§2. Cohomological study of projective morphisms

2.1. Explicit calculations of certain cohomology groups

2.1.1.

Let $X$ be a prescheme and $L$ an invertible $O_{X}$ -module; consider the graded ring $(0_{I}, 5.4.6)$

  S = Γ_*(X, ℒ) = ⊕_{n ∈ ℤ} Γ(X, ℒ^{⊗n}).                                    (2.1.1.1)

Let $(f_{i})_{1 \leq i \leq r}$ be a finite family of homogeneous elements of $S$ , with $f_{i} \in S_{d_{i}}$ ; set

  U_i = X_{f_i},   U = ⋃_i U_i,

and denote by $U$ the cover $(U_{i})$ of $U$ . For every quasi-coherent $O_{X}$ -module $F$ , set

  H^•(𝔘, ℱ(*)) = ⊕_{n ∈ ℤ} H^•(𝔘, ℱ ⊗ ℒ^{⊗n})                                (2.1.1.2)

  H^•(U, ℱ(*)) = ⊕_{n ∈ ℤ} H^•(U, ℱ ⊗ ℒ^{⊗n}).                               (2.1.1.3)

The abelian groups (2.1.1.2) and (2.1.1.3) are bigraded, by setting

  (H^•(𝔘, ℱ(*)))^{m,n} = H^m(𝔘, ℱ ⊗ ℒ^{⊗n})

and an analogous definition for (2.1.1.3). For the second degree, it is clear that these groups are graded $S$ -modules, as follows for instance from the fact that $F \mapsto H^{p} (U, F)$ and $F \mapsto H^{p} (U, F)$ are functors.

2.1.2.

Consider now the graded $S$ -module $(0_{I}, 5.4.6)$

  M = Γ_*(ℱ) = H^0(X, ℱ(*)) = ⊕_{n ∈ ℤ} Γ(X, ℱ ⊗ ℒ^{⊗n}).                    (2.1.2.1)

If $X$ is a prescheme whose underlying space is Noetherian, or a quasi-compact scheme, it follows from (I, 9.3.1) that, setting as usual $U_{i_{0} \dots i_{p}} = ⋂_{k = 0}^{p} U_{i_{k}}$ , we have, up to a canonical isomorphism,

  Γ(U_{i_0 … i_p}, ℱ(*)) = H^0(U_{i_0 … i_p}, ℱ(*)) = M_{f_{i_0} … f_{i_p}}.

One can also, with the notation of (1.2.2), identify $M_{f_{i_{0}} \dots f_{i_{p}}}$ with $lim M_{i_{0} \dots i_{p}}^{(n)}$ . This identification is an isomorphism of graded $S$ -modules, provided that one defines the degree of a homogeneous element $ξ \in lim M_{i_{0} \dots i_{p}}^{(n)}$ as follows: $ξ$ is the canonical image of a homogeneous element $x \in M_{i_{0} \dots i_{p}}^{(n)} = M$ of degree $m$ ; one then takes for the degree of $ξ$ the number $m - n (d_{i_{0}} + d_{i_{1}} + \dots + d_{i_{p}})$ . Taking the definition of the homomorphisms $ϕ_{nm} : M_{i_{0} \dots i_{p}}^{(n)} \to M_{i_{0} \dots i_{p}}^{(m)}$ (1.2.2) into account, one sees at once that this definition does not depend on the "representative" $x$ of $ξ$ chosen. Denoting, as in (1.2.2), by $C_{a}^{p} (M)$ the set of alternating maps from $[1, r]^{p + 1}$ to $M$ (for every $n$ ), one defines in the same way as above a structure of graded $S$ -module on $lim C_{a}^{p} (M)$ ; one has again as in (1.2.2)

  C^p(𝔘, ℱ(*)) = lim_→ C^p_a(M),                                              (2.1.2.2)

the isomorphism of the two sides preserving degrees. One then has, as in (1.2.2),

  C'^p(𝔘, ℱ(*)) = C^{p+1}_a(𝐟, M) = lim_→ K^{p+1}(𝐟^n, M),                    (2.1.2.3)

the isomorphism preserving degrees: the degree of an element of $lim K^{p + 1} (f^{n}, M)$ , the canonical image of a cochain $ζ \in K^{p + 1} (f^{n}, M)$ whose values $ζ (i_{0}, \dots, i_{p})$ lie in a single homogeneous component $M_{m}$ of $M$ , is $m - n (d_{i_{0}} + \dots + d_{i_{p}})$ , and is independent of the choice of this cochain as a representative of the element considered.

Since the preceding isomorphisms are compatible with the coboundary operators, we conclude, as in (1.2.2), the following.

Proposition (2.1.3).

Let $X$ be a prescheme whose underlying space is Noetherian, or a quasi-compact scheme. There exists a canonical isomorphism, functorial in $F$ ,

  H^p(𝔘, ℱ(*)) ⥲ H^{p+1}(𝐟, M)                              for every p ≥ 1.   (2.1.3.1)

Moreover, one has an exact sequence functorial in $F$

  0 → H^0(𝐟, M) → M → H^0(𝔘, ℱ(*)) → H^1(𝐟, M) → 0.                          (2.1.3.2)

Furthermore, all the homomorphisms introduced are of degree 0 for the graded $S$ -module structures ( $S$ being the ring (2.1.1.1)).

Corollary (2.1.4).

If $X$ is a quasi-compact scheme and the $U_{i} = X_{f_{i}}$ are affine, there exists a canonical isomorphism, functorial in $F$ , of degree 0,

  H^p(U, ℱ(*)) ⥲ H^{p+1}(𝐟, M)                              for p ≥ 1         (2.1.4.1)

and an exact sequence functorial in $F$

  0 → H^0(𝐟, M) → M → H^0(U, ℱ(*)) → H^1(𝐟, M) → 0                           (2.1.4.2)

where all homomorphisms are of degree 0.

Proof. It suffices to apply (1.4.1) to the result of (2.1.3).

The "local" proposition analogous to (2.1.3) is the following.

Proposition (2.1.5).

Let $S$ be a graded ring with positive degrees, $f_{i} (1 \leq i \leq r)$ a homogeneous element of $S_{+}$ of degree $d_{i}$ , $M$ a graded $S$ -module. Let $X = Proj (S)$ be the homogeneous prime spectrum of $S$ and set $U_{i} = D_{+} (f_{i})$ , $U = ⋃ U_{i}$ , $H^{∙} (U, \tilde{M} (*)) = \oplus_{n} H^{∙} (U, (M (n)) \sim)$ . There then exist canonical isomorphisms functorial in $M$ , of degree 0 for the graded $S$ -module structures,

  H^p(U, M̃(*)) ⥲ H^{p+1}(𝐟, M)                             for p ≥ 1         (2.1.5.1)

and an exact sequence functorial in $M$

  0 → H^0(𝐟, M) → M → H^0(U, M̃(*)) → H^1(𝐟, M) → 0                          (2.1.5.2)

where all homomorphisms are of degree 0.

Proof. We have $Γ (U_{i_{0} \dots i_{p}}, (M (n)) \sim) = (M_{f_{i_{0}} \dots f_{i_{p}}})_{n}$ by definition (II, 2.5.2), hence $Γ (U_{i_{0} \dots i_{p}}, \tilde{M} (*)) = M_{f_{i_{0}} \dots f_{i_{p}}}$ . The rest of the argument is then the same as for the proof of (2.1.4), taking into account that $X$ is a scheme.

Remarks (2.1.6).

(i) Under the conditions of (2.1.5), the functors $M \mapsto M_{f_{i_{0}} \dots f_{i_{p}}}$ are exact in $M$ , by virtue of $(0_{I}, 1.3.2)$ ; the same reasoning as in (1.2.5) then shows that if $0 \to M^{'} \to M \to M^{''} \to 0$ is an exact sequence of graded $S$ -modules (where the homomorphisms are of degree 0), one has commutative diagrams for every $p \geq 0$

  H^p(U, M̃''(*))    →    H^{p+1}(U, M̃'(*))

      ↓                          ↓

  H^{p+1}(𝐟, M'')   →    H^{p+2}(𝐟, M')                                      (2.1.6.1)

(ii) Proposition (2.1.5) will be especially interesting when $S$ is an $A$ -algebra generated by a finite number of elements of degree 1, $A$ being assumed Noetherian; for

then, every quasi-coherent $O_{X}$ -module is of the form $\tilde{M}$ (II, 2.7.7).

2.1.7.

We are going to apply (2.1.5) in the case $S = A [T_{0}, \dots, T_{r}]$ , where $A$ is an arbitrary ring, the $T_{i}$ are indeterminates, with $M = S$ and $f_{i} = T_{i}$ . One is therefore essentially reduced to computing $H^{∙} (T, S)$ , where $T = (T_{i})_{0 \leq i \leq r}$ .

Lemma (2.1.8).

If $S = A [T_{0}, \dots, T_{r}]$ , one has, with $T = (T_{i})_{0 \leq i \leq r}$ ,

  H^i(𝐓^n, S) = 0                                          if i ≠ r + 1      (2.1.8.1)

$H^{r + 1} (T^{n}, S) = S / (T^{n}) . (2.1.8.2)$

The $A$ -module $H^{r + 1} (T^{n}, S)$ thus has a basis over $A$ formed of the classes mod. $(T^{n})$ of the monomials $T^{p} = T_{0}^{p_{0}} \dots T_{r}^{p_{r}}$ with $p = (p_{0}, \dots, p_{r})$ , $0 \leq p_{i} < n$ for all $i$ .

Proof. This is an immediate consequence of (1.1.3.5) and Proposition (1.1.4), whose hypotheses are trivially verified.

2.1.9.

