^r$ and $\mathbf{g}=(g_i)\in A^r$; write $e_{\mathbf{g}}$ for the left multiplication by the vector $\mathbf{g}\in A^r$ on the exterior algebra $\Lambda (A^r)$. We have the homotopy formula

  i_𝐟 e_𝐠 + e_𝐠 i_𝐟 = ⟨𝐠, 𝐟⟩ · 1                                            (1.1.7.1)

on the $A$-module $\Lambda (A^r)$ (where 1 denotes the identity automorphism of $\Lambda (A^r)$); this relation says, equivalently, that in the complex $K_{\bullet }(\mathbf{f})$

  d e_𝐠 + e_𝐠 d = ⟨𝐠, 𝐟⟩ · 1.                                                (1.1.7.2)

If the ideal $(\mathbf{f})$ is equal to $A$, then there exists $\mathbf{g}\in A^r$ such that $⟨\mathbf{g},\mathbf{f}⟩={\Sigma }_{i=1}^r{g}_i{f}_i=1$. Therefore (G, I, 2.4):

Proposition (1.1.8).

Suppose the ideal $(\mathbf{f})$ generated by the $f_i$ is equal to $A$. Then the complex $K_{\bullet }(\mathbf{f})$ is homotopically trivial, and so are the complexes $K_{\bullet }(\mathbf{f},M)$ and $K^{\bullet }(\mathbf{f},M)$ for every $A$-module $M$.

Corollary (1.1.9).

If $(\mathbf{f})=A$, then $H^{\bullet }(\mathbf{f},M)=0$ and $H^{\bullet }((\mathbf{f}),M)=0$ for every $A$-module $M$.

Proof. We then have $({\mathbf{f}}^n)=A$ for every $n$.

Remark (1.1.10).

With the notation above, let $X=\mathrm{Spec}(A)$, and let $Y$ be the closed subscheme of $X$ defined by the ideal $(\mathbf{f})$. We shall prove in §9 that $H^{\bullet }((\mathbf{f}),M)$ is isomorphic to the cohomology $H_Y^{\bullet }(X,M)$ of $X$ with support in the filter of closed subsets of $Y$ (T, 3.2). We shall also show that Proposition (1.2.3), applied to $X$ and to $\mathcal{F}=M$, is a particular case of an exact cohomology sequence

  ⋯ → H_Y^p(X, ℱ) → H^p(X, ℱ) → H^p(X − Y, ℱ) → H_Y^{p+1}(X, ℱ) → ⋯.

1.2. Čech cohomology of an open cover

1.2.1. Notations.

Throughout this article we shall write:

• $X$ for a prescheme;
• $\mathcal{F}$ for a quasi-coherent ${\mathcal{O}}_X$-module;

• $A=\Gamma (X,{\mathcal{O}}_X)$, $M=\Gamma (X,\mathcal{F})$;
• $\mathbf{f}=(f_i{)}_{1\le i\le r}$ for a finite system of elements of $A$;
• $U_i=X_{f_i}$ for the open set $(0_I,\mathrm{5.5.2})$ of $x\in X$ with $f_i(x)\ne 0$;
• $U=\bigcup U_i$;
• $\mathfrak{U}$ for the open cover $(U_i{)}_{1\le i\le r}$ of $U$.

1.2.2.

Suppose $X$ is either a prescheme whose underlying space is Noetherian, or a scheme whose underlying space is quasi-compact. Then (I, 9.3.3) gives $\Gamma (U_i,\mathcal{F})=M_{f_i}$. Set

  U_{i_0, …, i_p} = ⋂_{k=0}^p U_{i_k} = X_{f_{i_0} ⋯ f_{i_p}}                (0_I, 5.5.3);

we then likewise have

  Γ(U_{i_0, …, i_p}, ℱ) = M_{f_{i_0} ⋯ f_{i_p}}.                             (1.2.2.1)

But $(0_I,\mathrm{1.6.1})$ identifies $M_{f_{i_0}\cdots f_{i_p}}$ with the inductive limit lim_→_n M_{i_0, …, i_p}^{(n)}, where $M_{i_0,\cdots ,i_p}^{(n)}=M$ and the transition map ${\varphi }_{n,m}:M_{i_0,\cdots ,i_p}^{(m)}\to M_{i_0,\cdots ,i_p}^{(n)}$ is multiplication by $(f_{i_0}\cdots f_{i_p}{)}^{n-m}$ for $m\le n$. Denote by $C_n^p(M)$ the set of alternating maps from $[1,r{]}^{p+1}$ to $M$ (for every $n$); these $A$-modules likewise form an inductive system under the ${\varphi }_{n,m}$. If $C^p(\mathfrak{U},\mathcal{F})$ is the group of alternating Čech $p$-cochains of $\mathcal{F}$ relative to the cover $\mathfrak{U}$ (G, II, 5.1), the foregoing gives

