§4. The fundamental theorem of proper morphisms; applications

4.1. The fundamental theorem

(4.1.1)

Let $X$, $Y$ be two usual Noetherian preschemes, $f:X\to Y$ a proper morphism, $Y^{\prime }$ a closed subset of $Y$, and $X^{\prime }$ its inverse image $f^{-1}(Y^{\prime })$. We denote by $\mathfrak{X}$ and $\mathfrak{Y}$ the formal preschemes $X_{/X^{\prime }}$ and $Y_{/Y^{\prime }}$, the completions of $X$ and $Y$ along $X^{\prime }$ and $Y^{\prime }$ respectively (I, 10.8.5); we write $\hat{f}$ for the extension of $f$ to these completions (I, 10.9.1), which is a morphism $\mathfrak{X}\to \mathfrak{Y}$ of formal preschemes. For every coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, we denote by $\hat{\mathcal{F}}$ its completion along $X^{\prime }$ (I, 10.8.4), which is a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module (I, 10.8.8).

(4.1.2)

Let $\mathcal{J}$ be a coherent ideal of ${\mathcal{O}}_Y$ such that $Supp({\mathcal{O}}_Y/\mathcal{J})=Y^{\prime }$ (I, 5.2.1); we know (I, 4.4.5) that $\mathcal{K}=f^{\ast }(\mathcal{J}){\mathcal{O}}_X$ is a coherent ideal of ${\mathcal{O}}_X$ such that

$$Supp({\mathcal{O}}_X/\mathcal{K})=X^{\prime }.$$

We consider, for every $k\ge 0$, the coherent ${\mathcal{O}}_X$-modules

  ℱ_k = ℱ / 𝒦^{k+1} ℱ.

The ${\mathcal{O}}_Y$-modules $R^n{f}_{\ast }(\mathcal{F})$ and $R^n{f}_{\ast }({\mathcal{F}}_k)$ are coherent for every $n$ (3.2.1). For every $k^{\prime }\ge k$ and every $n$, the canonical homomorphism ${\mathcal{F}}_{k^{\prime }}\to {\mathcal{F}}_k$ defines by functoriality a homomorphism

  R^n f_*(ℱ_{k'}) → R^n f_*(ℱ_k).                                            (4.1.2.1)

Moreover, since ${\mathcal{F}}_k$ is an ${\mathcal{O}}_X/{\mathcal{K}}^{k+1}$-module, $R^n{f}_{\ast }({\mathcal{F}}_k)$ is an ${\mathcal{O}}_Y/{\mathcal{J}}^{k+1}$-module $(0_{III},\mathrm{12.2.1})$, and one deduces from (4.1.2.1) a homomorphism

  R^n f_*(ℱ_{k'}) ⊗_{𝒪_Y} (𝒪_Y / 𝒥^{k+1}) → R^n f_*(ℱ_k).                    (4.1.2.2)

The two sides of (4.1.2.2) form two projective systems, and the projective limit of the first side is none other than the completion $(R^n{f}_{\ast }(\mathcal{F}){)}_{/Y^{\prime }}$ which we shall denote $(R^n{f}_{\ast }(\mathcal{F}){)}^{\wedge }$. Furthermore, it is immediate that the homomorphisms (4.1.2.2) form a projective system, whence by passage to the limit a canonical homomorphism

  φ_n : (R^n f_*(ℱ))^∧ → lim_← R^n f_*(ℱ_k).                                 (4.1.2.3)
                          k

Moreover, (4.1.2.2) is a homomorphism of $({\mathcal{O}}_Y/{\mathcal{J}}^{k+1})$-modules, and therefore (I, 10.8.3) may be considered as a continuous homomorphism of pseudo-discrete topological ${\mathcal{O}}_{\mathfrak{Y}}$-modules $(0_I,\mathrm{3.8.1})$. The homomorphism ${\varphi }_n$ is consequently a continuous homomorphism of topological ${\mathcal{O}}_{\mathfrak{Y}}$-modules.

(4.1.3)

Let $i:\mathfrak{X}\to X$ be the canonical morphism of ringed spaces defined in (I, 10.8.7), so that we have the commutative diagram

$$X_k{\to }^{h_k}X\downarrow f_k\downarrow f(\mathrm{4.1.3.1})Y_k\to Y$$

where $X_k$ is the closed subprescheme of $X$ defined by the ideal ${\mathcal{K}}^{k+1}$, $i_k$ the canonical injection, $h_k$ the morphism of ringed spaces corresponding to the identity on the underlying spaces and to the canonical homomorphism ${\mathcal{O}}_X\to {\mathcal{O}}_X/{\mathcal{K}}^{k+1}$ (I, 10.5.2). Moreover, we have $\hat{\mathcal{F}}=i^{\ast }(\mathcal{F})$ (I, 10.8.8) up to canonical isomorphism. We know that

  H^n(X_k, ℱ_k) = H^n(X, ℱ_k)                                                (4.1.3.2)

up to canonical isomorphism, since ${\mathcal{F}}_k=(h_k{)}_{\ast }((i_k{)}^{\ast }({\mathcal{F}}_k))$ $(0_I,\mathrm{4.9.1})$; the canonical homomorphism $H^n(X,\mathcal{F})\to H^n(X_k,{\mathcal{F}}_k)$ $(0_{III},\mathrm{12.1.3.5})$ thus also reads

  H^n(X, ℱ) → H^n(X, ℱ_k),                                                   (4.1.3.3)

and these homomorphisms obviously form a projective system, whence by passage to the limit a canonical homomorphism

  ψ_X : H^n(X, ℱ) → lim_← H^n(X, ℱ_k).                                       (4.1.3.4)
                      k

Replacing $X$ by an open set of the form $f^{-1}(V)$, where $V$ is an affine open set of $Y$, and taking (1.4.11) into account, we have homomorphisms

  ψ_V : H^n(X ∩ f^{-1}(V), ℱ) → lim_← Γ(V, R^n f_*(ℱ_k));                    (4.1.3.5)
                                  k

these homomorphisms obviously commute with restriction from $V$ to a smaller affine open set, and therefore finally define a canonical homomorphism of sheaves

  ψ : R^n f_*(ℱ) → lim_← R^n f_*(ℱ_k).                                       (4.1.3.6)
                     k

(4.1.4)

Let finally $j:\mathfrak{Y}\to Y$ be the canonical morphism of ringed spaces (I, 10.8.7); since $R^n{f}_{\ast }(\mathcal{F})$ is a coherent ${\mathcal{O}}_Y$-module (3.2.1), we have $j^{\ast }(R^n{f}_{\ast }(\mathcal{F}))=(R^n{f}_{\ast }(\mathcal{F}){)}^{\wedge }$ up to canonical isomorphism (I, 10.8.8), and we therefore have a canonical homomorphism

  ρ_n : (R^n f_*(ℱ))^∧ = j^*(R^n f_*(ℱ)) → R^n 𝑓̂_*(j^*(ℱ)) = R^n 𝑓̂_*(ℱ̂),     (4.1.4.1)

defined in general for ringed spaces (see the proof of (1.4.15)). We show that the diagram

  (R^n f_*(ℱ))^∧ ────→ R^n 𝑓̂_*(ℱ̂)
        ↓ φ_n             ↑ ψ_n                                              (4.1.4.2)
        lim_← R^n f_*(ℱ_k)
          k

is commutative. It clearly suffices to prove the commutativity of the corresponding diagram of homomorphisms of presheaves, so we may restrict to the case where $Y$ is affine, and everything reduces to proving that the diagram

  (H^n(X, ℱ))^∧ ────→ H^n(𝔛, ℱ̂)
        ↓ φ_n           ↑ ψ_{n,𝔛}                                            (4.1.4.3)
        lim_← H^n(X, ℱ_k)
          k

is commutative. But the commutativity of (4.1.3.1) and the relations seen in (4.1.3) between the cohomology groups give at once the commutative diagram

  H^n(X, ℱ) ────→ H^n(𝔛, ℱ̂) = H^n(𝔛, i^*(ℱ))
         ╲         ╱
          H^n(X_k, ℱ_k) = H^n(X, ℱ_k)

whence we deduce immediately the commutativity of (4.1.4.3).