Pass to the inductive limit on $n$ ; the relations (2.1.8.1) give $H^{i} (T, S) = 0$ for $i \neq = r + 1$ . For $i = r + 1$ , the inductive system is formed of the $S / (T^{n})$ , the homomorphism $ϕ_{nm} : S / (T^{n}) \to S / (T^{m})$ for $0 \leq n \leq m$ being multiplication by $(T_{0} \dots T_{r})^{m - n}$ . For $n \geq sup (p_{i})_{0 \leq i \leq r}$ , denote by $ξ_{p_{0}, \dots, p_{r}}^{(n)}$ the class of $T_{0}^{n - p_{0}} \dots T_{r}^{n - p_{r}}$ mod. $(T^{n})$ ; one then has $ϕ_{nm} (ξ_{p}^{(n)}) = ξ_{p}^{(m)}$ , and these elements thus have the same canonical image $ξ_{p} = ξ_{p_{0}, \dots, p_{r}}$ in the inductive limit $H^{r + 1} (T, S)$ ; by virtue of the definition of degree given in (2.1.2), the degree of $ξ_{p}$ is therefore equal to $- ∣ p ∣ = - (p_{0} + p_{1} + \dots + p_{r})$ . It is clear that the $ξ_{p}^{(n)}$ for $0 < p_{i} \leq n$ and $0 \leq i \leq r$ form a basis of $S / (T^{n})$ . One immediately deduces from (2.1.8):

Corollary (2.1.10).

With the notation of (2.1.8), one has

  H^i(𝐓, S) = 0                                            for i ≠ r + 1     (2.1.10.1)

and $H^{r + 1} (T, S)$ is a free $A$ -module with a basis formed of the elements $ξ_{p_{0}, \dots, p_{r}}$ such that $p_{i} > 0$ for $0 \leq i \leq r$ .

Remark (2.1.11).

Let $N$ be an arbitrary $A$ -module and set $M = S \otimes_{A} N$ ; the reasoning of (2.1.8) shows that one has more generally

  H^i(𝐓^n, M) = 0                                          if i ≠ r + 1      (2.1.11.1)

  H^{r+1}(𝐓^n, M) = (S/(𝐓^n)) ⊗_A N,                                         (2.1.11.2)

since the latter formula follows directly from (1.1.3.5), and on the other hand it is clear that $M / (T_{0}^{n} M + \dots + T_{r - 1}^{n} M)$ is identified with the tensor product $(S / (T_{0}^{n} S + \dots + T_{r - 1}^{n} S)) \otimes_{A} N$ , the ideal $T_{0}^{n} S + \dots + T_{r - 1}^{n} S$ being a direct factor in the $A$ -module $S$ ; this allows one to apply (1.1.4) to $M$ , and one thus obtains (2.1.11.1).

Combining (2.1.10) and (2.1.5), one obtains:

Proposition (2.1.12).

Let $A$ be a ring, $r$ an integer > 0, and $X = P_{A}^{r}$ (II, 4.1.1). Then:

(i) One has $H^{i} (X, O_{X} (*)) = 0$ for $i \neq = 0, r$ .

(ii) The canonical homomorphism $α : S \to H^{0} (X, O_{X} (*))$ (II, 2.6.2) is bijective.

(iii) $H^{r} (X, O_{X} (*))$ is a free $A$ -module having a basis formed of elements $ξ_{p_{0}, \dots, p_{r}}$ , where $p_{i} > 0$ for $0 \leq i \leq r$ , $ξ_{p_{0}, \dots, p_{r}}$ being of degree $- ∣ p ∣ = - (p_{0} + \dots + p_{r})$ , and the product $T_{i} \cdot ξ_{p_{0}, \dots, p_{r}}$ being equal to $ξ_{p_{0}, \dots, p_{i} - 1, \dots, p_{r}}$ .

Proof. Note that, in the exact sequence (2.1.5.2) applied to $M = S = A [T_{0}, \dots, T_{r}]$ , one has $H^{0} (T, S) = 0$ and $H^{1} (T, S) = 0$ by (2.1.10.1), and that Proposition (2.1.5) applies to $U = X$ , since $X$ is the union of the $D_{+} (T_{i})$ (II, 2.3.14). It remains to identify the map $S \to H^{0} (X, O_{X} (*))$ of the exact sequence (2.1.5.1) with the canonical map $α$ ; but this follows from the canonical identification of $H^{0} (U, O_{X} (*))$ and $H^{0} (U, O_{X} (*))$ .

Corollary (2.1.13).

The only values of $(i, n)$ for which one can have $H^{i} (X, O_{X} (n)) \neq = 0$ are the following: $i = 0$ and $n \geq 0$ ; $i = r$ and $n \leq - (r + 1)$ .

Note that if $A \neq = 0$ , one effectively has $H^{i} (X, O_{X} (n)) \neq = 0$ for the pairs enumerated in (2.1.13); this follows from (2.1.12), since $S_{n}$ is then $\neq = 0$ for all degrees $n \geq 0$ .

In the applications which will be made in this chapter, we shall mostly use the less precise result:

Corollary (2.1.14).

The $A$ -modules $H^{i} (X, O_{X} (n))$ are free of finite type; if $i > 0$ , they are zero for $n > 0$ .

Proposition (2.1.15).

Let $Y$ be a prescheme, $E$ an $O_{Y}$ -module locally free of rank $r + 1$ , $X = P (E)$ the projective bundle defined by $E$ , $f : X \to Y$ the structure morphism. The only values of $i$ and $n$ for which $R^{i} f_{*} (O_{X} (n)) \neq = 0$ are $i = 0$ and $n \geq 0$ , $i = r$ and $n \leq - (r + 1)$ ; in addition, the canonical homomorphism (II, 3.3.2)

  α : 𝐒_{𝒪_Y}(ℰ) → 𝚪_*(𝒪_X) = R^0 f_*(𝒪_X(*)) = ⊕_{n ∈ ℤ} f_*(𝒪_X(n))

is an isomorphism.

Proof. The question being local on $Y$ , we may suppose $Y$ affine with ring $A$ and $E = \tilde{E}$ , where $E = A^{r + 1}$ ; one is then immediately reduced to (2.1.12), taking (1.4.11) into account.

Remark (2.1.16).

We shall later complete the results of (2.1.15) by proving the following propositions: set $ω = f^{*} (⋀^{r + 1} E) (- r - 1)$ , which is an invertible $O_{X}$ -module. Then:

(i) One has a canonical isomorphism

  ρ : R^r f_*(ω) ⥲ 𝒪_Y.                                                      (2.1.16.1)

(ii) The cup-product pairing (0, 12.2.2)

  R^r f_*(𝒪_X(n)) × R^0 f_*(ω(−n)) → R^r f_*(ω)                              (2.1.16.2)

composed with the isomorphism $ρ^{- 1}$ , defines an isomorphism of $R^{r} f_{*} (O_{X} (n))$ onto the dual of the locally free $O_{Y}$ -module

  R^0 f_*(ω(−n)) = (⋀^{r+1} ℰ) ⊗_{𝒪_Y} (𝐒_{𝒪_Y}(ℰ))_{−n}.

2.2. The fundamental theorem of projective morphisms

Theorem (2.2.1) (Serre).

Let $Y$ be a Noetherian prescheme, $f : X \to Y$ a proper morphism, $L$ an invertible $O_{X}$ -module ample for $f$ . For every $O_{X}$ -module $F$ , set $F (n) = F \otimes_{O_{X}} L^{\otimes n}$ for every $n \in Z$ . Then, for every coherent $O_{X}$ -module $F$ :

(i) The $R^{q} f_{*} (F)$ are coherent $O_{Y}$ -modules.

(ii) There exists an integer $N$ such that for $n \geq N$ , one has $R^{q} f_{*} (F (n)) = 0$ for every $q > 0$ .

(iii) There exists an integer $N$ such that, for $n \geq N$ , the canonical homomorphism $f^{*} (f_{*} (F (n))) \to F (n)$ is surjective.

Proof. Let us first note that if the theorem is true when one replaces $L$ by $L^{\otimes d}$ ( $d > 0$ ), it is true in its initial form. Indeed, one may then write $F (n) = (F \otimes L^{\otimes r}) \otimes L^{\otimes h d}$ with $h \geq 0$ and $0 \leq r < d$ , and by hypothesis, for each $r$ there is an integer $N_{r}$ such that for $h \geq N_{r}$ , properties (ii) and (iii) hold for the $O_{X}$ -module $F \otimes L^{\otimes r}$ ; taking for $N$ the largest of the $d N_{r}$ , (ii) and (iii) will hold for $n \geq N$ . One may therefore suppose $L$ very ample relative to $f$ (II, 4.6.11); consequently, there is a dominant $Y$ -open immersion $i : X \to P$ , where $P = Proj (S)$ , with $S$ a quasi-coherent graded $O_{Y}$ -algebra with positive degrees, in which $S_{1}$ is of finite type and generates $S$ ; in addition, $L$ is isomorphic to $i^{*} (O_{P} (1))$ (II, 4.4.7). But since $f$ is proper, so is $i$ (II, 5.4.4), so $i$ is an isomorphism $X \sim P$ . One may thus restrict to the case where $X = Proj (S)$ and $L = O_{X} (1)$ . Theorem (2.2.1) is then a consequence of the following.

Proposition (2.2.2).

Let $A$ be a Noetherian ring, $S$ an $A$ -algebra graded with positive degrees, in which S_1 is an $A$ -module having a system of $r + 1$ generators, and which generates the algebra $S$ . Let $X = Proj (S)$ . For every coherent $O_{X}$ -module $F$ :

(i) The $A$ -modules $H^{q} (X, F)$ are of finite type.

(ii) One has $H^{q} (X, F) = 0$ for $q > r$ .

(iii) There exists an integer $N$ such that for $n \geq N$ , one has $H^{q} (X, F (n)) = 0$ for every $q > 0$ .

(iv) There exists an integer $N$ such that for $n \geq N$ , $F (n)$ is generated by its sections over $X$ .

Proof. Let us first show how (2.2.2) entails (2.2.1): in (2.2.1) (reduced to the particular case $X = Proj (S)$ considered above), $Y$ is quasi-compact, so can be covered by a finite number of affine open sets, with Noetherian rings, such that the restriction of $S_{1}$ to each of these open sets $U_{α}$ is generated by a finite number of sections of $S_{1}$ over $U_{α}$ . Assuming (2.2.2) proved, it then suffices to take for $N$ in parts (ii) and (iii) of (2.2.1) the largest of the analogous integers corresponding to the $U_{α}$ (taking (1.4.11) and (II, 3.4.7) into account).