  C^p(𝔘, ℱ) = lim_→ C^p_n(M).                                                 (1.2.2.2)
                n

With the notation of (1.1.2), $C_n^p(M)$ is identified with $K^{p+1}({\mathbf{f}}^n,M)$, and the map ${\varphi }_{n,m}$ with the map ${\varphi }_{{\mathbf{f}}^{n-m}}$ defined in (1.1.6). Thus, for every $p\ge 0$, there is a canonical functorial-in-$\mathcal{F}$ isomorphism

  C^p(𝔘, ℱ) ≅ C^p((𝐟), M).                                                   (1.2.2.3)

Moreover, the formula (1.1.2.3) and the definition of the cohomology of a cover (G, II, 5.1) show that the isomorphisms (1.2.2.3) are compatible with the coboundary operators.

Proposition (1.2.3).

If $X$ is a prescheme whose underlying space is Noetherian, or a scheme whose underlying space is quasi-compact, then there is a canonical functorial-in-$\mathcal{F}$ isomorphism

  H^p(𝔘, ℱ) ≅ H^{p+1}((𝐟), M)              for every p ≥ 1.                  (1.2.3.1)

In addition, there is a functorial-in-$\mathcal{F}$ exact sequence

  0 → H^0((𝐟), M) → M → H^0(𝔘, ℱ) → H^1((𝐟), M) → 0.                         (1.2.3.2)

Proof. The relations (1.2.3.1) are an immediate consequence of (1.2.2). On the other hand, $C^0(\mathfrak{U},\mathcal{F})=C^1((\mathbf{f}),M)$, so $H^0(\mathfrak{U},\mathcal{F})$ identifies with the subgroup of 1-cocycles of $C^{\bullet }((\mathbf{f}),M)$; since $M=C^0((\mathbf{f}),M)$, the exact sequence (1.2.3.2) is the one that comes from the definition of the cohomology groups $H^0((\mathbf{f}),M)$ and $H^1((\mathbf{f}),M)$.

Corollary (1.2.4).

Suppose the $X_{f_i}$ are quasi-compact and that there exist $g_i\in \Gamma (U,\mathcal{F})$ such that ${\Sigma }_i{g}_i(f_i\mid U)=1_U$. Then, for every quasi-coherent $({\mathcal{O}}_X\mid U)$-module $\mathcal{F}$, we have $H^p(\mathfrak{U},\mathcal{F})=0$ for $p>0$; if in addition $U=X$, the canonical homomorphism (1.2.3.2) $M\to H^0(\mathfrak{U},\mathcal{F})$ is bijective.

Proof. Since by hypothesis the $U_i=X_{f_i}$ are quasi-compact, so is $U$, so we may reduce to the case $U=X$; the hypothesis then implies $H^p((\mathbf{f}),M)=0$ for every $p\ge 0$ (1.1.9). The corollary follows from (1.2.3.1) and (1.2.3.2).

Note that since $H^0(\mathfrak{U},\mathcal{F})=H^0(U,\mathcal{F})$ (G, II, 5.2.2), the result (I, 1.3.7) is recovered here as a special case.

Remark (1.2.5).

Suppose $X$ is an affine scheme; then the $U_i=X_{f_i}=D(f_i)$ are affine open sets, as are the $U_{i_0,\cdots ,i_p}$ (though $U$ is not necessarily affine). In this case the functors $\Gamma (X,\cdot )$ and $\Gamma (U_{i_0,\cdots ,i_p},\cdot )$ are exact in $\mathcal{F}$ (I, 1.3.11). Given an exact sequence $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ of quasi-coherent ${\mathcal{O}}_X$-modules, the sequence of complexes

  0 → C^•(𝔘, ℱ') → C^•(𝔘, ℱ) → C^•(𝔘, ℱ'') → 0

is exact, and yields a long exact sequence in cohomology

  ⋯ → H^p(𝔘, ℱ') → H^p(𝔘, ℱ) → H^p(𝔘, ℱ'') → H^{p+1}(𝔘, ℱ') → ⋯.

On the other hand, setting $M^{\prime }=\Gamma (X,{\mathcal{F}}^{\prime })$, $M^{\prime \prime }=\Gamma (X,{\mathcal{F}}^{\prime \prime })$, the sequence $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ is exact; as $C^{\bullet }((\mathbf{f}),M)$ is an exact functor of $M$, we also obtain the exact sequence in cohomology

  ⋯ → H^p((𝐟), M') → H^p((𝐟), M) → H^p((𝐟), M'') → H^{p+1}((𝐟), M') → ⋯.