Theorem (4.1.5).

Let $f:X\to Y$ be a proper morphism of Noetherian preschemes, $Y^{\prime }$ a closed subset of $Y$, $X^{\prime }=f^{-1}(Y^{\prime })$. Then, for every coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, $R^n{\hat{f}}_{\ast }(\hat{\mathcal{F}})$ is a coherent ${\mathcal{O}}_{\mathfrak{Y}}$-module, and the homomorphisms ${\varphi }_n$, ${\psi }_n$, and ${\rho }_n$ of the diagram (4.1.4.2) are topological isomorphisms.

Proof. It clearly suffices to prove that ${\varphi }_n$ and ${\psi }_n$ are isomorphisms; since $R^n{f}_{\ast }(\mathcal{F})$ is coherent (3.2.1), it will follow that $(R^n{f}_{\ast }(\mathcal{F}){)}^{\wedge }$ is coherent (I, 10.8.8), and the bicontinuity of ${\varphi }_n$, ${\psi }_n$, and ${\rho }_n$ is then automatic (I, 10.11.6).

Remarks (4.1.6).

(i) If we set ${\hat{\mathcal{F}}}_k=\hat{\mathcal{F}}/{\hat{\mathcal{K}}}^{k+1}\hat{\mathcal{F}}$, it is immediate that ${\hat{\mathcal{F}}}_k=i^{\ast }({\mathcal{F}}_k)$, and the canonical homomorphism (4.1.3.6) is none other than the homomorphism already defined in (3.4.2.2)

  R^n 𝑓̂_*(ℱ̂) → lim_← R^n 𝑓̂_*(ℱ̂_k);                                          (4.1.6.1)
                 k

consequently, the fact that ${\psi }_n$ is an isomorphism is a particular case of (3.4.3). But we shall give below a direct proof, avoiding the delicate considerations on projective limits of spectral sequences $(0_{III},13.7)$ on which the general theorem (3.4.3) rests.

(ii) Taking account of the fact that the ${\psi }_n$ are isomorphisms, it is equivalent to say that the ${\varphi }_n$ or the ${\rho }_n={\psi }_n\circ {\varphi }_n^{-1}$ are isomorphisms. Theorem (4.1.5) expresses, among other things, that the formation of $R^n{f}_{\ast }(\mathcal{F})$ commutes with completion, and may be called the first comparison theorem between the "algebraic" and "formal" theories.

We shall begin by establishing the affine form of (4.1.5):

Corollary (4.1.7).

The hypotheses being those of (4.1.5), suppose in addition that $Y=\mathrm{Spec}(A)$, where $A$ is Noetherian, and $\mathcal{J}=\tilde{\mathfrak{J}}$, where $\mathfrak{J}$ is an ideal of $A$, so that ${\mathcal{F}}_k=\mathcal{F}/{\mathfrak{J}}^{k+1}\mathcal{F}$. The canonical homomorphism

  φ_n : (H^n(X, ℱ))^∧ → lim_← H^n(X, ℱ_k)                                    (4.1.7.1)
                          k

(where the first member is the Hausdorff completion of $H^n(X,\mathcal{F})$ for the $\mathfrak{J}$-preadic topology) is an isomorphism. The projective system $(H^n(X,{\mathcal{F}}_k){)}_{k\ge 0}$ satisfies condition (ML) for every $n$, and the canonical homomorphism

  ψ_n : H^n(X, ℱ) → lim_← H^n(X, ℱ_k)                                        (4.1.7.2)
                       k

is an isomorphism. Finally, the filtration on $H^n(X,\mathcal{F})$ defined by the kernels of the canonical homomorphisms

$H^n(X,\mathcal{F})\to H^n(X,{\mathcal{F}}_k)$ is $\mathfrak{J}$-good $(0_{III},\mathrm{13.7.7})$, and ${\psi }_n$ is a topological isomorphism (1).

(1) The following proof, simpler than the original proof, and the complement on the filtration of $H^n(X,>\mathcal{F})$, were communicated to us by J.-P. Serre.

Proof. The integer $n\ge 0$ being fixed in this proof, we shall set for simplicity

  H = H^n(X, ℱ),    H_k = H^n(X, ℱ_k),                                       (4.1.7.3)
  R_k = Ker(H → H_k),     a sub-`A`-module of `H`.                           (4.1.7.4)

The exact sequence of cohomology

  H^n(X, 𝔍^{k+1} ℱ) → H^n(X, ℱ) → H^n(X, ℱ_k) → H^{n+1}(X, 𝔍^{k+1} ℱ) → H^{n+1}(X, ℱ)

shows that we also have $R_k=Im(H^n(X,{\mathfrak{J}}^{k+1}\mathcal{F})\to H^n(X,\mathcal{F}))$; we shall set

  Q_k = Ker(H^{n+1}(X, 𝔍^{k+1} ℱ) → H^{n+1}(X, ℱ))
      = Im(H^n(X, ℱ_k) → H^{n+1}(X, 𝔍^{k+1} ℱ)).                             (4.1.7.5)

We thus have the exact sequence

  0 → R_k → H → H_k → Q_k → 0.                                               (4.1.7.6)

(4.1.7.7). Let $x$ be an element of ${\mathfrak{J}}^r$ ($r\ge 0$); multiplication by $x$ in ${\mathfrak{J}}^k\mathcal{F}$ is a homomorphism ${\mathfrak{J}}^k\mathcal{F}\to {\mathfrak{J}}^{k+r}\mathcal{F}$ and consequently gives rise to a homomorphism

  μ_x : H^n(X, 𝔍^k ℱ) → H^n(X, 𝔍^{k+r} ℱ).                                   (4.1.7.8)

If we denote by $S$ the graded $A$-algebra $\oplus {\mathfrak{J}}^k$, we know that the multiplications ${\mu }_x$ endow $E=\oplus H^n(X,{\mathfrak{J}}^k\mathcal{F})$ with a structure of graded module of finite type over the graded ring $S$ (3.3.2), which is Noetherian (II, 2.1.5).

Lemma (4.1.7.9). The submodules $R_k$ of $H$ define on $H$ a $\mathfrak{J}$-good filtration.