To prove (2.2.2), note that $X$ identifies with a closed subscheme of $P = P_{A}^{r}$ (II, 3.6.2); in addition, if $j : X \to P$ is the canonical injection, $j_{*} (F)$ is a coherent $O_{P}$ -module, and one has $j_{*} (F (n)) = (j_{*} (F)) (n)$ (II, 3.4.5 and 3.5.2). Taking (G, II, cor. of th. 4.9.1) into account, one is therefore reduced to proving (2.2.2) in the particular case where $X = P_{A}^{r}$ and $S = A [T_{0}, \dots, T_{r}]$ . As $X$ is covered by the affine opens

$D_{+} (T_{i})$ in number $r + 1$ , (ii) results from (1.4.12). Note on the other hand that (iv) has already been proved (II, 2.7.9).

We shall prove (i) and (iii) simultaneously. Note that these assertions are true for $F = O_{X} (m)$ (2.1.13); they are therefore also true when $F$ is a direct sum of a finite number of $O_{X}$ -modules of the form $O_{X} (m_{j})$ . On the other hand, (i) and (iii) are trivially true for $q > r$ by virtue of (ii). We shall proceed by descending induction on $q$ . We know that $F$ is isomorphic to a quotient of a direct sum $E$ of a finite number of sheaves $O_{X} (m_{j})$ (II, 2.7.10); in other words, one has an exact sequence $0 \to R \to E \to F \to 0$ , where $R$ is coherent $(0_{I}, 5.3.3)$ and $E$ satisfies (i) and (iii). Since $F \mapsto F (n)$ is an exact functor in $F$ , one also has the exact sequence

$0 \to R (n) \to E (n) \to F (n) \to 0$

for every $n \in Z$ . One deduces the exact cohomology sequence

  H^{q−1}(X, ℰ(n)) → H^{q−1}(X, ℱ(n)) → H^q(X, ℛ(n)).

Since $E (n)$ is a direct sum of the $O_{X} (n + m_{j})$ (II, 2.5.14), $H^{q - 1} (X, E (n))$ is of finite type, and so is $H^{q} (X, R (n))$ by the induction hypothesis; since $A$ is Noetherian, one concludes that $H^{q - 1} (X, F (n))$ is of finite type for every $n \in Z$ , and in particular for $n = 0$ . On the other hand, by the induction hypothesis, there exists an integer $N$ such that for $n \geq N$ one has $H^{q} (X, R (n)) = 0$ ; furthermore, one may also suppose $N$ chosen so that $H^{q - 1} (X, E (n)) = 0$ for $n \geq N$ , since $E$ satisfies (iii); one concludes that $H^{q - 1} (X, F (n)) = 0$ for $n \geq N$ , which completes the proof.

Corollary (2.2.3).

Under the hypotheses of (2.2.1), let $F \to G \to H$ be an exact sequence of coherent $O_{X}$ -modules. There then exists an integer $N$ such that for $n \geq N$ , the sequence

$f_{*} (F (n)) \to f_{*} (G (n)) \to f_{*} (H (n))$

is exact.

Proof. Let $F^{'}$ , $G^{'}$ , $G^{''}$ be the kernel, image, and cokernel of $F \to G$ ; $G^{'}$ is the kernel and $G^{''}$ the image of $G \to H$ , let $H^{''}$ be the cokernel of this homomorphism; all these $O_{X}$ -modules are coherent $(0_{I}, 5.3.4)$ . Since $F \mapsto F (n)$ is an exact functor in $F$ , it suffices to show that for $n$ large enough, each of the sequences

$0 \to f_{*} (F^{'} (n)) \to f_{*} (F (n)) \to f_{*} (G^{'} (n)) \to 00 \to f_{*} (G^{'} (n)) \to f_{*} (G (n)) \to f_{*} (G^{''} (n)) \to 00 \to f_{*} (G^{''} (n)) \to f_{*} (H (n)) \to f_{*} (H^{''} (n)) \to 0$

is exact; consequently, one may assume that $0 \to F \to G \to H \to 0$ is exact. One then has the exact cohomology sequence

  0 → f_*(ℱ(n)) → f_*(𝒢(n)) → f_*(ℋ(n)) → R^1 f_*(ℱ(n)) → ⋯

and the conclusion follows from (2.2.1, (ii)).

Corollary (2.2.4).

Let $Y$ be a Noetherian prescheme, $f : X \to Y$ a morphism of finite type, $L$ an invertible $O_{X}$ -module ample for $f$ ; for every $O_{X}$ -module $F$ , set $F (n) = F \otimes_{O_{X}} L^{\otimes n}$ (for $n \in Z$ ). Let $F \to G \to H$ be an exact sequence of coherent $O_{X}$ -modules,

such that the supports of $F$ and $H$ are proper over $Y$ (II, 5.4.10). There then exists an integer $N$ such that, for $n \geq N$ , the sequence

$f_{*} (F (n)) \to f_{*} (G (n)) \to f_{*} (H (n))$

is exact.

Proof. The same reasoning as at the beginning of (2.2.1) shows that if the corollary is true for $L^{\otimes d}$ ( $d > 0$ ), it is also true for $L$ ; one may therefore restrict to the case where $L$ is very ample for $f$ (II, 4.6.11), and consequently one may identify $X$ with an open set in a $Y$ -scheme $Z = Proj (S)$ , where $S$ is a quasi-coherent graded $O_{Y}$ -algebra with positive degrees, in which $S_{1}$ is of finite type and generates $S$ , so that $L = i^{*} (O_{Z} (1))$ , where $i$ is the canonical immersion $X \to Z$ (II, 4.4.7). This being so, as $S u pp (G)$ is closed in $X$ and contained in $S u pp (F) \cap S u pp (H)$ , it is proper over $Y$ ; the supports of $F$ , $G$ , $H$ are thus closed in $Z$ (II, 5.4.10). The sheaves $F^{'} = i_{*} (F)$ , $G^{'} = i_{*} (G)$ , $H^{'} = i_{*} (H)$ are therefore coherent $O_{Z}$ -modules, and the sequence $F^{'} \to G^{'} \to H^{'}$ is exact; in addition, if $g : Z \to Y$ is the structure morphism, one has $f = g \circ i$ , and it is clear that $F (n) = i^{*} (F^{'} (n))$ and similarly for $G$ and $H$ ; the conclusion thus follows from (2.2.3) applied to $F^{'}$ , $G^{'}$ , $H^{'}$ .

Remarks (2.2.5).

(i) Assertion (i) of (2.2.1) is still true when one supposes only that $Y$ is locally Noetherian; indeed, the property is evidently local on $Y$ ; on the other hand, the hypotheses of (2.2.1) imply that for every open $U \subset Y$ , the restriction of $f$ to $f^{- 1} (U)$ is a projective morphism $f^{- 1} (U) \to U$ (II, 5.5.5, (iii)) and $L ∣ f^{- 1} (U)$ is ample for this morphism (II, 4.6.4).

(ii) Assertion (iii) of (2.2.1) is still valid, as we have seen, when one supposes only that $X$ is a quasi-compact scheme or a prescheme whose underlying space is Noetherian, and $f : X \to Y$ a quasi-compact morphism (II, 4.6.8). But it should be noted that even when one supposes that $Y$ is the spectrum of a field $K$ and that $f$ is quasi-projective, assertion (ii) of (2.2.1) is no longer necessarily verified. For example, let $X^{'} = Spec (K [T_{0}, \dots, T_{r}])$ and let $X$ be the union of the affine opens $D (T_{i})$ of $X^{'}$ ( $0 \leq i \leq r$ ); since the immersion $X \to X^{'}$ is quasi-compact, the structure morphism $f : X \to Y$ is quasi-affine (II, 5.1.10), so $O_{X}$ is very ample for $f$ (II, 5.1.6). But the ring $Γ (X, O_{X})$ identifies with the intersection of the rings of fractions $(K [T_{0}, \dots, T_{r}])_{T_{i}}$ for $0 \leq i \leq r$ (I, 8.2.1.1), that is, with $K [T_{0}, \dots, T_{r}]$ . Consequently, it follows from formulas (1.4.3.1) and (1.1.3.5) that one has $H^{r} (X, O_{X}^{\otimes n}) = H^{r} (X, O_{X}) = A \neq = 0$ for every $n$ .

2.3. Application to graded sheaves of algebras and of modules

Theorem (2.3.1).

Let $Y$ be a Noetherian prescheme, $S$ a quasi-coherent graded $O_{Y}$ -algebra of finite type with positive degrees, $X = Proj (S)$ , $q : X \to Y$ the structure morphism, $M$ a quasi-coherent graded $S$ -module satisfying condition (TF). Then there exists an integer $N$ such that, for $n \geq N$ , the canonical homomorphism (II, 8.14.5.1)

  α_n : ℳ_n → q_*(𝒫roj(ℳ(n))) = q_*((𝒫roj(ℳ))_n)

is bijective. In other words, the canonical homomorphism

$α : M \to Γ_{*} (P ro j (M))$

is a (TN)-isomorphism.

Proof. One may restrict to the case where $M$ is an $S$ -module of finite type (II, 3.4.2).

As $Y$ is quasi-compact, there exists an integer $d > 0$ such that $S^{(d)}$ is generated by the quasi-coherent $O_{Y}$ -module $S_{d}$ , the latter being of finite type (II, 3.1.10), hence coherent since $Y$ is Noetherian. Note now that $M$ is a direct sum of the $M^{(d, k)}$ for $0 \leq k < d$ , and that each of the $M^{(d, k)}$ is a quasi-coherent $S^{(d)}$ -module of finite type, as follows from (II, 2.1.6, (iii)), the question being local on $Y$ . Now, it obviously suffices to prove that each of the canonical homomorphisms $α : M^{(d, k)} \to Γ_{*} (P ro j (M))^{(d, k)}$ is a (TN)-isomorphism. Taking (II, 8.14.13) into account (and notably the diagram (8.14.13.4)), one sees that one is reduced to proving the theorem when $S$ is generated by $S_{1}$ and $S_{1}$ is a coherent $O_{Y}$ -module. Since $Y$ is Noetherian, the same reasoning as at the beginning of (2.2.2) shows that one may restrict to the case where $Y = Spec (A)$ , $S = S$ , $M = M$ , $A$ being a Noetherian ring, S_1 an $A$ -module of finite type and $M$ a graded $S$ -module of finite type. Let us show that it then suffices to prove the theorem when $M = S$ . Indeed, in the general case, one has an exact sequence $L^{'} \to L \to M \to 0$ , where $L$ and $L^{'}$ are direct sums of graded modules of the form $S (m)$ . If the result is true for $M = S$ , it is also true for $M = S (m)$ , hence for $L$ and $L^{'}$ . Consider then the commutative diagram

$L_{n}^{'} \to L_{n} \to M_{n} \to 0 ↓ α_{n} ↓ α_{n} ↓ α_{n} q_{*} (L^{'} (n)) \to q_{*} (L (n)) \to q_{*} (M (n)) \to 0$

The second line is exact by virtue of (2.2.3) as soon as $n$ is large enough; as the same holds for the first, and as the two left vertical arrows are isomorphisms, so is the third.