Since the diagram

  0 → C^•(𝔘, ℱ')     → C^•(𝔘, ℱ)     → C^•(𝔘, ℱ'')     → 0
        ↓                 ↓                 ↓
  0 → C^•((𝐟), M')   → C^•((𝐟), M)   → C^•((𝐟), M'')   →

is commutative, the diagrams

              H^p(𝔘, ℱ'')   →   H^{p+1}(𝔘, ℱ')
                  ↓                  ↓                                       (1.2.5.1)
              H^{p+1}((𝐟), M'') → H^{p+1}((𝐟), M')

are commutative for every $p$ (G, I, 2.1.1).

1.3. Cohomology of an affine scheme

Theorem (1.3.1).

Let $X$ be an affine scheme. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, we have $H^p(X,\mathcal{F})=0$ for every $p>0$.

Proof. Let $\mathfrak{U}$ be a finite cover of $X$ by affine open sets $X_{f_i}=D(f_i)$ ($1\le i\le r$); the ideal of $A=\Gamma (X,{\mathcal{O}}_X)$ generated by the $f_i$ is then $A$. By (1.2.4) we have ${\check{H}}^p(\mathfrak{U},\mathcal{F})=0$ for $p>0$. Since there are arbitrarily fine finite affine open covers of $X$ (I, 1.1.10), the definition of Čech cohomology (G, II, 5.8) shows that ${\check{H}}^p(X,\mathcal{F})=0$ for $p>0$. The same argument applies to every $X_f$ for $f\in A$ (I, 1.3.6), so ${\check{H}}^p(X_f,\mathcal{F})=0$ for $p>0$. As $X_f\cap X_g=X_{fg}$, we conclude that $H^p(X,\mathcal{F})=0$ for every $p>0$ by virtue of (G, II, 5.9.2).

Corollary (1.3.2).

Let $Y$ be a prescheme and $f:X\to Y$ an affine morphism (II, 1.6.1). For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, we have $R^q{f}_{\ast }(\mathcal{F})=0$ for $q>0$.

Proof. By definition, $R^q{f}_{\ast }(\mathcal{F})$ is the ${\mathcal{O}}_Y$-module associated to the presheaf $U\mapsto H^q(f^{-1}(U),\mathcal{F})$, where $U$ ranges over the open sets of $Y$. The affine open sets form a base of $Y$, and for such a $U$ the preimage $f^{-1}(U)$ is affine (II, 1.3.2), hence $H^q(f^{-1}(U),\mathcal{F})=0$ by (1.3.1), proving the corollary.

Corollary (1.3.3).

Let $Y$ be a prescheme and $f:X\to Y$ an affine morphism. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the canonical homomorphism

  H^p(Y, f_*(ℱ)) → H^p(X, ℱ)                                   (0, 12.1.3.1)

is bijective for every $p$.

Proof. By (0, 12.1.7) it suffices to show that the edge homomorphisms ${}^{\prime \prime }{E}_2^{p,0}=H^p(Y,f_{\ast }(\mathcal{F}))\to H^p(X,\mathcal{F})$ of the second spectral sequence of the composite functor $\Gamma f_{\ast }$ are bijective. The E_2 term of this sequence is ${}^{\prime \prime }{E}_2^{p,q}=H^p(Y,R^q{f}_{\ast }(\mathcal{F}))$ (G, II, 4.17.1), and (1.3.2) gives ${}^{\prime \prime }{E}_2^{p,q}=0$ for $q>0$; the spectral sequence therefore degenerates. The assertion follows from (0, 11.1.6).

Corollary (1.3.4).

Let $f:X\to Y$ be an affine morphism and $g:Y\to Z$ a morphism.

For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the canonical homomorphism $R^p{g}_{\ast }(f_{\ast }(\mathcal{F}))\to R^p(g\circ f{)}_{\ast }(\mathcal{F})$ (0, 12.2.5.1) is bijective for every $p$.

Proof. Note that, by (1.3.3), for every affine open $W$ of $Z$, the canonical homomorphism $H^p(g^{-1}(W),f_{\ast }(\mathcal{F}))\to H^p(f^{-1}(g^{-1}(W)),\mathcal{F})$ is bijective; this shows that the presheaf homomorphism defining the canonical homomorphism $R^p{g}_{\ast }(f_{\ast }(\mathcal{F}))\to R^p(g\circ f{)}_{\ast }(\mathcal{F})$ is bijective (0, 12.2.5).

1.4. Application to the cohomology of arbitrary preschemes

Proposition (1.4.1).

Let $X$ be a scheme and $\mathfrak{U}=(U_i)$ a cover of $X$ by affine open sets. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the cohomology modules $H^{\bullet }(X,\mathcal{F})$ and $H^{\bullet }(\mathfrak{U},\mathcal{F})$ (over $\Gamma (X,{\mathcal{O}}_X)$) are canonically isomorphic.

Proof. Since $X$ is a scheme, every finite intersection $V$ of open sets of the cover $\mathfrak{U}$ is affine (I, 5.5.6), so $H^q(V,\mathcal{F})=0$ for $q\ge 1$ by (1.3.1). The proposition follows from Leray's theorem (G, II, 5.4.1).