Proof. First, we show that we have

$$\mathfrak{J}{R}_k\subset R_{k+1},(\mathrm{4.1.7.10})$$

multiplication in $H=H^n(X,\mathcal{F})$ by an element $x\in \mathfrak{J}$ being therefore the map ${\mu }_x$ for $r=1$. For every $x\in \mathfrak{J}$, the diagram

  𝔍^{k+1} ℱ ────→ 𝔍^{k+2} ℱ
       ↓             ↓
       ℱ   ────→     ℱ

(where the horizontal arrows are multiplication by $x$, and the vertical arrows the canonical injections) is commutative; hence the corresponding diagram

  H^n(X, 𝔍^{k+1} ℱ) ──^{μ_{x,n}}──→ H^n(X, 𝔍^{k+2} ℱ)
           ↓                                ↓                                (4.1.7.11)
         H^n(X, ℱ)  ──^{μ_{x,0}}──→  H^n(X, ℱ)

is commutative, which, taking into account the interpretation of $R_k$ as the image of $H^n(X,{\mathfrak{J}}^{k+1}\mathcal{F})\to H^n(X,\mathcal{F})$, proves (4.1.7.10) and shows in addition that the graded $S$-module $R=\oplus R_k$ is a quotient of the sub-$S$-module $M=\oplus H^n(X,{\mathfrak{J}}^{k+1}\mathcal{F})$ of $E$; the remark made above thus shows that $R$ is an $S$-module of finite type, which is equivalent to the assertion of (4.1.7.9) (Bourbaki, Alg. comm., chap. III, § 3, n° 1, th. 1).

(4.1.7.12). Consider now the graded $S$-module $N=\oplus H^{n+1}(X,{\mathfrak{J}}^{k+1}\mathcal{F})$ defined as in (4.1.7.8); it is again an $S$-module of finite type by virtue of (3.3.2); we have $Q_k\subset N_k$ for every $k$ by (4.1.7.5), and the diagram (4.1.7.11) where we replace $n$ by $n+1$ shows that ${\mathfrak{J}}^r{Q}_k\subset Q_{k+r}$. In other words, $Q=\oplus Q_k$ is a graded sub-$S$-module of $N$, and is therefore of finite type.

(4.1.7.13). Denote by $a_k$ the canonical injection ${\mathfrak{J}}^k\to A$, which we may write $a_k:S_k\to S_0$. Since ${\mathfrak{J}}^{k+1}\mathcal{F}$ is annihilated by $S_{k+1}$, the $A$-module $H^n(X,{\mathfrak{J}}^{k+1}\mathcal{F})$ is annihilated by $S_{k+1}$; since $Q_k$ is the image of the $A$-homomorphism $H^n(X,{\mathcal{F}}_k)\to H^{n+1}(X,{\mathfrak{J}}^{k+1}\mathcal{F})$, $Q_k$, as an $A$-module, is also annihilated by $S_{k+1}$. This still means that, in the $S$-module $Q$, we have

$$a_{k+1}(S_{k+1})Q_k=0.(\mathrm{4.1.7.14})$$

Since $Q$ is an $S$-module of finite type, there exist an integer $k_0$ and an integer $h$ such that $Q_{k+h}=S_h{Q}_k$ for $k\ge k_0$ (II, 2.1.6, (ii)); from this relation and (4.1.7.14), one deduces that there exists an integer $r>0$ such that

$$a_r(S_r)Q=0.(\mathrm{4.1.7.15})$$

(4.1.7.16). Note now that the canonical injection ${\mathfrak{J}}^{k+1}\mathcal{F}\to {\mathfrak{J}}^k\mathcal{F}$ gives, on passage to cohomology, an $A$-homomorphism

  v_k : H^{n+1}(X, 𝔍^{k+1} ℱ) → H^{n+1}(X, 𝔍^k ℱ),                           (4.1.7.17)

and, for every $x\in \mathfrak{J}$, we have the obvious factorization

  μ_{x,0} : H^{n+1}(X, ℱ) → H^{n+1}(X, 𝔍^{k+1} ℱ) →^{v_k} H^{n+1}(X, 𝔍^k ℱ), (4.1.7.18)

from which we conclude that, for every sub-$A$-module $P$ of $H^{n+1}(X,{\mathfrak{J}}^{k+1}\mathcal{F})$, we have, in the $S$-module $N$,

  v_k(S_1 P) = a_1(S_1) P.                                                   (4.1.7.19)

Lemma (4.1.7.20). There exists an integer $m>0$ such that $v_k(Q_{k+m})=0$ for every $k\ge k_0$.

Proof. Take for $m$ a multiple of $h$ that is $\ge r$; since $Q_{k+m}=S_m{Q}_k$ for $k\ge k_0$, we have by virtue of (4.1.7.19) and (4.1.7.15) $v_k(Q_{k+m})=a_m(S_m)Q\subset a_r(S_r)Q=0$.

(4.1.7.21). Note that from the commutative diagram

  H^n(X, ℱ) ──→ H^n(X, ℱ_k) ──→ H^{n+1}(X, 𝔍^{k+1} ℱ) ──→ H^{n+1}(X, ℱ)
      ║              ↓                       ↓                  ║
  H^n(X, ℱ) ──→ H^n(X, ℱ_{k+m}) → H^{n+1}(X, 𝔍^{k+m+1} ℱ) → H^{n+1}(X, ℱ)

itself coming from the commutative diagram

  0 → 𝔍^{k+1} ℱ ──→ ℱ ──→ ℱ_k ──→ 0
        ↑          ║      ↑
  0 → 𝔍^{k+m+1} ℱ → ℱ → ℱ_{k+m} → 0

where the vertical arrows are the canonical maps, one deduces a commutative diagram

  0 → R_k    ──→ H ──→ H_k    ──→ Q_k    → 0
       ↓        id      ↓           ↓
  0 → R_{k+m} → H → H_{k+m} ──→ Q_{k+m} → 0

where the rows are exact. Since the last vertical arrow is zero for $k\ge k_0$ (4.1.7.20), the image of $H_{k+m}$ in $H_k$ is contained in $Ker(H_k\to Q_k)=Im(H\to H_k)$, but moreover it contains $Im(H\to H_k)$ by the commutativity of the diagram, so it is equal to it; the same therefore holds for the images in $H_k$ of the $H_{k^{\prime }}$ for $k^{\prime }\ge k+m$, which proves condition (ML) for the projective system $(H_k{)}_{k\ge 0}$. Moreover, for every affine open set $U$ of $X$, we have $H^i(U,\mathcal{F})=0$ for $i>0$ (1.3.1), and for $m>0$, the map $H^0(U,{\mathcal{F}}_{k+m})\to H^0(U,{\mathcal{F}}_k)$ is surjective (I, 1.3.9). We may therefore apply $(0_{III},\mathrm{13.3.1})$, and the canonical homomorphism $H^n(X,\mathcal{F})\to \underleftarrow{\lim }{H}^n(X,{\mathcal{F}}_k)$ is bijective for every $n\ge 0$.

Since the projective system $(H/R_k{)}_{k\ge 0}$ is strict, we may pass to the projective limit in the exact sequences

$$0\to H/R_k\to H_k\to Q_k\to 0(\mathrm{4.1.7.22})$$

$(0_{III},\mathrm{13.2.2})$; since $v_k(Q_{k+m})=0$, we have $\underleftarrow{\lim }{Q}_k=0$, whence a topological isomorphism $\underleftarrow{\lim }(H/R_k)\stackrel{\sim }{\to }\underleftarrow{\lim }{H}_k$. But since the filtration $(R_k)$ of $H$ is $\mathfrak{J}$-good, it defines on $H$ the $\mathfrak{J}$-preadic topology; therefore $\underleftarrow{\lim }(H/R_k)$ is the Hausdorff completion of $H$ for the $\mathfrak{J}$-preadic topology, which completes the proof of (4.1.7).