This being so, to prove the theorem when $M = S$ , suppose first that $S = A [T_{0}, \dots, T_{r}]$ ( $T_{i}$ indeterminates); in this case, our assertion is none other than (2.1.11, (ii)). In the general case, $S$ identifies with a quotient of a ring $S^{'} = A [T_{0}, \dots, T_{r}]$ by a graded ideal, hence $X$ with a closed subscheme of $X^{'} = P_{A}^{r}$ (II, 2.9.2). If $j$ is the canonical injection $X \to X^{'}$ , $j_{*} (S (n))$ is none other than the $O_{X^{'}}$ -module $(P ro j (S)) (n)$ where $S$ is considered as a graded $S^{'}$ -module; this follows immediately from (II, 2.8.7). As $j_{*} (S (n))$ is an $O_{X^{'}}$ -module satisfying (TF), the canonical homomorphism $α_{n} : S_{n} \to Γ (X^{'}, j_{*} (S (n)))$ is bijective for $n$ large enough, by virtue of what precedes; this completes the proof, since $Γ (X^{'}, j_{*} (S (n))) = Γ (X, S (n))$ .

Corollary (2.3.2).

Under the hypotheses of (2.3.1), let $S_{X} = \oplus_{n \in Z} O_{X} (n)$ , and let $F$ be a quasi-coherent graded $S_{X}$ -module of finite type. Then $Γ_{*} (F)$ satisfies condition (TF).

Proof. We have seen in the proof of (2.3.1) that $X$ , which is isomorphic to $Proj (S^{(d)})$ (II, 3.1.8), is of finite type over $Y$ (II, 3.4.1). It then follows from (II, 8.14.9) that $F$ is isomorphic to an $O_{X}$ -graded module of the form $P ro j (M)$ , where $M$ is a quasi-coherent graded $S$ -module of finite type. By virtue of (2.3.1), $Γ_{*} (F)$ is (TN)-isomorphic to $M$ , and consequently satisfies (TF).

Scholium (2.3.3).

Let $Y$ be a Noetherian prescheme, $S$ an $O_{Y}$ -algebra graded satisfying the conditions of (2.3.1) and $X = Proj (S)$ . Let $K_{S}$ be the abelian category of quasi-coherent graded $S$ -modules satisfying (TF), $K_{S}^{'}$ the subcategory of $K_{S}$ formed of $S$ -modules satisfying (TN); finally, let $K_{X}$ be the category of quasi-coherent graded $S_{X}$ -modules of finite type $F$ (which amounts to saying, since $S_{X}$ is periodic (II, 8.14.4 and 8.14.12), that the $F_{i}$ are coherent $O_{X}$ -modules). Then the functors $M \mapsto P ro j (M)$ in $K_{S}$ and $F \mapsto Γ_{*} (F)$ in $K_{X}$ define, by virtue of (II, 8.14.8 and 8.14.10) and (2.3.2), an equivalence (T, I, 1.2) of the quotient category $K_{S} / K_{S}^{'}$ (T, I, 1.11) with the category $K_{X}$ . When $S$ is generated by $S_{1}$ , one may replace $K_{X}$ by the category of coherent $O_{X}$ -modules (II, 8.14.12).

Proposition (2.3.4).

Let $Y$ be a Noetherian prescheme.

(i) Let $S$ be a quasi-coherent graded $O_{Y}$ -algebra of finite type with positive degrees. Let $X = Proj (S)$ and $S_{X} = P ro j (S) = \oplus_{n \in Z} O_{X} (n)$ . Then $S_{X}$ is a periodic graded $O_{Y}$ -algebra (II, 8.14.12) whose homogeneous components $(S_{X})_{n} = O_{X} (n)$ are coherent $O_{X}$ -modules, and if $d > 0$ is a period of $S_{X}$ , $(S_{X})_{d} = O_{X} (d)$ is an invertible $O_{X}$ -module $Y$ -ample. In addition, the canonical homomorphism $α : S \to Γ_{*} (S_{X})$ is a (TN)-isomorphism.

(ii) Conversely, let $q : X \to Y$ be a projective morphism, and let $S^{'}$ be a graded $O_{X}$ -algebra whose homogeneous components $S_{n}^{'}$ ( $n \in Z$ ) are coherent $O_{X}$ -modules, and which admits a period $d > 0$ such that $S_{d}^{'}$ is an invertible $O_{X}$ -module ample for $q$ . Then $S = \oplus_{n \geq 0} q_{*} (S_{n}^{'})$ is a quasi-coherent graded $O_{Y}$ -algebra with positive degrees of finite type, and there exists a $Y$ -isomorphism $r : X \sim Proj (S)$ such that $r^{*} (P ro j (S))$ is isomorphic (as a graded $O_{X}$ -algebra) to $S^{'}$ .

Proof. (i) All the assertions have essentially already been proved, the last being none other than a particular case of (2.3.2). The fact that $S_{X}$ is periodic has been seen in (II, 8.14.14), and the fact that there is a period $d > 0$ such that $O_{X} (d)$ is invertible and $Y$ -ample is none other than (II, 4.6.18). Finally, for $0 \leq k < d$ , $(S_{X})^{(d, k)}$ is a $(S_{X})^{(d)}$ -module of finite type (II, 8.14.14), so each of the $(S_{X})_{n}$ is a quasi-coherent $S_{X}$ -module of finite type by virtue of (II, 2.1.6, (ii)), the question being local; as $S_{X}$ is coherent, so are the $(S_{X})_{n}$ .

(ii) Up to replacing the period $d$ by one of its multiples, one may suppose that $L = S_{d}^{'}$ is an $O_{X}$ -module very ample relative to $q$ (II, 4.6.11). We have in addition $S^{' (d)} = \oplus L^{\otimes n}$ by hypothesis, so $S^{(d)} = \oplus q_{*} (L^{\otimes n})$ ; we know (II, 3.1.8 and 3.2.9) that there is a $Y$ -isomorphism $s$ of $X^{'} = Proj (S)$ onto $X^{''} = Proj (S^{(d)})$ such that

$s^{*} (O_{X^{''}} (n)) = O_{X^{'}} (n d)$ . One will therefore establish the existence of a $Y$ -isomorphism $X \sim X^{'}$ if one proves the following.

Proposition (2.3.4.1).

Let $Y$ be a Noetherian prescheme, $q : X \to Y$ a projective morphism, $L$ an invertible $O_{X}$ -module very ample for $q$ . Then $S = \oplus_{n \geq 0} q_{*} (L^{\otimes n})$ is a quasi-coherent graded $O_{Y}$ -algebra of finite type, such that $S_{n} = S_{1}^{n}$ for $n$ large enough, and there exists a $Y$ -isomorphism $r : X \sim P = Proj (S)$ such that $L = r^{*} (O_{P} (1))$ .

Proof. As $q$ is a projective morphism, it follows from (II, 5.4.4 and 4.4.7) that there exists a $Y$ -isomorphism $r^{'} : X \sim P^{'} = Proj (T)$ , where $T$ is a quasi-coherent $O_{Y}$ -algebra such that $T_{1}$ is an $O_{Y}$ -module of finite type and generates $T$ , and one has $L = r^{' *} (O_{P^{'}} (1))$ . One then has $S = \oplus_{n \geq 0} q_{*}^{'} (O_{P^{'}} (n))$ , where $q^{'} : P^{'} \to Y$ is the structure morphism, and it follows from (2.3.1) that for $n$ large enough, the canonical homomorphism $α_{n} : T_{n} \to S_{n} = q_{*}^{'} (O_{P^{'}} (n))$ is bijective; as $T_{n} = T_{1}^{n}$ , one has a fortiori $S_{n} = S_{1}^{n}$ as soon as $n$ is large enough. In addition, as the canonical homomorphism $α : T \to S$ of graded $O_{Y}$ -algebras is a (TN)-isomorphism, Φ = Proj(α) : Proj(𝒮) → Proj(𝒯) is an isomorphism (II, 3.6.1) and one has $Φ_{*} (O_{P} (n)) = (O_{P^{'}} (n))_{[α]}$ (II, 3.5.2); but since the $T$ -graded modules $(S (n))_{[α]}$ and $T (n)$ are (TN)-isomorphic, one has $Φ_{*} (O_{P} (n)) = O_{P^{'}} (n)$ for every $n$ (II, 3.4.2); to complete the proof of (2.3.4.1), it remains to show that $S$ is an $O_{Y}$ -algebra of finite type; now the $S_{n} = q_{*}^{'} (O_{P^{'}} (n))$ are coherent $O_{Y}$ -modules by virtue of (2.2.1) and, since $S_{n} = S_{1}^{n}$ for $n \geq n_{0}$ , $S$ is generated by $\oplus_{i \leq n_{0}} S_{i}$ , which is coherent, whence our assertion (I, 9.6.2).