Remark (1.4.2).

The conclusion of (1.4.1) remains valid whenever the finite intersections of the $U_i$ are affine, even if $X$ is not assumed to be a scheme.

Corollary (1.4.3).

Let $X$ be a scheme whose underlying space is quasi-compact, set $A=\Gamma (X,{\mathcal{O}}_X)$, and let $\mathbf{f}=(f_i{)}_{1\le i\le r}$ be a finite sequence of elements of $A$ such that the $X_{f_i}$ (notation of (1.2.1)) are affine. Then, with the notation of (1.2.1), for every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ there is a canonical functorial-in-$\mathcal{F}$ isomorphism

  H^q(U, ℱ) ≅ H^{q+1}((𝐟), M)              for q ≥ 1                         (1.4.3.1)

and a functorial-in-$\mathcal{F}$ exact sequence

  0 → H^0((𝐟), M) → M → H^0(U, ℱ) → H^1((𝐟), M) → 0.                         (1.4.3.2)

Proof. This follows from (1.4.1) and (1.2.3).

1.4.4.

If $X$ is an affine scheme, (1.2.5) and (1.4.1) show that, for every $q\ge 0$, the diagrams

              H^q(𝔘, ℱ'')   →   H^{q+1}(𝔘, ℱ')
                  ↓                  ↓                                       (1.4.4.1)
              H^{q+1}((𝐟), M'') → H^{q+1}((𝐟), M')

corresponding to an exact sequence $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ of quasi-coherent ${\mathcal{O}}_X$-modules (with the notation of (1.2.5)) are commutative.

Proposition (1.4.5).

Let $X$ be a quasi-compact scheme, $\mathcal{L}$ an invertible ${\mathcal{O}}_X$-module, and consider the graded ring $A_{\bullet }={\Gamma }_{\ast }(\mathcal{L})$ (II, 5.4.6). Then $H^{\bullet }(X,\mathcal{F})={\oplus }_n{H}^{\bullet }(X,\mathcal{F}\otimes {\mathcal{L}}^{\otimes n})$ is a graded $A_{\bullet }$-module, and for every $f\in A_d$ there is a canonical isomorphism

  H^•(X_f, ℱ) ≅ (H^•(X, ℱ))_{(f)}                                            (1.4.5.1)

of $(A_{\bullet }{)}_{(f)}$-modules.

Proof. Since $X$ is a quasi-compact scheme, the cohomology of all the ${\mathcal{O}}_X$-modules $\mathcal{F}\otimes {\mathcal{L}}^{\otimes n}$ can be computed with one and the same finite cover $\mathfrak{U}=(U_i)$ by affine open sets such that the restriction $\mathcal{L}\mid U_i$ is isomorphic to ${\mathcal{O}}_X\mid U_i$ for each $i$ (1.4.1). It is then immediate that the $U_i\cap X_f$ are affine open sets (I, 1.3.6), so the cohomology $H^{\bullet }(X_f,\mathcal{F}\otimes {\mathcal{L}}^{\otimes n})$ can also be computed with the cover $\mathfrak{U}\mid X_f=(U_i\cap X_f)$ (1.4.1). It is immediate that for every $f\in A_d$ multiplication by $f$ defines a homomorphism $C^{\bullet }(\mathfrak{U},\mathcal{F}\otimes {\mathcal{L}}^{\otimes n})\to C^{\bullet }(\mathfrak{U},\mathcal{F}\otimes {\mathcal{L}}^{\otimes (n+d)})$, hence a homomorphism $H^{\bullet }(\mathfrak{U},\mathcal{F}\otimes {\mathcal{L}}^{\otimes n})\to H^{\bullet }(\mathfrak{U},\mathcal{F}\otimes {\mathcal{L}}^{\otimes (n+d)})$, which establishes the first assertion. On the other hand, for $f\in A_d$, (I, 9.3.2) gives an isomorphism of complexes of $(A_{\bullet }{)}_{(f)}$-modules

  C^•(𝔘 ∣ X_f, ℱ) ≅ (C^•(𝔘, ℱ ⊗ ℒ^{⊗•}))_{(f)},

taking (I, 1.3.9, (ii)) into account. Passing to the cohomology of both complexes, one obtains the isomorphism (1.4.5.1), using that the functor $M\mapsto M_{(f)}$ is exact on the category of graded $A_{\bullet }$-modules.

Corollary (1.4.6).

Under the hypotheses of (1.4.5), suppose in addition that $\mathcal{L}={\mathcal{O}}_X$. Setting $A=\Gamma (X,{\mathcal{O}}_X)$, then, for every $f\in A$, there is a canonical isomorphism $H^{\bullet }(X_f,\mathcal{F})\cong (H^{\bullet }(X,\mathcal{F}){)}_f$ of $A_f$-modules.