(4.1.8)

We now pass to the proof of (4.1.5). For every affine open set $V$ of $Y$, $\Gamma (V,(R^n{f}_{\ast }(\mathcal{F}){)}^{\wedge })$ is the Hausdorff completion of $\Gamma (V,R^n{f}_{\ast }(\mathcal{F}))$ for the $\mathfrak{J}$-preadic topology (if $\mathcal{J}\mid V=\tilde{\mathfrak{J}}$) since $R^n{f}_{\ast }(\mathcal{F})$ is a coherent ${\mathcal{O}}_Y$-module (I, 10.8.4), and $\Gamma (V,\underleftarrow{\lim }{R}^n{f}_{\ast }({\mathcal{F}}_k))=\underleftarrow{\lim }\Gamma (V,R^n{f}_{\ast }({\mathcal{F}}_k))$ $(0_I,\mathrm{3.2.6})$; the fact that ${\varphi }_n$ is a topological isomorphism then results from (4.1.7) and (1.4.11). On the other hand (again by virtue of (1.4.11)), it follows from (4.1.7) that the homomorphism ${\psi }_n$ of (4.1.3.3) is an isomorphism, hence ${\psi }_n$ is an isomorphism by definition of $R^n{\hat{f}}_{\ast }(\hat{\mathcal{F}})$.

Corollary (4.1.9).

Under the hypotheses of (4.1.4), for every affine open set $V$ of $Y$, the canonical homomorphism

  H^n(𝔛 ∩ 𝑓̂^{-1}(V), ℱ̂) → Γ(𝔜 ∩ V, R^n 𝑓̂_*(ℱ̂))

is bijective.

Remark (4.1.10).

Let $f:X\to Y$ be a morphism of finite type of (usual) Noetherian preschemes, and let $\mathcal{F}$ be a coherent ${\mathcal{O}}_X$-module whose support is proper over $Y$ (II, 5.4.10). We then know (3.2.4) that $R^n{f}_{\ast }(\mathcal{F})$ is a coherent ${\mathcal{O}}_Y$-module for every $n\ge 0$. Moreover, we may always assume that $\mathcal{F}=u_{\ast }(\mathcal{G})$, where $\mathcal{G}=u^{\ast }(\mathcal{F})$ is a coherent ${\mathcal{O}}_Z$-module, $Z$ denoting a suitable closed subprescheme of $X$ whose underlying space is $Supp(\mathcal{F})$, and $u:Z\to X$ the canonical injection (I, 9.3.5). If we set ${\mathcal{G}}_k=\mathcal{G}/{\mathfrak{J}}^{k+1}\mathcal{G}$ (with $\mathfrak{J}=u^{\ast }(\mathcal{J}){\mathcal{O}}_Z$), we have ${\mathcal{F}}_k=u_{\ast }({\mathcal{G}}_k)$, $R^n{f}_{\ast }(\mathcal{F})=R^n(f\circ u{)}_{\ast }(\mathcal{G})$, and $R^n{f}_{\ast }({\mathcal{F}}_k)=R^n(f\circ u{)}_{\ast }({\mathcal{G}}_k)$ (1.3.4), and finally, taking (I, 10.9.5) into account,

  R^n 𝑓̂_*(ℱ̂) = R^n (f ∘ u)^∧_*(𝒢̂).

We may then apply (4.1.5) to $\mathcal{G}$ and to the proper morphism $f\circ u$, and we conclude that under these hypotheses, the results of (4.1.5) are valid for $\mathcal{F}$ and $f$.

4.2. Particular cases and variants

The most useful form of the comparison theorem (4.1.5) is the following:

Proposition (4.2.1).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a proper morphism, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module. Then, for every $y\in Y$ and every $p$, $(R^p{f}_{\ast }(\mathcal{F}){)}_y$ is

an ${\mathcal{O}}_y$-module of finite type, hence separated for the ${\mathfrak{m}}_y$-preadic topology, and we have a canonical topological isomorphism

  ((R^p f_*(ℱ))_y)^∧ ⥲ lim_← H^p(f^{-1}(y), ℱ ⊗_{𝒪_Y} (𝒪_y / 𝔪_y^{k+1}))      (4.2.1.1)
                         k

where the first member is the completion of $(R^p{f}_{\ast }(\mathcal{F}){)}_y$ for the ${\mathfrak{m}}_y$-preadic topology, and at the second member $f^{-1}(y)$ is considered, for every $k\ge 0$, as the underlying space of the prescheme $X{\times }_Y\mathrm{Spec}({\mathcal{O}}_y/{\mathfrak{m}}_y^{k+1})$ (I, 3.6.1).

Proof. Since ${\mathcal{O}}_y$ is a Noetherian local ring and $(R^p{f}_{\ast }(\mathcal{F}){)}_y$ is a finitely generated ${\mathcal{O}}_y$-module (3.2.1), the ${\mathfrak{m}}_y$-preadic topology on $(R^p{f}_{\ast }(\mathcal{F}){)}_y$ is separated $(0_I,\mathrm{7.3.5})$. The other assertions are consequences of (4.1.7) when $Y$ is Noetherian and the point $y$ is closed, on replacing $Y$ by an affine neighbourhood of $y$ and taking $Y^{\prime }=y$, in view of (G, II, 4.9.1). In the general case, set $Y_1=\mathrm{Spec}({\mathcal{O}}_y)$, $X_1=X{\times }_Y{Y}_1$, ${\mathcal{F}}_1=\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_y$, and let $f_1=f{\times }_Y{1}_{Y_1}:X_1\to Y_1$; Y_1 is Noetherian and $f_1$ is proper (II, 5.4.2, (iii)), and ${\mathcal{F}}_1$ is coherent $(0_I,\mathrm{5.3.11})$. Let $y_1$ be the unique closed point of Y_1; the proposition is valid for $f_1$, $y_1$, and ${\mathcal{F}}_1$; we have ${\mathcal{O}}_{y_1}={\mathcal{O}}_y$, $f_1^{-1}(y_1)=f^{-1}(y)$ (I, 3.6.5), the preschemes $X{\times }_Y\mathrm{Spec}({\mathcal{O}}_y/{\mathfrak{m}}_y^{k+1})$ and $X_1{\times }_{Y_1}\mathrm{Spec}({\mathcal{O}}_{y_1}/{\mathfrak{m}}_{y_1}^{k+1})$ being canonically identified (I, 3.3.9); moreover, ${\mathcal{F}}_1{\otimes }_{{\mathcal{O}}_{y_1}}({\mathcal{O}}_{y_1}/{\mathfrak{m}}_{y_1}^{k+1})$ is identified with $\mathcal{F}{\otimes }_{{\mathcal{O}}_y}({\mathcal{O}}_y/{\mathfrak{m}}_y^{k+1})$ (I, 9.1.6). It remains to see that $R^p{f}_{1\ast }({\mathcal{F}}_1)$ is canonically isomorphic to $R^p{f}_{\ast }(\mathcal{F}){\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_y$, which results from (1.4.15), the local morphism $\mathrm{Spec}({\mathcal{O}}_y)\to Y$ being flat $(0_I,\mathrm{6.7.1}andI,\mathrm{2.4.2})$.

The following corollary uses the terminology of dimension theory (chap. IV) and will not be applied before chap. IV.