Let us return to the proof of (2.3.4), whose notation we resume. We have proved the existence of a $Y$ -isomorphism $r^{''} : X \sim X^{''}$ such that $r^{'' *} (L^{\otimes n}) = O_{X^{''}} (n d)$ for every $n \in Z$ ; we shall denote by q'' the structure morphism $X^{''} \to Y$ . Note now that $S^{'}$ is a direct sum of the $S^{' (d)}$ -graded modules $S^{' (d, k)}$ ; each of the latter is a quasi-coherent $S^{' (d)}$ -module of finite type, by virtue of the periodicity of $S^{'}$ and the hypothesis that the $S_{n}^{'}$ are $O_{X}$ -modules of finite type (II, 8.14.12). Set $F^{(k)} = r_{*}^{''} (S^{' (d, k)})$ , so the $F^{(k)}$ are quasi-coherent graded $S_{X^{''}}$ -modules of finite type; consequently (II, 8.14.8), the canonical homomorphism $β : P ro j (Γ_{*} (F^{(k)})) \to F^{(k)}$ is an isomorphism of $S_{X^{''}}$ -modules. But one has $q_{*}^{''} ((F^{(k)})_{n}) = q_{*} ((S^{' (d, k)})_{n})$ and for $n \geq 0$ , this last $O_{Y}$ -module is by definition equal to $(S^{(d, k)})_{n}$ . In other words, the canonical injection $S^{(d, k)} \to Γ_{*} (F^{(k)})$ is a (TN)-isomorphism, hence (II, 3.4.2) one has $P ro j (S^{(d, k)}) = P ro j (Γ_{*} (F^{(k)}))$ , and consequently $r^{'' *} (P ro j (S^{(d, k)})) = S^{' (d, k)}$ . It remains to note that $P ro j (S^{(d, k)}) = (P ro j (S))^{(d, k)}$ up to a canonical isomorphism (II, 8.14.13.1) in order to have proved the isomorphism of $r^{*} (P ro j (S))$ and $S^{'}$ . Finally, by virtue of (2.3.2), each of the $Γ_{*} (F^{(k)})$ satisfies condition (TF), so the same holds for each of the $S^{(d, k)}$ ; in addition, since the $S_{n}^{'}$ are coherent, the same holds for the $S_{n} = q_{*} (S_{n}^{'})$ by (2.2.1), and one concludes at once that the $S_{n}$ are $O_{Y}$ -modules of finite type. As we have seen in (2.3.4.1) that $S^{(d)}$ is an $O_{Y}$ -algebra of finite type, one concludes that $S$ is also an $O_{Y}$ -algebra of finite type.

Proposition (2.3.5).

Let $Y$ be an integral Noetherian prescheme, $X$ an integral prescheme, $f : X \to Y$ a birational projective morphism. There then exists a coherent fractional ideal sheaf $J \subset R (Y)$ (II, 8.1.2) such that $X$ is $Y$ -isomorphic to the prescheme obtained by blowing up $J$ (II, 8.1.3). In addition, there exists an open set $U$ of $Y$ such that the restriction of $f$ to $f^{- 1} (U)$ is an isomorphism of $f^{- 1} (U)$ onto $U$ (cf. (I, 6.5.5)), and such that $J ∣ U$ is invertible.

Proof. As there exists an invertible $O_{X}$ -module $L$ very ample for $f$ (II, 4.4.2 and 5.3.2), one may apply (2.3.4.1), and one sees that $X$ identifies with $Proj (S)$ , where $S = \oplus f_{*} (L^{\otimes n})$ . We know in addition that the $f_{*} (L^{\otimes n})$ are torsion-free $O_{Y}$ -modules (I, 7.4.5), so the same holds for the $O_{Y}$ -module $S$ , and consequently $S$ identifies canonically with a sub- $O_{Y}$ -module of $S \otimes_{O_{Y}} R (Y)$ (I, 7.4.1); the latter is a simple sheaf (I, 7.3.6), which is known when one knows its restriction to a nonempty open set, for instance to a nonempty open $U^{'} \subset U$ such that $L ∣ f^{- 1} (U^{'})$ is isomorphic to $O_{X} ∣ f^{- 1} (U^{'})$ . Since by hypothesis the $f_{*} (L^{\otimes n}) ∣ U^{'}$ are then isomorphic to $O_{Y} ∣ U^{'}$ , one sees that $S \otimes_{O_{Y}} R (Y)$ is an $R (Y)$ -module isomorphic to $R (Y) [T]$ , where $T$ is an indeterminate, and $S$ is (TN)-isomorphic to the sub- $O_{Y}$ -algebra generated by the canonical image of $f_{*} (L)$ in $S \otimes R (Y)$ (2.3.4.1); but if one identifies $S \otimes R (Y)$ with $R (Y) [T]$ , the image of $f_{*} (L)$ identifies with $J \cdot T$ , where $J$ is a coherent sub- $O_{Y}$ -module (2.2.1) of $R (Y)$ , whose restriction to $U^{'}$ is isomorphic to $O_{Y} ∣ U^{'}$ , and which consequently is such that $J ∣ U$ is invertible. One then sees that $S$ is (TN)-isomorphic to $\oplus J^{n}$ , which completes the proof.

Corollary (2.3.6).

Under the hypotheses of (2.3.5), suppose in addition that for every coherent sub- $O_{Y}$ -module $J \neq = 0$ of $R (Y)$ , there exists an invertible $O_{Y}$ -module $L$ such that $Γ (Y, L \otimes_{O_{Y}} H o m (J, O_{Y})) \neq = 0$ ; then, in the statement of (2.3.5), one may suppose that $J$ is an ideal of $O_{Y}$ . This additional condition is always satisfied if there exists an ample $O_{Y}$ -module.

Proof. Indeed, one has $(0_{I}, 5.4.2)$

  ℒ ⊗ ℋom(𝒥, 𝒪_Y) = ℋom(ℒ^{−1}, ℋom(𝒥, 𝒪_Y)) = ℋom(𝒥 ⊗ ℒ^{−1}, 𝒪_Y);

the hypothesis thus signifies that there is a nonzero homomorphism $u$ of $J \otimes L^{- 1}$ into $O_{Y}$ . As, for every $y \in Y$ , $(J \otimes L^{- 1})_{y}$ identifies with a sub- $O_{y}$ -module of the field of fractions $(R (Y))_{y}$ of $O_{y}$ (I, 7.1.5), $u_{y}$ is necessarily injective, so $u$ is an isomorphism of $J \otimes L^{- 1}$ onto an ideal $J^{'}$ of $O_{Y}$ . But since $Proj (\oplus_{n \geq 0} J^{n})$ and $Proj (\oplus_{n \geq 0} (J \otimes L^{- 1})^{n})$ are $Y$ -isomorphic (II, 3.1.8), this proves the first assertion of the corollary. To prove the second, note that $F = H o m_{O_{Y}} (J, O_{Y})$ is coherent and $\neq = 0$ , since there exists an open set $U$ of $Y$ such that $J ∣ U$ is invertible. If $L$ is an ample $O_{Y}$ -module, there exists an integer $n$ such that $F (n) = F \otimes L^{\otimes n}$ is generated by its sections over $Y$ (II, 4.5.5); a fortiori, one has $Γ (Y, F (n)) \neq = 0$ , whence the conclusion.

Corollary (2.3.7).

Let $X$ and $Y$ be two integral schemes, projective over a field $k$ , and let $f : X \to Y$ be a birational $k$ -morphism. Then $X$ is $k$ -isomorphic to a $k$ -scheme obtained by blowing up a closed subscheme $Y^{'}$ (not necessarily reduced) of $Y$ .

Proof. Indeed, $f$ is projective (II, 5.5.5, (v)), and as $Y$ is projective over $k$ , the additional condition of (2.3.6) is satisfied; it then suffices to consider the closed subscheme $Y^{'}$ of $Y$ defined by the coherent ideal $J$ of cor. (2.3.6).

Remark (2.3.8).

In Chap. IV, in studying the notion of divisor, we shall see that if, in the statement of (2.3.5), one supposes that the rings $O_{y}$ ( $y \in Y$ ) are factorial (which is the case for instance if $Y$ is non-singular), then $X$ may be deduced from $Y$ by blowing up a closed sub-prescheme $Y^{'}$ of $Y$ whose underlying space is contained in $Y - U$ .

2.4. A generalization of the fundamental theorem

Theorem (2.4.1).

Let $Y$ be a Noetherian prescheme, $S$ a quasi-coherent $O_{Y}$ -algebra of finite type. Let $f : X \to Y$ be a projective morphism, $S^{'} = f^{*} (S)$ , $M$ a quasi-coherent $S^{'}$ -module of finite type. Then:

(i) For every $p \in Z$ , $R^{p} f_{*} (M)$ is an $S$ -module of finite type.

(ii) Let in addition $L$ be an invertible $O_{X}$ -module ample for $f$ , and set $M (n) = M \otimes L^{\otimes n}$ for every $n \in Z$ . There exists an integer $N$ such that, for $n \geq N$ , one has

$R^{p} f_{*} (M (n)) = 0 (2.4.1.1)$

for every $p > 0$ , and the canonical homomorphism $f^{*} (f_{*} (M (n))) \to M (n)$ $(0_{I}, 4.4.3)$ is surjective.

Proof. Set $Y^{'} = Spec (S)$ , $X^{'} = Spec (S^{'})$ , so that $X^{'} = X \times_{Y} Y^{'}$ (II, 1.5.5); let $g : Y^{'} \to Y$ , $g^{'} : X^{'} \to X$ be the structure morphisms, which are affine by definition, and $f^{'} = f_{(Y^{'})} : X^{'} \to Y^{'}$ ; one therefore has a commutative diagram

        g'
   X ←──── X'
   |       |
 f |  ↘ h  | f'
   ↓       ↓
   Y ←──── Y'
        g

and the morphism $f^{'}$ is projective (II, 5.5.5, (iii)); set $h = f \circ g^{'} = g \circ f^{'}$ .