Corollary (1.4.7).

Let $X$ be a quasi-compact scheme and $f\in \Gamma (X,{\mathcal{O}}_X)$.

(i) Suppose the open set $X_f$ is affine. Then, for every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, every $i>0$, and every $\xi \in H^i(X,\mathcal{F})$, there exists an integer $n>0$ such that $f^n\xi =0$.

(ii) Conversely, suppose $X_f$ is quasi-compact and that for every quasi-coherent sheaf of ideals $\mathcal{J}$ of ${\mathcal{O}}_X$ and every $\xi \in H^1(X,\mathcal{J})$ there exists $n>0$ such that $f^n\xi =0$. Then $X_f$ is affine.

Proof. (i) If $X_f$ is affine, then $H^i(X_f,\mathcal{F})=0$ for every $i>0$ (1.3.1), so the assertion follows from (1.4.6).

(ii) By Serre's criterion (II, 5.2.1), it suffices to show that for every quasi-coherent sheaf of ideals ${\mathcal{J}}^{\prime }$ of ${\mathcal{O}}_{X_f}$ we have $H^1(X_f,{\mathcal{J}}^{\prime })=0$. Since $X_f$ is a quasi-compact open set in the quasi-compact scheme $X$, there is a quasi-coherent sheaf of ideals $\mathcal{J}$ of ${\mathcal{O}}_X$ such that ${\mathcal{J}}^{\prime }=\mathcal{J}\mid X_f$ (I, 9.4.2). By (1.4.6), $H^1(X_f,{\mathcal{J}}^{\prime })=(H^1(X,\mathcal{J}){)}_f$, and the hypothesis forces the right side to be zero, whence the conclusion.

Remark (1.4.8).

Note that (1.4.7, (i)) yields a simpler proof of the relation (II, 4.5.13.2).

Lemma (1.4.9).

Let $X$ be a quasi-compact scheme, $\mathfrak{U}=(U_i{)}_{1\le i\le n}$ a finite cover of $X$ by affine open sets, and $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-module. The complex of sheaves ${\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F})$ defined by the cover $\mathfrak{U}$ (G, II, 5.2) is then a quasi-coherent ${\mathcal{O}}_X$-module.

Proof. By the definitions (G, II, 5.2), ${\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F})$ is a direct sum of sheaves obtained as direct images of the $\mathcal{F}\mid U_{i_0,\cdots ,i_p}$ under the canonical injections $U_{i_0,\cdots ,i_p}\to X$. The hypothesis that $X$ is a scheme implies that these injections are affine morphisms (I, 5.5.6), so the ${\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F})$ are quasi-coherent (II, 1.2.6).

Proposition (1.4.10).

Let $u:X\to Y$ be a separated and quasi-compact morphism. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the $R^q{u}_{\ast }(\mathcal{F})$ are quasi-coherent ${\mathcal{O}}_Y$-modules.

Proof. The question being local on $Y$, we may suppose $Y$ affine. Then $X$ is a finite union of affine open sets $U_i$ ($1\le i\le n$); let $\mathfrak{U}=(U_i)$. Since $Y$ is a scheme, (I, 5.5.10) shows that for every affine open $V\subseteq Y$ the canonical injection $u^{-1}(V)\to X$ is an affine morphism; therefore (by (1.4.1) and (G, II, 5.2)) there is a canonical isomorphism

  H^•(u^{-1}(V), ℱ) ≅ H^•(Γ(V, 𝒢)),                                          (1.4.10.1)

where $\mathcal{G}=u_{\ast }({\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F}))$. By (1.4.9) and (I, 9.2.2), $\mathcal{G}$ is a quasi-coherent ${\mathcal{O}}_Y$-module; moreover, since ${\mathcal{C}}^{\bullet }(\mathfrak{U},\mathcal{F})$ is a complex of sheaves, so is $\mathcal{G}$. The definition of the cohomology ${\mathcal{H}}^{\bullet }(\mathcal{G})$ (G, II, 4.1) then shows that the latter consists of quasi-coherent ${\mathcal{O}}_Y$-modules (I, 4.1.1). Since, for $V$ affine in $Y$, the functor $\Gamma (V,\cdot )$ is exact in $\mathcal{G}$ on the category of quasi-coherent ${\mathcal{O}}_Y$-modules, (G, II, 4.1) gives

  H^•(Γ(V, 𝒢)) = Γ(V, 𝓗^•(𝒢)).                                               (1.4.10.2)

Finally, the definition of the canonical homomorphism $H^{\bullet }(\mathfrak{U},\mathcal{F})\to H^{\bullet }(X,\mathcal{F})$ given in (G, II, 5.2) shows that, if $V^{\prime }\subseteq V$ is a second affine open set in $Y$, the diagram

  H^•(u^{-1}(V), ℱ)   ≅   H^•(Γ(V, 𝒢))
        ↓                       ↓
  H^•(u^{-1}(V'), ℱ)  ≅   H^•(Γ(V', 𝒢))

is commutative. We conclude from the foregoing that the isomorphisms (1.4.10.1) define an isomorphism of ${\mathcal{O}}_Y$-modules

$$R^{\bullet }{u}_{\ast }(\mathcal{F})\cong {\mathcal{H}}^{\bullet }(\mathcal{G}),(\mathrm{1.4.10.3})$$

and hence $R^q{u}_{\ast }(\mathcal{F})$ is quasi-coherent. The relations (1.4.10.3), (1.4.10.2), (1.4.10.1) also yield:

Corollary (1.4.11).