Corollary (4.2.2).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a proper morphism, $y$ a point of $Y$, $r$ the dimension of $f^{-1}(y)$. Then, for every coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the sheaves $R^p{f}_{\ast }(\mathcal{F})$ are zero in a neighbourhood of $y$ for every $p>r$.

Proof. Indeed, we then have $H^p(f^{-1}(y),\mathcal{F}\otimes ({\mathcal{O}}_y/{\mathfrak{m}}_y^{k+1}))=0$ (G, II, 4.15.2) for every $k$, hence (4.2.1) the Hausdorff completion of $(R^p{f}_{\ast }(\mathcal{F}){)}_y$ for the ${\mathfrak{m}}_y$-preadic topology is zero, and as this topology is separated, we also have $(R^p{f}_{\ast }(\mathcal{F}){)}_y=0$; whence the conclusion, since $R^p{f}_{\ast }(\mathcal{F})$ is coherent $(0_I,\mathrm{5.2.2})$.

(4.2.3). The result (4.2.1) is principally used for $p=0$; we thus obtain the following corollary:

Corollary (4.2.4).

Under the hypotheses of (4.2.1), we have a canonical topological isomorphism

  ((f_*(ℱ))_y)^∧ ⥲ lim_← Γ(f^{-1}(y), ℱ ⊗_{𝒪_Y} (𝒪_y / 𝔪_y^{k+1})).           (4.2.4.1)
                     k

4.3. Zariski's connection theorem

The results of this section and of the next generalize well-known theorems of Zariski, and may all be deduced from (4.2.4). They are consequences of the following theorem:

Theorem (4.3.1) (Connection theorem).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a proper morphism. Then $\mathcal{B}(X)=f_{\ast }({\mathcal{O}}_X)$ is a coherent ${\mathcal{O}}_Y$-algebra.

Let $Y^{\prime }$ be the finite $Y$-scheme over $Y$ such that $\mathcal{A}(Y^{\prime })=f_{\ast }({\mathcal{O}}_X)$, determined up to $Y$-isomorphism (II, 1.3.1 and 6.1.3); if $f^{\prime }=\nu (e)$ is the $Y$-morphism $X\to Y^{\prime }$ deduced from the identity isomorphism $e:\mathcal{A}(Y^{\prime })\stackrel{\sim }{\to }{f}_{\ast }({\mathcal{O}}_X)$ (II, 1.2.7), then $f^{\prime }$ is proper, $f_{\ast }^{\prime }({\mathcal{O}}_X)$ is isomorphic to ${\mathcal{O}}_{Y^{\prime }}$, and the fibres $f^{\prime -1}(y^{\prime })$ of the morphism $f^{\prime }$ are connected and non-empty for every $y^{\prime }\in Y^{\prime }$.

Proof. Let $g:Y^{\prime }\to Y$ be the structure morphism. To prove that the homomorphism $\theta :{\mathcal{O}}_{Y^{\prime }}\to f_{\ast }^{\prime }({\mathcal{O}}_X)$ entering the definition of the morphism $f^{\prime }$ is bijective, it suffices, since $Y^{\prime }$ is affine over $Y$, to prove that $g_{\ast }(\theta ):g_{\ast }({\mathcal{O}}_{Y^{\prime }})\to g_{\ast }(f_{\ast }^{\prime }({\mathcal{O}}_X))=f_{\ast }({\mathcal{O}}_X)$ is the identity (II, 1.4.2); but this results from the definitions since $g_{\ast }({\mathcal{O}}_{Y^{\prime }})=\mathcal{A}(Y^{\prime })$ and $f_{\ast }({\mathcal{O}}_X)=\mathcal{B}(X)$. The fact that $\mathcal{B}(X)$ is coherent is a particular case of the finiteness theorem (3.2.1). Since $f$ is proper and $g$ separated, $f^{\prime }$ is proper (II, 5.4.3, (i)); to complete the proof of (4.3.1), it suffices therefore to prove the

Corollary (4.3.2).

Under the hypotheses of (4.3.1), suppose in addition that $f_{\ast }({\mathcal{O}}_X)$ is isomorphic to ${\mathcal{O}}_Y$. Then the fibres $f^{-1}(y)$ of $f$ are connected and non-empty for every $y\in Y$.

Proof. The hypothesis that $f_{\ast }({\mathcal{O}}_X)$ is isomorphic to ${\mathcal{O}}_Y$ already implies that $f$ is dominant, hence surjective since $f$ is a closed map. We may reduce, as in (4.2.1), to the case where $y$ is closed in $Y$; $f^{-1}(y)$, being a Noetherian space, has a finite number of connected components, and it is the underlying space of the completion $\mathfrak{X}$ along $f^{-1}(y)$. If $Z_i$ ($1\le i\le n$) are these connected components, it is clear that $\Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$ is the direct sum of the rings $\Gamma (Z_i,{\mathcal{O}}_{\mathfrak{X}})$, and each of these is not reduced to 0, since the unit section is distinct from 0 at each point of $X$. Now, if we apply (4.1.5) to $\mathcal{F}={\mathcal{O}}_X$, whose completion along $f^{-1}(y)$ is ${\mathcal{O}}_{\mathfrak{X}}$, we see that $\Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$ is isomorphic to the Hausdorff ${\mathfrak{m}}_y$-adic completion ${\hat{\mathcal{O}}}_y$ of the local ring ${\mathcal{O}}_y$; it is therefore a local ring which cannot be a direct sum of several rings not reduced to 0 (otherwise it would have several distinct maximal ideals). We thus have $n=1$, which proves the corollary.

Corollary (4.3.3).

Under the hypotheses of (4.3.1), for every $y\in Y$, the set of connected components of the fibre $f^{-1}(y)$ is in one-to-one correspondence with the finite set of points of the fibre $g^{-1}(y)$, where $g:Y^{\prime }\to Y$ is the structure morphism (in other words, the set of maximal ideals of $(f_{\ast }({\mathcal{O}}_X){)}_y$).

Proof. Since $Y^{\prime }$ is finite over $Y$, we know indeed that $g^{-1}(y)$ is a finite discrete space (II, 6.1.7). Since $f^{-1}(y)=f^{\prime -1}(g^{-1}(y))$, the corollary follows from this remark and from (4.3.1).

We thus have a remarkable interpretation of the $Y$-prescheme $Y^{\prime }$ defined in (4.3.1). The factorization $f=g\circ f^{\prime }$ of the proper morphism $f$ is analogous to the factorization obtained by K. Stein for holomorphic maps of analytic spaces, and we shall call it henceforth the Stein factorization of $f$.

Remark (4.3.4).

Let $k$ be an extension of the field $\kappa (y)$: if the prescheme $f^{-1}(y){\otimes }_{\kappa (y)}k=f^{-1}(y){\times }_Y\mathrm{Spec}(k)$ is connected, then so is $f^{-1}(y)$, which is its image under a projection morphism (I, 3.4.7). We shall say that, for a morphism $f:X\to Y$ of preschemes and a point $y\in Y$, the fibre $f^{-1}(y)$ is geometrically connected if, for every extension $k$ of $\kappa (y)$, the prescheme $f^{-1}(y){\otimes }_{\kappa (y)}k=f^{-1}(y){\times }_Y\mathrm{Spec}(k)$ is connected.