(i) Let $M$ be the $O_{X^{'}}$ -module associated to the quasi-coherent $S^{'}$ -module $M$ , when $X^{'}$ is considered as an affine $X$ -scheme (II, 1.4.3), so that one has $M = g_{*}^{'} (M)$ ; as $M$ is an $S^{'}$ -module of finite type, $M$ is an $O_{X^{'}}$ -module of finite type (II, 1.4.5); as $h$ is of finite type, since $g$ and $f^{'}$ are (II, 1.3.7 and I, 6.3.4, (ii)), $X^{'}$ is Noetherian (I, 6.3.7) and $M$ is consequently coherent. This being so, as $g^{'}$ is affine, the canonical homomorphism $R^{p} f_{*} (M) \to R^{p} h_{*} (\tilde{M})$ is bijective (1.3.4). In addition, this homomorphism is a homomorphism of $S$ -modules; indeed, from the canonical homomorphism

  g^*(𝒮) ⊗_{𝒪_{X'}} g'^*(ℳ) → g'^*(ℳ)                                       (2.4.1.2)

which defines the $S^{'}$ -module structure of $M$ (recalling that $S^{'} = g^{' *} (O_{X^{'}})$ ), one canonically deduces a homomorphism

  f_*(g^*(𝒮)) ⊗ R^p f_*(g'^*(ℳ)) → R^p f_*(g'^*(ℳ))

(0, 12.2.2), and since (2.4.1.2) itself comes (by application of $(0_{I}, 4.2.2.1)$ ) from the homomorphism $O_{X^{'}} \otimes M \to M$ defining the $O_{X^{'}}$ -module structure of $\tilde{M}$ , the diagram

  f_*(g'_*(𝒪_{X'})) ⊗ R^p f_*(g'_*(ℳ̃))    →    R^p f_*(g'_*(ℳ̃))

           ↓                                            ↓

  h_*(𝒪_{X'}) ⊗ R^p h_*(ℳ̃)                →    R^p h_*(ℳ̃)

is commutative (0, 12.2.6); composing the horizontal arrows with the homomorphism coming from the canonical homomorphism $S \to f_{*} (f^{*} (S)) = f_{*} (S^{'}) = f_{*} (g_{*}^{'} (O_{X^{'}})) = h_{*} (O_{X^{'}})$ , one obtains our assertion. On the other hand, since $g$ is affine and $f^{'}$ is separated and quasi-compact, the canonical homomorphism $R^{p} h_{*} (M) \to g_{*} (R^{p} f_{*}^{'} (M))$ is bijective (1.4.14), and one shows as above that it is an isomorphism of $S$ -modules (this time using the commutativity of (0, 12.2.6.2)). Now, since $f^{'}$ is projective and $M$ is coherent, $R^{p} f_{*}^{'} (M)$ is a coherent $O_{Y^{'}}$ -module by virtue of (2.2.1); one concludes that $g_{*} (R^{p} f_{*}^{'} (\tilde{M}))$ is an $S$ -module of finite type (II, 1.4.5).

(ii) Let $L^{'} = g^{' *} (L)$ , which is an invertible $O_{X^{'}}$ -module; for every $n \in Z$ , one has $g_{*}^{'} (M \otimes L^{' \otimes n}) = g_{*}^{'} (M) \otimes L^{\otimes n} = M (n)$ $(0_{I}, 5.4.10)$ up to an isomorphism; one may apply to $M \otimes L^{' \otimes n}$ the reasoning made in (i) for $M$ , which proves that $R^{p} f_{*} (M (n))$ is isomorphic to $g_{*} (R^{p} f_{*}^{'} (M \otimes L^{' \otimes n}))$ . Now $L^{'}$ is ample for $f^{'}$ (II, 4.6.13, (iii)), so it follows from (2.2.1) that there exists an integer $N$ such that $R^{p} f_{*}^{'} (M \otimes L^{' \otimes n}) = 0$ for every $p$ and every $n \geq N$ , which proves (2.4.1.1). Finally, it follows again from (2.2.1) that one may suppose $N$ chosen so that for $n \geq N$ , the canonical homomorphism $f^{' *} (f_{*}^{'} (M \otimes L^{' \otimes n})) \to M \otimes L^{' \otimes n}$ is surjective; as $g_{*}^{'}$ is an exact functor (II, 1.4.4), the corresponding homomorphism

  g'_*(f'^*(f'_*(ℳ̃ ⊗ ℒ'^{⊗n}))) → g'_*(ℳ̃ ⊗ ℒ'^{⊗n}) = ℳ(n)

is surjective. Now, one has $g_{*}^{'} (f^{' *} (f_{*}^{'} (M \otimes L^{' \otimes n}))) = f^{*} (g_{*} (f_{*}^{'} (M \otimes L^{' \otimes n})))$ (II, 1.5.2) and since $g \circ f^{'} = f \circ g^{'}$ , one finally sees that one has

  g'_*(f'^*(f'_*(ℳ̃ ⊗ ℒ'^{⊗n}))) = f^*(f_*(g'_*(ℳ̃ ⊗ ℒ'^{⊗n}))) = f^*(f_*(ℳ(n))),

which completes the proof.

2.4.2.

We shall in particular have to apply (2.4.1) when $S$ is an $O_{Y}$ -graded algebra with positive degrees, $M = \oplus_{k \in Z} M_{k}$ an $S^{'}$ -graded module. Then (with the

same hypotheses of finiteness on $S$ and $M$ ) one concludes from (2.4.1) that $\oplus_{k \in Z} R^{p} f_{*} (M_{k})$ is an $S$ -module of finite type for every $p$ , and (under the additional hypotheses of (2.4.1, (ii))) that there exists $N$ such that for $n \geq N$ , one has $R^{p} f_{*} (M_{k} (n)) = 0$ for every $p > 0$ and every $k \in Z$ , and that the canonical homomorphism $f^{*} (f_{*} (M_{k} (n))) \to M_{k} (n)$ is surjective for every $k \in Z$ .

2.5. Euler–Poincaré characteristic and the Hilbert polynomial

2.5.1.

Let $A$ be an Artinian ring, $X$ an $A$ -scheme projective over $Y = Spec (A)$ . For every coherent $O_{X}$ -module $F$ , the $H^{i} (X, F)$ ( $i \geq 0$ ) are $A$ -modules of finite type (2.2.1), hence here of finite length since $A$ is Artinian. One knows in addition (2.2.1) that $H^{i} (X, F) = 0$ except for a finite number of values of $i \geq 0$ ; the integer

  χ_A(ℱ) = Σ_{i=0}^∞ (−1)^i long(H^i(X, ℱ))                                  (2.5.1.1)

is thus defined for every coherent $O_{X}$ -module $F$ . When $A$ is an Artinian local ring, one says that $χ_{A} (F)$ is the Euler–Poincaré characteristic of $F$ (with respect to the ring $A$ ). For $F = O_{X}$ , one says that $χ_{A} (O_{X})$ is the arithmetic genus of $X$ (with respect to $A$ ).

Proposition (2.5.2).

Let $0 \to F^{'} \to F \to F^{''} \to 0$ be an exact sequence of coherent $O_{X}$ -modules; one then has

$χ_{A} (F) = χ_{A} (F^{'}) + χ_{A} (F^{''}) . (2.5.2.1)$

Proof. As the cohomology modules of $F^{'}$ , $F$ , $F^{''}$ are zero except for a finite number of them, there is an integer $r > 0$ such that the exact cohomology sequence is written

  0 → H^0(X, ℱ') → H^0(X, ℱ) → H^0(X, ℱ'') → H^1(X, ℱ') → ⋯
       ⋯ → H^r(X, ℱ') → H^r(X, ℱ) → H^r(X, ℱ'') → 0.

Now, we know that in an exact sequence of $A$ -modules of finite length, with 0 at both ends, the alternating sum of the lengths is zero (0, 11.10.1); applying this result, one immediately finds the formula (2.5.2.1).

One notes that the result of (2.5.2) applies whenever one knows that $X$ is a quasi-compact $A$ -scheme and that the $A$ -modules $H^{i} (X, F)$ are of finite type for every coherent $O_{X}$ -module $F$ (1.4.12).

Theorem (2.5.3).

Let $A$ be an Artinian local ring, $X$ a scheme projective over $Y = Spec (A)$ , $L$ an invertible $O_{X}$ -module very ample relative to $Y$ , $F$ a coherent $O_{X}$ -module; set $F (n) = F \otimes_{O_{X}} L^{\otimes n}$ for every $n \in Z$ .

(i) There exists a unique polynomial $P \in Q [T]$ such that $χ_{A} (F (n)) = P (n)$ for every $n \in Z$ (one says that $P$ is the Hilbert polynomial of $F$ with respect to $A$ ).

(ii) For $n$ large enough, one has $χ_{A} (F (n)) = l o n g_{A} Γ (X, F (n))$ .

(iii) The leading coefficient of $χ_{A} (F (n))$ is $\geq 0$ .

Let us add that in Chap. IV, in the paragraph devoted to the notion of dimension, we shall in addition show that the degree of $χ_{A} (F (n))$ is equal to the dimension of the support of $F$ .

Proof. As one has $H^{i} (X, F (n)) = 0$ for every $i > 0$ as soon as $n$ is large enough (2.2.1),

one has $χ_{A} (F (n)) = l o n g H^{0} (X, F (n)) = l o n g Γ (X, F (n))$ for $n$ large enough, whence (ii); this implies $χ_{A} (F (n)) \geq 0$ for $n$ large enough, and (iii) thus follows from (i); as moreover the assertion of uniqueness in (i) is immediate, it remains to prove the existence of the polynomial $P$ .

Let us first show that one may suppose $m F = 0$ , where $m$ is the maximal ideal of $A$ . Indeed, there exists an integer $s > 0$ such that $m^{s} = 0$ , and $F (n)$ thus admits a finite filtration

  ℱ(n) ⊃ 𝔪 ℱ(n) ⊃ ⋯ ⊃ 𝔪^{s−1} ℱ(n) ⊃ 0.

By induction, one deduces from (2.5.2.1) that

  χ_A(ℱ(n)) = Σ_{k=1}^s χ_A(𝔪^{k−1} ℱ(n) / 𝔪^k ℱ(n));

since $m^{k - 1} F (n) / m^{k} F (n) = F_{k}^{'} (n)$ , where $F_{k}^{'} = m^{k - 1} F / m^{k} F$ , this proves our assertion.