Under the hypotheses of (1.4.10), for every affine open set $V$ of $Y$ the canonical homomorphism

  H^q(u^{-1}(V), ℱ) → Γ(V, R^q u_*(ℱ))                                       (1.4.11.1)

is an isomorphism for every $q\ge 0$.

Corollary (1.4.12).

Under the hypotheses of (1.4.10), suppose in addition that $Y$ is quasi-compact. Then there exists an integer $r>0$ such that for every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ and every integer $q>r$ we have $R^q{u}_{\ast }(\mathcal{F})=0$. If $Y$ is affine, one may take for $r$ any integer such that $X$ admits an affine open cover of cardinality $r$.

Proof. Since $Y$ can be covered by finitely many affine open sets, it suffices, by (1.4.11), to prove the second assertion. If $\mathfrak{U}$ is a cover of $X$ by $r$ affine open sets, then $C^q(\mathfrak{U},\mathcal{F})=0$ for $q>r$, since the Čech cochains in $C^q(\mathfrak{U},\mathcal{F})$ are alternating; the conclusion follows from (1.4.1).

Corollary (1.4.13).

Under the hypotheses of (1.4.10), suppose in addition that $Y=\mathrm{Spec}(A)$ is affine. Then, for every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ and every $f\in A$,

  Γ(Y_f, R^q u_*(ℱ)) = (Γ(Y, R^q u_*(ℱ)))_f

up to canonical isomorphism.

Proof. This follows from the fact that $R^q{u}_{\ast }(\mathcal{F})$ is a quasi-coherent ${\mathcal{O}}_Y$-module (I, 1.3.7).

Proposition (1.4.14).

Let $f:X\to Y$ be a separated quasi-compact morphism, and $g:Y\to Z$ an affine morphism. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the canonical homomorphism

  R^p(g ∘ f)_*(ℱ) → g_*(R^p f_*(ℱ))                                  (0, 12.2.5.2)

is bijective for every $p$.

Proof. For every affine open set $W$ of $Z$, $g^{-1}(W)$ is an affine open set of $Y$. The presheaf homomorphism defining the canonical homomorphism $R^p(g\circ f{)}_{\ast }(\mathcal{F})\to g_{\ast }(R^p{f}_{\ast }(\mathcal{F}))$ (0, 12.2.5) is therefore bijective by (1.4.11).

Proposition (1.4.15).

Let $u:X\to Y$ be a separated morphism of finite type, $v:Y^{\prime }\to Y$ a flat morphism of preschemes $(0_I,\mathrm{6.7.1})$. Set $u^{\prime }=u_{(Y^{\prime })}$, so that one has the commutative diagram

       X  ←  X' = X ×_Y Y'
       ↓        ↓                                                            (1.4.15.1)
       Y  ←  Y'

Then, for every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ and every $q\ge 0$, $R^q{u}_{\ast }^{\prime }({\mathcal{F}}^{\prime })$, where ${\mathcal{F}}^{\prime }=v^{\prime \ast }(\mathcal{F})=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_{X^{\prime }}$, is canonically isomorphic to $R^q{u}_{\ast }(\mathcal{F}){\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_{Y^{\prime }}=v^{\ast }(R^q{u}_{\ast }(\mathcal{F}))$.

Proof. The canonical homomorphism $\rho :\mathcal{F}\to v_{\ast }^{\prime }(v^{\prime \ast }(\mathcal{F}))$ $(0_I,\mathrm{4.4.3.2})$ defines by functoriality a homomorphism

  R^q u_*(ℱ) → R^q u_*(v'_*(ℱ')).                                            (1.4.15.2)

On the other hand, setting $w=u\circ v^{\prime }=v\circ u^{\prime }$, one has the canonical homomorphisms (0, 12.2.5.1) and (0, 12.2.5.2)

  R^q u_*(v'_*(ℱ')) → R^q w_*(ℱ') → v_*(R^q u'_*(ℱ')).                       (1.4.15.3)

Composing (1.4.15.3) and (1.4.15.2) gives a homomorphism

  ψ : R^q u_*(ℱ) → v_*(R^q u'_*(ℱ'))

and hence the canonical homomorphism (whose definition uses no hypothesis on $v$)

  Φ : v^*(R^q u_*(ℱ)) → R^q u'_*(ℱ'),                                        (1.4.15.4)

which one must show is an isomorphism when $v$ is flat. The question is local on $Y$ and $Y^{\prime }$, so we may suppose $Y=\mathrm{Spec}(A)$ and $Y^{\prime }=\mathrm{Spec}(B)$. We use the following lemma.