Under the hypotheses of (4.3.2), one may then strengthen its conclusion: the fibres $f^{-1}(y)$ are in fact geometrically connected. To see this, observe that for every extension $k$ of $\kappa (y)$, there exists a Noetherian local ring $A$ and a local homomorphism $\varphi :{\mathcal{O}}_y\to A$ which makes $A$ a flat ${\mathcal{O}}_y$-module and such that the residue field of $A$ is $\kappa (y)$-isomorphic to $k$ $(0_{III},\mathrm{10.3.1})$. Let then $Y_1=\mathrm{Spec}(A)$ and let $h:Y_1\to Y$ be the local morphism corresponding to $\varphi$, transforming the unique closed point of Y_1 into $y$ (I, 2.4.1); set $X_1=X{\times }_Y{Y}_1$ and $f_1=f{\times }_Y{1}_{Y_1}$; $f_1$ is proper (II, 5.4.2, (iii)) and $f_1^{-1}(y_1)$ is a $\kappa (y_1)$-prescheme isomorphic to $X{\times }_Y\mathrm{Spec}(k)$. It thus suffices to show that $f_{1\ast }({\mathcal{O}}_{X_1})={\mathcal{O}}_{Y_1}$ in order to apply (4.3.2) to $f_1$. Now, $h$ is a flat morphism, as follows from (I, 2.4.2) and (1.4.15.5); we therefore have $f_{1\ast }({\mathcal{O}}_{X_1})=h^{\ast }(f_{\ast }({\mathcal{O}}_X))=h^{\ast }({\mathcal{O}}_Y)={\mathcal{O}}_{Y_1}$ by virtue of (1.4.15) applied for $q=0$.

In the general case (4.3.1), the same reasoning shows that we have (with the notation of (4.3.1)) $f_{1\ast }({\mathcal{O}}_{X_1})=h^{\ast }(f_{\ast }({\mathcal{O}}_X))$, and the Stein factorization $f_1=g_1\circ f_1^{\prime }$ of $f_1$ is such that $g_1=g{\times }_Y{1}_{Y_1}$ (II, 1.5.2), the corresponding finite Y_1-scheme being $Y_1^{\prime }=Y^{\prime }{\times }_Y{Y}_1$. Taking the transitivity of fibres (I, 3.6.4) into account, we therefore see that the number of connected components of $f_1^{-1}(y_1)$ is, by virtue of (4.3.3), equal to the number of elements of $g_1^{-1}(y_1)=g^{-1}(y){\otimes }_{\kappa (y)}k$. If we take for $k$ an algebraically closed extension of $\kappa (y)$, this number is independent of the algebraically closed extension considered and equal to the geometric number of points of $g^{-1}(y)$ (I, 6.4.7), or again to the sum of the separable ranks $[\kappa (y_i^{\prime }):\kappa (y){]}_s$ where $y_i^{\prime }$ runs over the finite set $g^{-1}(y)$. We also say that this number is the geometric number of connected components of $f^{-1}(y)$. Note that the $\kappa (y_i^{\prime })$ are none other than the residue fields of the semi-local ring $(f_{\ast }({\mathcal{O}}_X){)}_y$.

Proposition (4.3.5).

Let $X$ and $Y$ be two locally Noetherian integral preschemes and $f:X\to Y$ a proper dominant morphism. For every $y\in Y$, the number of connected components of $f^{-1}(y)$ is at most equal to the number of maximal ideals of the integral closure ${\mathcal{O}}_y^{\prime }$ of ${\mathcal{O}}_y$ in the field of rational functions $R(X)$.

Proof. Indeed, for every open set $U$ of $Y$, $\Gamma (U,f_{\ast }({\mathcal{O}}_X))=\Gamma (f^{-1}(U),{\mathcal{O}}_X)$ is the intersection of the local rings ${\mathcal{O}}_x$ such that $x\in f^{-1}(U)$ (I, 8.2.1.1). We thus conclude immediately that the stalk $(f_{\ast }({\mathcal{O}}_X){)}_y$ is a subring of $R(X)$ containing ${\mathcal{O}}_y$. Moreover, since $f_{\ast }({\mathcal{O}}_X)$ is a coherent ${\mathcal{O}}_Y$-module, $(f_{\ast }({\mathcal{O}}_X){)}_y$ is a finitely generated ${\mathcal{O}}_y$-module, and is therefore contained in ${\mathcal{O}}_y^{\prime }$; we know ([13], vol. I, p. 257 and 259) that every maximal ideal of such a ring $A$ is the intersection of $A$ and a maximal ideal of ${\mathcal{O}}_y^{\prime }$, whence the proposition.

Definition (4.3.6).

We say that an integral local ring is unibranch if its integral closure is a local ring. We say that a point $y$ of an integral prescheme $Y$ is unibranch if the local ring ${\mathcal{O}}_y$ is unibranch (which is in particular the case when $Y$ is normal at the point $y$).

Let $A$ be an integral local ring, and let $K$ be its field of fractions; for $A$ to be unibranch, it is necessary and sufficient that every subring A_1 of $K$ containing $A$ and which is a finite $A$-algebra be a local ring. Indeed, let $A^{\prime }$ be the integral closure of $A$; it follows from the first Cohen–Seidenberg theorem (Bourbaki, Alg. comm., chap. V, § 2, n° 1, th. 1) that every maximal ideal of A_1 is the trace of a maximal ideal of $A^{\prime }$, so if $A^{\prime }$ is local, then so is A_1. Conversely, $A^{\prime }$ is the inductive limit

of the increasing filtered family of finite sub-$A$-algebras A_1 of $A^{\prime }$, and if each of the A_1 is a local ring, the maximal ideal of A_1 is the trace on A_1 of that of A_2, for $A_1\subset A_2$, by the same reasoning as above, so $A^{\prime }$ is a local ring $(0_I,\mathrm{10.3.1.3})$.

Note that if the completion of a Noetherian local ring $A$ is integral (which we express by saying that $A$ is analytically integral), then $A$ is unibranch. Indeed, let $\mathfrak{m}$ be the maximal ideal of $A$, $K$ its field of fractions, $K^{\prime }$ the field of fractions of Â; we then have $K^{\prime }=K{\otimes }_A\hat{A}$. Let A_1 be a finite sub-$A$-algebra of $K$. The subring B_1 of $K^{\prime }$ generated by Â and A_1 is isomorphic to $A_1{\otimes }_A\hat{A}$; it is an Â-module of finite type, the completion of A_1 for the $\mathfrak{m}$-adic topology $(0_I,\mathrm{7.3.3}and\mathrm{7.3.6})$. Since A_1 is a semi-local ring (Bourbaki, Alg. comm., chap. IV, § 2, n° 5, cor. 3 of prop. 9) and its completion is integral, A_1 can have only one maximal ideal ${\mathfrak{m}}_1$, and we have ${\hat{\mathfrak{m}}}_1\cap A=\mathfrak{m}$; whence our assertion.

Corollary (4.3.7).

Under the hypotheses of (4.3.5), suppose that the algebraic closure of $R(Y)$ in $R(X)$ is of separable degree $n$, and that $y\in Y$ is unibranch. Then the fibre $f^{-1}(y)$ has at most $n$ connected components. In particular, if the algebraic closure of $R(Y)$ in $R(X)$ is radicial over $R(Y)$, then $f^{-1}(y)$ is connected.