So suppose $m F = 0$ ; if $X^{'}$ is the closed subscheme of $X$ , inverse image by the structure morphism $X \to Spec (A)$ of the unique closed point of $Spec (A)$ , and $j : X^{'} \to X$ the canonical injection, one has $F = j_{*} (F^{'})$ , where $F^{'}$ is a coherent $O_{X^{'}}$ -module; $X^{'}$ is a scheme projective over $Spec (K)$ , where $K = A / m$ . If $L^{'} = j^{*} (L)$ , $L^{'}$ is very ample relative to $Spec (K)$ (II, 4.4.10), and one has $F (n) = j_{*} (F^{'} (n))$ , where $F^{'} (n) = F^{'} \otimes_{O_{X^{'}}} L^{' \otimes n}$ $(0_{I}, 5.4.10)$ . One concludes that $χ_{A} (F (n)) = χ_{K} (F^{'} (n))$ (G, II, 4.9.1), and one is thus reduced to the case where $A$ is a field.

Note now that $X$ can be considered as a closed subscheme of $P = P_{A}^{r}$ for a suitable $r$ (II, 5.5.4, (ii)); if $i : X \to P$ is the canonical injection, one sees as above that one has $χ_{A} (F (n)) = χ_{A} (i_{*} (F) (n))$ , so that one may restrict to the case where $X = P_{A}^{r} = Proj (S)$ with $S = A [T_{0}, \dots, T_{r}]$ , $A$ being a field.

This being so, one has $F = \tilde{M}$ , where $M$ is a graded $S$ -module of finite type (II, 2.7.8); there exists consequently a finite resolution of $M$ by graded free $S$ -modules of finite type

  0 → L_q → L_{q−1} → ⋯ → L_1 → M → 0

by virtue of Hilbert's syzygy theorem (M, VIII, 6.5); as $M \mapsto \tilde{M}$ is an exact functor in $M$ (II, 2.5.4), one also has an exact sequence

  0 → L̃_q → L̃_{q−1} → ⋯ → L̃_1 → M̃ → 0

and consequently, for every $n \in Z$ , the sequence

  0 → L̃_q(n) → L̃_{q−1}(n) → ⋯ → L̃_1(n) → M̃(n) → 0

is exact; applying by induction on $q$ Proposition (2.5.1), it comes

  χ_A(M̃(n)) = Σ_{i=1}^q (−1)^{i+1} χ_A(L̃_i(n))

and to prove (i), one is therefore reduced to the case where $M$ is free and graded of finite type, hence to the case where $M = S (h)$ for an $h \in Z$ . As we then have $\tilde{M} (n) = (M (n)) \sim= (S (n + h)) \sim$ (II, 2.5.15), one finally sees that the theorem will follow from the following.

Lemma (2.5.3.1).

Let $A$ be a field, $r$ an integer > 0, and $X = P_{A}^{r}$ ; one then has $χ_{A} (O_{X} (n)) = (n + rc h ooser)$ for every $n \in Z$ .

Proof. Indeed, for $n \geq 0$ , one has $χ_{A} (O_{X} (n)) = l o n g H^{0} (X, O_{X} (n))$ , which is the number of monomials in the $T_{i}$ of total degree $n$ , that is, $(n + rc h ooser)$ (2.1.12). For $n \leq - r - 1$ , one has similarly $χ_{A} (O_{X} (n)) = (- 1)^{r} l o n g H^{r} (X, O_{X} (n))$ ; if $n = - r - h$ , the dimension of $H^{r} (X, O_{X} (n))$ over $A$ is the number of sequences $(p_{i})_{0 \leq i \leq r}$ of integers $p_{i} > 0$ such that $Σ_{i = 0}^{r} p_{i} = r + h$ (2.1.12), or equivalently the number of sequences of integers $q_{i} \geq 0$ ( $0 \leq i \leq r$ ) such that $Σ_{i = 0}^{r} q_{i} = h - 1$ ; this is therefore the number $(h - 1 + rc h ooser) = (- 1)^{r} (n + rc h ooser)$ . Finally, for $- r \leq n \leq 0$ , one has $(n + rc h ooser) = 0$ and on the other hand $H^{i} (X, O_{X} (n)) = 0$ for every $i \geq 0$ (2.1.12), which proves the lemma.

Corollary (2.5.4).

Let $A$ be an Artinian local ring, $S$ a graded $A$ -algebra of finite type generated by S_1, $M$ a graded $S$ -module of finite type, $X = Proj (S)$ . One then has $χ_{A} (\tilde{M} (n)) = l o n g M_{n}$ for $n$ large enough.

Proof. This follows from the fact that $M_{n}$ and $Γ (X, \tilde{M} (n))$ are isomorphic for $n$ large enough (2.3.1).

2.6. Application: ampleness criteria

Proposition (2.6.1).

Let $Y$ be a Noetherian prescheme, $f : X \to Y$ a proper morphism, $L$ an invertible $O_{X}$ -module. The following conditions are equivalent:

a) $L$ is ample for $f$ .

b) For every coherent $O_{X}$ -module $F$ , there exists an integer $N$ such that for $n \geq N$ , one has $R^{q} f_{*} (F \otimes L^{\otimes n}) = 0$ for every $q > 0$ .

c) For every coherent ideal sheaf $J$ of $O_{X}$ , there exists an integer $N$ such that for $n \geq N$ one has $R^{1} f_{*} (J \otimes L^{\otimes n}) = 0$ .

Proof. We have seen that a) entails b) (2.2.1, (ii)). It is trivial that b) entails c), and it remains to prove that c) implies a). One may restrict to the case where $Y$ is affine (II, 4.6.4), and prove in this case that $L$ is ample; it will suffice to show that as $h$ runs over the set of sections of the $L^{\otimes n}$ ( $n > 0$ ) over $X$ , those of the $X_{h}$ which are affine form a cover of $X$ (II, 4.5.2). For this, let us show that for every closed point $x$ of $X$ and every affine open neighbourhood $U$ of $x$ , there exist an $n$ and an $h \in Γ (X, L^{\otimes n})$ such that $x \in X_{h} \subset U$ ; $X_{h}$ will necessarily be affine (I, 1.3.6) and the union of these $X_{h}$ will be an open set of $X$ containing all the closed points of $X$ , and consequently $X$ itself, since $X$ is Noetherian $(I, 6.3.7 an d 0_{I}, 2.1.3)$ . Let $J$ (resp. $J^{'}$ ) be the quasi-coherent ideal sheaf of $O_{X}$ defining the closed reduced subprescheme of $X$ having for underlying space $X - U$ (resp. $(X - U) \cup x$ ) (I, 5.2.1); it is clear that $J$ and $J^{'}$ are coherent (I, 6.1.1), that $J^{'} \subset J$ , and that $J^{''} = J / J^{'}$ is a coherent $O_{X}$ -module $(0_{I}, 5.3.3)$ with support $x$ and such that $J_{x}^{''} = κ (x)$ . As $L$ is locally free, the sequence $0 \to J^{'} \otimes L^{\otimes n} \to J \otimes L^{\otimes n} \to J^{''} \otimes L^{\otimes n} \to 0$ is exact for every $n$ , and by hypothesis there exists $n$ large enough such that $H^{1} (X, J^{'} \otimes L^{\otimes n}) = 0$ ; the exact cohomology sequence

therefore proves that the homomorphism $Γ (X, J \otimes L^{\otimes n}) \to Γ (X, J^{''} \otimes L^{\otimes n})$ is surjective. A section $g$ of $J^{''} \otimes L^{\otimes n}$ over $X$ such that $g (x) \neq = 0$ is therefore the image of a section $h \in Γ (X, J \otimes L^{\otimes n}) \subset Γ (X, L^{\otimes n})$ (since by virtue of $(0_{I}, 5.4.1)$ , $J \otimes L^{\otimes n}$ is a sub- $O_{X}$ -module of $L^{\otimes n}$ ); one has by definition $h (x) \neq = 0$ and $h (z) = 0$ for $z \in / U$ , which completes the proof.

Proposition (2.6.2).

Let $Y$ be a Noetherian prescheme, $f : X \to Y$ a morphism of finite type, $g : X^{'} \to X$ a finite surjective morphism, $L$ an invertible $O_{X}$ -module and $L^{'} = g^{*} (L)$ . Suppose the following condition is satisfied: there exists a subset $Z$ of $X$ , proper over $Y$ (II, 5.4.10), such that for every $x \in X - Z$ , either $X$ is normal at the point $x$ , or $(g_{*} (O_{X^{'}}))_{x}$ is a free $O_{x}$ -module. Under these conditions, for $L$ to be ample for $f$ , it is necessary and sufficient that $L^{'}$ be ample for $f \circ g$ .

Proof.

(2.6.2.1)

Since $g$ is affine, the condition is necessary (II, 5.1.12). To see that it is sufficient, one may suppose $Y$ affine (II, 4.6.4). Let us further show that one may restrict to the case where $X$ is reduced. Indeed, let $j : X_{re d} \to X$ be the canonical injection, and set $X_{1} = X_{re d}$ , $X_{1}^{'} = X^{'} \times_{X} X_{1}$ , so that one has the commutative diagram

         j'
    X' ←───── X'_1
    |          |
  g |          | g_1
    ↓          ↓
    X  ←───── X_1
         j

                                                                              (2.6.2.2)

The morphism $f \circ j$ is then of finite type (I, 6.3.4) and $g_{1}$ is a finite morphism (II, 6.1.5, (iii)); if $L^{'}$ is ample for $f \circ g$ , $j^{' *} (L^{'})$ is ample for $f \circ g \circ j^{'}$ since $j^{'}$ is a closed immersion (II, 5.1.12 and I, 4.3.2). If one sets $Z_{1} = j^{- 1} (Z)$ , Z_1 is proper over $Y$ (II, 5.4.10); on the other hand, if $X$ is normal at a point $x$ , the same is evidently true of $X_{re d}$ ; finally, if $(g_{*} (O_{X^{'}}))_{x}$ is a free $O_{x}$ -module, it follows at once from (II, 1.5.2) that $((g_{1})_{*} (O_{X_{1}^{'}}))_{x}$ is a free $O_{X_{1}, x}$ -module. Finally, since $X$ is Noetherian (I, 6.3.7), if $j^{*} (L)$ is ample, $L$ is ample (II, 4.5.14), and as $j^{' *} (L^{'}) = g_{1}^{*} (j^{*} (L))$ , this completes the reduction announced. We therefore suppose henceforth $Y$ affine and $X$ reduced.