Lemma (1.4.15.5).

Let $\varphi :A\to B$ be a ring homomorphism, $Y=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(B)$, $f:X\to Y$ the morphism corresponding to $\varphi$, and $M$ a $B$-module. The ${\mathcal{O}}_X$-module $\tilde{M}$ is $f$-flat $(0_I,\mathrm{6.7.1})$ if and only if $M$ is a flat $A$-module. In particular, the morphism $f$ is flat if and only if $B$ is a flat $A$-module.

Proof. This follows from the definition $(0_I,\mathrm{6.7.1})$ and from $(0_I,\mathrm{6.3.3})$, taking (I, 1.3.4) into account.

That being so, by (1.4.11.1) and the definitions of the homomorphisms (1.4.15.3) (cf. 0, 12.2.5), $\Phi$ corresponds to the composite morphism

  H^q(X, ℱ) →^ρ H^q(X, v'_*(ℱ')) →^σ H^q(X', v'^*(v'_*(ℱ'))) →^θ H^q(X', ℱ'),

where $\rho$ and $\sigma$ are the homomorphisms corresponding in cohomology to the canonical morphisms $\rho$ and $\sigma :v^{\prime \ast }(v_{\ast }^{\prime }({\mathcal{F}}^{\prime }))\to {\mathcal{F}}^{\prime }$, and $\theta$ is the $\varphi$-morphism (0, 12.1.3.1) relative to the ${\mathcal{O}}_X$-module $v_{\ast }^{\prime }(v^{\prime \ast }(\mathcal{F}))$. By the functoriality of $\theta$, one has the commutative diagram

   H^q(X, ℱ)              →^{ρ_φ}              H^q(X, v'_*(v'^*(ℱ)))
        ↓                                          ↓
   H^q(X', v'^*(ℱ))                  H^q(X', v'^*(v'_*(v'^*(ℱ))))

and since by definition $(0_I,\mathrm{4.4.3})$ $v^{\prime \ast }(\rho )$ is the inverse of $\sigma$, the composite morphism considered above is none other than ${\theta }_{\varphi }$; thus $\Phi$ is the $B$-homomorphism associated to $H^q(X,\mathcal{F}){\otimes }_{A}B\to H^q(X^{\prime },{\mathcal{F}}^{\prime })$. Since $u$ is of finite type, $X$ is a finite union of affine open sets $U_i$ ($1\le i\le r$); let $\mathfrak{U}=(U_i)$. Moreover, $v$ is an affine morphism, hence so is $v^{\prime }$ (II, 1.6.2, (iii)), and the $U_i^{\prime }=v^{\prime -1}(U_i)$ form a finite affine open cover ${\mathfrak{U}}^{\prime }$ of $X^{\prime }$. We know (0, 12.1.4.2) that the diagram

   H^•(𝔘, ℱ)    →    H^•(𝔘', ℱ')
        ↓                  ↓
   H^•(X, ℱ)    →    H^•(X', ℱ')

is commutative, and the vertical arrows are isomorphisms, since $X$ and $X^{\prime }$ are schemes (1.4.1). It thus suffices to show that the canonical $\varphi$-morphism $\theta :H^{\bullet }(\mathfrak{U},\mathcal{F})\to H^{\bullet }({\mathfrak{U}}^{\prime },{\mathcal{F}}^{\prime })$ is such that the associated $B$-homomorphism $H^{\bullet }(\mathfrak{U},\mathcal{F}){\otimes }_{A}B\to H^{\bullet }({\mathfrak{U}}^{\prime },{\mathcal{F}}^{\prime })$ is an isomorphism. For every sequence $s=(i_h{)}_{0\le h\le p}$ of $p+1$ indices in [1, r], set $U_s={\bigcap }_h{U}_{i_h}$, $U_s^{\prime }={\bigcap }_h{U}_{i_h}^{\prime }=v^{\prime -1}(U_s)$, $M_s=\Gamma (U_s,\mathcal{F})$, $M_s^{\prime }=\Gamma (U_s^{\prime },{\mathcal{F}}^{\prime })$. The canonical map $M_s{\otimes }_{A}B\to M_s^{\prime }$ is an isomorphism (I, 1.6.5), so the canonical map $C^{\bullet }(\mathfrak{U},\mathcal{F}){\otimes }_{A}B\to C^{\bullet }({\mathfrak{U}}^{\prime },{\mathcal{F}}^{\prime })$ is an isomorphism, under which $d\otimes 1$ is identified with the coboundary operator $C^{\bullet }({\mathfrak{U}}^{\prime },{\mathcal{F}}^{\prime })\to C^{\bullet +1}({\mathfrak{U}}^{\prime },{\mathcal{F}}^{\prime })$. Since $B$ is a flat $A$-module, the definition of the cohomology modules immediately gives that the canonical map $H^{\bullet }(\mathfrak{U},\mathcal{F}){\otimes }_{A}B\to H^{\bullet }({\mathfrak{U}}^{\prime },{\mathcal{F}}^{\prime })$ is an isomorphism $(0_I,\mathrm{6.1.1})$. This result will be generalized later (§6).