Proof. Indeed, let ${\mathcal{O}}_y^{\prime }$ be the integral closure of ${\mathcal{O}}_y$; the integral closure ${\mathcal{O}}_y$ of ${\mathcal{O}}_y$ in $R(X)$ is also that of ${\mathcal{O}}_y^{\prime }$; but we know that if ${\mathcal{O}}_y^{\prime }$ is a local ring, then ${\mathcal{O}}_y$ is a semi-local ring whose number of maximal ideals is at most equal to $n$ ([13], vol. I, p. 289, th. 22).

This corollary is essentially the form in which Zariski states his "connection theorem" for algebraic schemes.

Remark (4.3.8).

If one adds to the hypotheses of (4.3.7) the hypothesis that $Y$ is normal at $y$, the fibre $f^{-1}(y)$ is geometrically connected, since (with the notation of (4.3.4)) $g^{-1}(y)$ is reduced to a point $y^{\prime }$ and $\kappa (y^{\prime })$ is radicial over $\kappa (y)$.

Definition (4.3.9).

Given a locally Noetherian prescheme $Y$, we say that a morphism of finite type $f:X\to Y$ is universally open if, for every irreducible locally Noetherian prescheme $Y^{\prime }$ and every dominant morphism $g:Y^{\prime }\to Y$, every irreducible component of $X^{\prime }=X{\times }_Y{Y}^{\prime }$ dominates $Y^{\prime }$.

If $Y$ is irreducible, this comes to saying that if $\eta$, ${\eta }^{\prime }$ are the generic points of $Y$ and $Y^{\prime }$ respectively (so that $g({\eta }^{\prime })=\eta$), and if we set $f^{\prime }=f_{(Y^{\prime })}$, every irreducible component of $X^{\prime }$ meets $f^{\prime -1}({\eta }^{\prime })$ $(0_I,\mathrm{2.1.8})$; this implies that for every open set $U$ of $Y$, the morphism $f^{-1}(U)\to U$, restriction of $f$, is universally open.

Corollary (4.3.10).

Let $X$, $Y$ be two locally Noetherian integral preschemes, $f:X\to Y$ a proper dominant universally open morphism. If the algebraic closure of $R(Y)$ in $R(X)$ is radicial over $R(Y)$, every fibre $f^{-1}(y)$ ($y\in Y$) is geometrically connected.

Proof. We may restrict to the case where $Y=\mathrm{Spec}(B)$, $B$ being an integral Noetherian ring. It then follows from (II, 7.1.7) that there exists an integrally closed Noetherian local ring $A$ which dominates ${\mathcal{O}}_y$ and has $R(Y)$ for field of fractions. Let $Y^{\prime }=\mathrm{Spec}(A)$, and let $h:Y^{\prime }\to Y$ be the morphism corresponding to the canonical injection $B\to A$, which is birational (hence dominant); moreover, if $y^{\prime }$ is the unique closed point of $Y^{\prime }$, we have $h(y^{\prime })=y$. Let

$X^{\prime }=X{\times }_Y{Y}^{\prime }$, $f^{\prime }=f{\times }_Y{1}_{Y^{\prime }}$; denote by $\eta$, ${\eta }^{\prime }$, $\xi$ the generic points of $Y$, $Y^{\prime }$, and $X$ respectively, so that $f(\xi )=\eta$ and $h^{-1}(\eta )={\eta }^{\prime }$; moreover, $\kappa ({\eta }^{\prime })=\kappa (\eta )=R(Y)$, so $f^{\prime -1}({\eta }^{\prime })$ is isomorphic to $f^{-1}(\eta )$ (I, 3.6.4), and in particular, since $\xi$ is the generic point of $f^{-1}(\eta )$ $(0_I,\mathrm{2.1.8})$, $f^{\prime -1}({\eta }^{\prime })$ has a single generic point. But by hypothesis, every irreducible component of $X^{\prime }$ has its generic point in $f^{\prime -1}({\eta }^{\prime })$, so $X^{\prime }$ is necessarily irreducible, its generic point ${\xi }^{\prime }$ is the generic point of $f^{\prime -1}({\eta }^{\prime })$, and we have $\kappa ({\xi }^{\prime })=\kappa (\xi )$. Set $X^{\prime \prime }=X_{red}^{\prime }$; X'' is then integral and Noetherian, f'' is proper (II, 5.4.6) and the underlying spaces of the fibres $f^{\prime \prime -1}(y^{\prime })$ and $f^{\prime -1}(y^{\prime })$ are the same; moreover, $R(X^{\prime \prime })=\kappa ({\xi }^{\prime })=R(X)$, so f'' satisfies the hypotheses of (4.3.8), and $f^{\prime \prime -1}(y^{\prime })$ is geometrically connected. Now let $k$ be an arbitrary extension of $\kappa (y)$; there exists an extension $k_1$ of $\kappa (y^{\prime })$ such that $\kappa (y)$ and $k$ can be considered as subextensions of $k_1$ (Bourbaki, Alg., chap. V, § 4, prop. 2). By hypothesis, $f^{\prime \prime -1}(y^{\prime }){\times }_{Y^{\prime }}\mathrm{Spec}(k_1)$ is connected, and it has the same reduced prescheme as $f^{\prime -1}(y^{\prime }){\times }_{Y^{\prime }}\mathrm{Spec}(k_1)$ (I, 5.1.8), so the latter is connected, and since it is isomorphic to $f^{-1}(y){\times }_Y\mathrm{Spec}(k_1)$ (I, 3.6.4), we conclude that the latter is connected; a fortiori, the same holds for $f^{-1}(y){\times }_Y\mathrm{Spec}(k)$ by the remark at the beginning of (4.3.4), which completes the proof.

Remarks (4.3.11).

(i) The preceding reasoning is due in substance to Zariski [20], except that he can take for $A$ the integral closure of ${\mathcal{O}}_y$, the latter being a Noetherian ring for the local rings of classical algebraic geometry. On the other hand, Zariski proves that if $Y$ is the Chow variety of a projective space $P_k^r$ over a field $k$, and if $X$ is the closed part of $P_k^r\times Y$ which defines the Chow correspondence between $P_k^r$ and $Y$, then the projection $X\to Y$ is a universally open morphism (loc. cit., lemma on p. 82). It appears indeed to be the only formal property of "Chow coordinates" that has been used in certain applications; consequently, in such a situation, it is of interest to substitute the language of fibres of a proper morphism (possibly assumed universally open or subject to other analogous restrictions of local regularity) for the language of specialization of cycles in projective space.

(ii) In chap. IV, we shall see that a universally open morphism $f:X\to Y$ may also be defined in the following way (which justifies the terminology): for every morphism $Z\to Y$, the morphism $f_{(Z)}:X_{(Z)}\to Z$ is open. One may moreover show that if $f$ satisfies the hypotheses of (4.3.10), then if $y$, $y^{\prime }$ are two points of $Y$ such that $y^{\prime }$ is a specialization of $y$, the geometric number of connected components of $f^{-1}(y^{\prime })$ is at most equal to that of the connected components of $f^{-1}(y)$.

Corollary (4.3.12).

Under the hypotheses of (4.3.5), suppose in addition that $R(Y)$ is algebraically closed in $R(X)$, and let $y$ be a normal point of $Y$. Then $f^{-1}(y)$ is geometrically connected, and there exists an open neighbourhood $U$ of $y$ in $Y$ such that $f_{\ast }({\mathcal{O}}_X)(f^{-1}(U))$ is isomorphic to ${\mathcal{O}}_Y\mid U$. More particularly, if we assume $Y$ normal (and $R(Y)$ algebraically closed in $R(X)$), then $f_{\ast }({\mathcal{O}}_X)$ is isomorphic to ${\mathcal{O}}_Y$.