The hypotheses of (II, 6.6.11) then being verified, there exists a reduced $Y$ -prescheme X_2 and a $Y$ -morphism $h : X_{2} \to X$ finite and birational such that the restriction of $h$ to $h^{- 1} (X - Z)$ is an isomorphism onto $X - Z$ and that $h^{*} (L)$ is ample. Replacing $X^{'}$ by X_2, one sees that one is reduced to proving the proposition supposing in addition that $g$ has the properties just enumerated for $h$ . We shall again denote by $Z$ a sub-prescheme of $X$ having $Z$ for underlying space, which is proper over $Y$ (II, 5.4.10).

(2.6.2.3)

Let now X_1 be a closed sub-prescheme of $X$ , $j : X_{1} \to X$ the canonical injection, $X_{1}^{'} = g^{- 1} (X_{1}) = X^{'} \times_{X} X_{1}$ its inverse image, $j^{'} : X_{1}^{'} \to X^{'}$ the canonical injection, so that one has the commutative diagram (2.6.2.2); set $L_{1} = j^{*} (L)$ , $L_{1}^{'} = j^{' *} (L^{'}) = g_{1}^{*} (L_{1})$ , so that $L_{1}^{'}$ is ample for $f \circ g \circ j^{'}$ (II, 5.1.12). If one sets $Z_{1} = j^{- 1} (Z)$ , the closed sub-prescheme Z_1 of X_1 is proper over $Y$ (II, 5.4.2, (ii)). In other words, the hypotheses of (2.6.2) are verified for X_1, $L_{1}$ , $g_{1}$ , and Z_1.

This will allow us to prove (2.6.2) by Noetherian induction $(0_{I}, 2.2.2)$ in the case where the restriction of $g$ to $g^{- 1} (X - Z)$ is an isomorphism onto $X - Z$ (which is sufficient for our purpose, as one has seen in (2.6.2.2)): it will suffice to establish that if, for every closed sub-prescheme X_1 of $X$ whose underlying space is $\neq = X$ , the conclusion of (2.6.2) is true for the sheaf $L_{1}$ , then it is also true for the sheaf $L$ .

(2.6.2.4)

Let then $A = O_{X}$ , $B = g_{*} (O_{X^{'}})$ , so that $B$ is a sub- $A$ -algebra of $R (X)$ , which is a coherent $A$ -module; in addition, the restriction $B ∣ (X - Z)$ is equal to $A ∣ (X - Z)$ . Let $K$ be the conductor of $B$ over $A$ , that is, the largest sub- $A$ -module quasi-coherent of $A$ such that $B \cdot K \subset A$ (or equivalently the annihilator of the $A$ -module $B / A$ $(0_{I}, 5.3.7)$ ), which entails $B \cdot K = K$ . It is clear that $B_{z} = A_{z}$ at every point admitting a neighbourhood $W_{z}$ such that $g$ is an isomorphism of $g^{- 1} (W_{z})$ onto $W_{z}$ , and in particular at every point of $X - Z$ and in a neighbourhood of every generic point of an irreducible component of $X$ . Consider then the closed sub-prescheme $Z_{1} = Spec (A / K)$ of $X$ defined by $K$ ; it is still proper over $Y$ , since the sub-space Z_1 is closed in $Z$ (II, 5.4.10). Moreover, the definition of $K$ shows that $B ∣ (X - Z_{1}) = A ∣ (X - Z_{1})$ ; one thus sees that one can always reduce to the case where $Z = Spec (A / K)$ , and as we have seen that $X - Z_{1}$ is a nonempty open set of $X$ , one may always suppose that the space $Z$ is distinct from $X$ .

(2.6.2.5)

Consider $X^{'}$ as equal to $Spec (B)$ (since $g$ is affine) and let $K^{'} = \tilde{K}$ , a coherent ideal sheaf of $O_{X^{'}}$ such that $g_{*} (K^{'}) = K$ (II, 1.4.1); the closed sub-prescheme $Z^{'} = g^{- 1} (Z) = Z \times_{X} X^{'}$ of $X^{'}$ is defined by $K^{'}$ and equal to $Spec (B / K)$ (II, 1.4.10); as $h : Z^{'} \to Z$ is a finite morphism (II, 6.1.5, (iii)), $Z^{'}$ is proper over $Y$ (II, 6.1.11 and 5.4.2, (ii)).

This being so, we must prove that for every $x \in X$ and every open neighbourhood $U$ of $x$ , there exists a section $s$ of an $L^{\otimes n}$ ( $n > 0$ ) over $X$ such that $x \in X_{s} \subset U$ (II, 4.5.2); we distinguish two cases:

1° One has $x \in X - Z$ ; one may evidently then suppose that one also has $U \subset X - Z$ , so the open set $U^{'} = g^{- 1} (U)$ does not meet $Z^{'}$ . As $L^{'}$ is ample by hypothesis, there exist an $n > 0$ and a section $s^{'}$ of $L^{' \otimes n}$ over $X^{'}$ such that $x^{'} = g^{- 1} (x) \in X_{s^{'}}^{'} \subset g^{- 1} (U)$ (II, 4.5.2). In addition, one may suppose that $K^{'} \otimes L^{' \otimes n}$ is generated by its sections over $X^{'}$ (II, 4.5.5), so, since $K_{x^{'}}^{'} = O_{x^{'}}$ , there is a section s'' of these such that $s^{''} (x^{'}) \neq = 0$ ; multiplying it by $s^{'}$ (which amounts to replacing $n$ by 2n), one sees that one may also suppose that $x^{'} \in X_{s^{''}}^{'} \subset g^{- 1} (U)$ . This being so, it follows from $(0_{I}, 5.4.10)$ that one has a canonical isomorphism

  Γ(X, 𝒦 ⊗ ℒ^{⊗n}) ⥲ Γ(X', 𝒦' ⊗ ℒ'^{⊗n}).

The section $s$ of $K \otimes L^{\otimes n}$ corresponding to s'' under this isomorphism evidently has the desired properties.

2° One has $x \in Z$ . Let $J$ be the coherent ideal sheaf of $O_{X}$ defining the closed reduced sub-prescheme of $X$ having for underlying space $X - U$ , and consider in $B$ the coherent ideals

$J B$ and $J_{1} = J ∣ (J B \cap A) = J (J B)$ , so that one has the diagram of inclusions

   𝒥 ℬ    →    ℬ
   ↑           ↑
   𝒥     →    𝒜
   ↑           ↑
   𝒥 𝒦 ℬ = 𝒥 𝒦  →   𝒦                                                       (2.6.2.6)

Let $J^{'}$ be the coherent ideal sheaf $(J B) \sim$ of $O_{X^{'}}$ , so that $J B = g_{*} (J^{'})$ , $J K = (J K^{'}) \sim$ , and consequently $J^{'} / J^{'} K^{'} = (J B / J K B) \sim$ (II, 1.4.4). As $J ∣ V = J K ∣ V$ for every open set $V$ not meeting $Z$ , one sees that the support of $J^{'} / J^{'} K^{'}$ is contained in $Z^{'}$ . As $Z^{'}$ is proper over $Y$ , one may apply (2.2.4) and one sees that for $n$ large enough, the canonical map

  Γ(X', 𝒥' ⊗ ℒ'^{⊗n}) → Γ(X', (𝒥' / 𝒥' 𝒦') ⊗ ℒ'^{⊗n})

is surjective.

But by virtue of $(0_{I}, 5.4.10)$ , one concludes that the canonical map

  Γ(X, 𝒥 ℬ ⊗ ℒ^{⊗n}) → Γ(X, (𝒥 ℬ / 𝒥 𝒦 ℬ) ⊗ ℒ^{⊗n})

is surjective.

This being so, let $i : Z \to X$ be the canonical injection, $i^{'} : Z^{'} \to X^{'}$ the canonical injection, so that one has the commutative diagram

         i'
    X' ←───── Z'
    |          |
  g |          | h
    ↓          ↓
    X  ←───── Z
         i

Let $M = i^{*} (L)$ , $M^{'} = i^{' *} (L^{'})$ ; as $L^{'}$ is ample, $M^{'}$ is ample (II, 5.1.12), and on the other hand $M^{'} = h^{*} (M)$ ; one concludes therefore from the hypothesis of Noetherian induction (since $Z \neq = X$ ) that $M$ is ample. Consequently $i^{*} ((J B / J K) \otimes L^{\otimes n})$ is generated by its sections over $Z$ for $n$ large enough (II, 4.5.5). As $J B / J K = i_{*} (i^{*} (J B / J K))$ , one deduces again from $(0_{I}, 5.4.10)$ that there exists a section $s$ of $(J B / J K) \otimes L^{\otimes n}$ over $X$ (for $n$ large enough) such that $s (x) \neq = 0$ , since one has $B_{x} = A_{x}$ by definition of $Z$ and $K_{x} \neq = A_{x}$ by hypothesis. The diagram (2.6.2.6) shows that $s$ is also a section of $(J B / J K B) \otimes L^{\otimes n}$ over $X$ , hence $s$ is the canonical image of a section $t$ of $J B \otimes L^{\otimes n}$ over $X$ . But by definition, the canonical image $s$ of $t$ mod $(J K B) \otimes L^{\otimes n}$ is in $(J \otimes L^{\otimes n}) / ((J K B) \otimes L^{\otimes n})$ , hence by virtue of (2.6.2.6), this implies that $t$ is a section of $J \otimes L^{\otimes n}$ over $X$ , and a fortiori a section of $L^{\otimes n}$ . One has seen above that $t (x) \neq = 0$ , so $x \in X_{t}$ , and by definition of $J$ , $t (y) = 0$ in $X - U$ which is the support of $O_{X} / J$ ; thus $X_{t} \subset U$ , which completes the proof.

Remark (2.6.3).

When $X$ is proper over $Y$ , one can prove (2.6.2) more simply, by reasoning as in Chevalley's theorem (II, 6.7.1), using (2.6.1) and Lemma (II, 6.7.1.1).

Éléments de Géométrie Algébrique — English