Corollary (1.4.16).

Let $A$ be a ring, $X$ an $A$-scheme of finite type, $B$ an $A$-algebra faithfully flat over $A$. Then $X$ is affine if and only if $X{\otimes }_{A}B$ is affine.

Proof. The condition is evidently necessary (I, 3.2.2); we show it is sufficient. Since $X$ is separated over $A$ and the morphism $\mathrm{Spec}(B)\to \mathrm{Spec}(A)$ is flat, (1.4.15) gives

  H^i(X ⊗_A B, ℱ ⊗_A B) = H^i(X, ℱ) ⊗_A B                                    (1.4.16.1)

for every $i\ge 0$ and every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$. If $X{\otimes }_{A}B$ is affine, the left side of (1.4.16.1) is zero for $i=1$, hence so is $H^1(X,\mathcal{F})$, since $B$ is faithfully flat over $A$. As $X$ is a quasi-compact scheme, we conclude by Serre's criterion (II, 5.2.1).

Proposition (1.4.17).

Let $X$ be a prescheme and $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ an exact sequence of ${\mathcal{O}}_X$-modules. If ${\mathcal{F}}^{\prime }$ and ${\mathcal{F}}^{\prime \prime }$ are quasi-coherent, so is $\mathcal{F}$.

Proof. The question is local on $X$, so we may suppose $X=\mathrm{Spec}(A)$ affine; it then suffices to show that $\mathcal{F}$ satisfies the conditions $(d_1)$ and $(d_2)$ of (I, 1.4.1) (with $V=X$). The verification of $(d_1)$ is immediate: if $t\in \Gamma (X,\mathcal{F})$ has zero restriction to $D(f)$, then so does its image $v(t)\in \Gamma (X,{\mathcal{F}}^{\prime \prime })$; thus there exists $m>0$ such that $f^{m}v(t)=v(f^{m}t)=0$ (I, 1.4.1). Since ${\mathcal{F}}^{\prime }$ is exact at left, $f^{m}t=u(s)$ for some $s\in \Gamma (X,{\mathcal{F}}^{\prime })$; since $u$ is injective, the restriction of $s$ to $D(f)$ is zero, hence by (I, 1.4.1) there is $n>0$ with $f^{n}s=0$, and finally $f^{m+n}t=u(f^{n}s)=0$.

Now verify $(d_2)$. Let $t^{\prime }\in \Gamma (D(f),\mathcal{F})$. Since ${\mathcal{F}}^{\prime \prime }$ is quasi-coherent, there exists an integer $m$ such that $f^{m}v(t^{\prime })=v(f^m{t}^{\prime })$ extends to a section $t^{\prime \prime }\in \Gamma (X,{\mathcal{F}}^{\prime \prime })$ (I, 1.4.1). By (1.3.1) (or (I, 5.1.9.2)) applied to the quasi-coherent ${\mathcal{O}}_X$-module ${\mathcal{F}}^{\prime \prime }$, the sequence $\Gamma (X,\mathcal{F})\to \Gamma (X,{\mathcal{F}}^{\prime \prime })\to 0$ is exact, so there exists $\tau \in \Gamma (X,\mathcal{F})$ with $v(\tau )=t^{\prime \prime }$. We see that $v(f^m{t}^{\prime }-\tau \mid D(f))=0$; writing t''' for the restriction of $\tau$ to $D(f)$, we have $f^m{t}^{\prime }-t^{\prime \prime \prime }=u(s^{\prime })$ for some $s^{\prime }\in \Gamma (D(f),{\mathcal{F}}^{\prime })$. Since ${\mathcal{F}}^{\prime }$ is quasi-coherent, there exists $n>0$ such that $f^n{s}^{\prime }$ extends to a section $s\in \Gamma (X,{\mathcal{F}}^{\prime })$; then $f^{m+n}{t}^{\prime }-f^n{t}^{\prime \prime \prime }=u(f^n{s}^{\prime })$, so $f^{m+n}{t}^{\prime }$ is the restriction to $D(f)$ of the section $f^n\tau +u(f^{n}s)\in \Gamma (X,\mathcal{F})$, completing the proof.