Proof. The first assertion relative to $f^{-1}(y)$ is a particular case of (4.3.8). We

deduce that if $f:X\to Y^{\prime }\to Y$ is the Stein factorization of $f$ (4.3.3), $g^{-1}(y)$ is reduced to a single point $y^{\prime }$; moreover, we have ${\mathcal{O}}_y\subset {\mathcal{O}}_{y^{\prime }}=(f_{\ast }({\mathcal{O}}_X){)}_y\subset R(X)$, and since ${\mathcal{O}}_{y^{\prime }}$ is finite over ${\mathcal{O}}_y$ (and a fortiori over $R(Y)$), it is contained in $R(Y)$ by virtue of the hypothesis; since $y$ is normal, we necessarily have ${\mathcal{O}}_{y^{\prime }}={\mathcal{O}}_y$, from which we conclude that $g$ is a local isomorphism at the point $y^{\prime }$ (I, 6.5.4), which completes the proof of the first part of the corollary. The second results from the first, for the additional hypothesis implies that $g$ is bijective and a local isomorphism in the neighbourhood of every point of $Y^{\prime }$, hence an isomorphism.

The fact that (4.3.7) is established in the framework of schemes permits applications such as the following:

Proposition (4.3.13).

Let $A$ be a Noetherian unibranch local ring, $\mathfrak{a}$ an ideal of definition of $A$, $A_0=A/\mathfrak{a}$, $S=g{r}_{\mathfrak{a}}(A)$ the graded ring associated to $A$ for the $\mathfrak{a}$-preadic filtration; $S$ is a graded A_0-algebra generated by S_1, S_1 being a finitely generated A_0-module. Then $\mathrm{Proj}(S)$ is a connected A_0-scheme.

Proof. Let $\mathfrak{m}$ be the maximal ideal of $A$; $Y=\mathrm{Spec}(A)$ is an integral scheme whose point corresponding to $\mathfrak{m}$ is the unique closed point. By hypothesis, we have ${\mathfrak{m}}^p\subset \mathfrak{a}\subset \mathfrak{m}$ for an integer $p$, so $V(\mathfrak{a})=\mathfrak{m}$. Let $S^{\prime }={\oplus }_{n\ge 0}{\mathfrak{a}}^n$, and let $X=\mathrm{Proj}(S^{\prime })$, which is the $Y$-scheme obtained by blowing up the ideal $\mathfrak{a}$; $X$ is integral and the structure morphism $f:X\to Y$ is birational (II, 8.1.4) and obviously projective. Consequently, (4.3.7) is applicable and shows that $f^{-1}(\mathfrak{m})$ is connected; but the space $f^{-1}(\mathfrak{m})$ is the underlying space of $\mathrm{Proj}(S^{\prime }{\otimes }_A{A}_0)$ (I, 3.6.1 and II, 2.8.10); since $S^{\prime }{\otimes }_A{A}_0=S$ by definition, the proposition is proved.

4.4. Zariski's "main theorem"

Proposition (4.4.1).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a proper morphism. Let $X^{\prime }$ be the set of points $x\in X$ which are isolated in their fibre $f^{-1}(f(x))$. Then the set $X^{\prime }$ is open in $X$, and if $f=g\circ f^{\prime }$ is the Stein factorization of $f$ (4.3.3), the restriction of $f^{\prime }$ to $X^{\prime }$ is an isomorphism of $X^{\prime }$ onto a subprescheme induced on an open set $U$ of $Y^{\prime }$, and we have $X^{\prime }=f^{\prime -1}(U)$.

Proof. Since $g^{-1}(f(x))$ is finite and discrete (4.3.3 and II, 6.1.7), for $x$ to be isolated in $f^{-1}(f(x))$, it is necessary and sufficient that it be isolated in $f^{\prime -1}(f^{\prime }(x))$; we may thus restrict to the case where $f^{\prime }=f$, hence $f_{\ast }({\mathcal{O}}_X)={\mathcal{O}}_Y$. Then, if $x\in X^{\prime }$, $f^{-1}(f(x))$, which is connected (4.3.2), is necessarily reduced to the point $x$. Since $f$ is closed, for every open neighbourhood $V$ of $x$ in $X$, $f(X-V)$ is closed in $Y$ and does not contain $y=f(x)$, since $f^{-1}(y)=x$; if $U$ is the complement of $f(X-V)$ in $Y$, we have $f^{-1}(U)\subset V$, and we conclude that the inverse images by $f$ of a fundamental system of open neighbourhoods of $y$ form a fundamental system of open neighbourhoods of $x$. The hypothesis $f_{\ast }({\mathcal{O}}_X)={\mathcal{O}}_Y$ and the definition of the direct image of a sheaf $(0_I,\mathrm{3.4.1}and\mathrm{4.2.1})$ then imply that, if $f=(\psi ,\theta )$, the homomorphism ${\theta }_x^{♯}:{\mathcal{O}}_y\to {\mathcal{O}}_x$ is an isomorphism. We conclude that there exists an open neighbourhood $V$ of $x$ and an open neighbourhood $U$ of $y$ such that the restriction of $f$ to $V$ is an isomorphism of $V$ onto $U$ (I, 6.5.4); furthermore, by what we have just seen,

we may suppose $f^{-1}(U)=V$, whence we conclude immediately, by definition, that $V\subset X^{\prime }$, which completes the proof.

The following proposition was proved by Chevalley in the case of algebraic schemes:

Proposition (4.4.2).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism. The following conditions are equivalent:

a) $f$ is finite.

b) $f$ is affine and proper.

c) $f$ is proper and, for every $y\in Y$, $f^{-1}(y)$ is a finite set.

Proof. We know that a) implies b) (II, 6.1.2 and 6.1.11). If $f$ is proper and affine, the same holds for the morphism $f^{-1}(y)\to \mathrm{Spec}(\kappa (y))$ (II, 1.6.2, (iii) and 5.4.2, (iii)), and the finiteness theorem (3.2.1) applied to the structure sheaf of $f^{-1}(y)$ shows that $f^{-1}(y)=\mathrm{Spec}(A)$, where $A$ is a finite $\kappa (y)$-algebra; hence $f^{-1}(y)$ is a finite set (II, 6.1.7), and we see that b) implies c). Finally, since $f^{-1}(y)$ is an algebraic prescheme over $\kappa (y)$, the hypothesis that the set $f^{-1}(y)$ is finite implies that the space $f^{-1}(y)$ is discrete (I, 6.4.4). With the notation of (4.4.1), we therefore have $X^{\prime }=X$, and $f^{\prime }:X\to Y^{\prime }$ is an isomorphism; since $g$ is a finite morphism, we see that c) implies a).

Theorem (4.4.3) ("Main theorem" of Zariski).

Let $Y$ be a Noetherian prescheme, $f:X\to Y$ a quasi-projective morphism, $X^{\prime }$ the set of points $x\in X$ which are isolated in their fibre $f^{-1}(f(x))$. Then $X^{\prime }$ is an open part of $X$, and the subprescheme induced $X^{\prime }$ is isomorphic to a prescheme induced on an open part of a $Y$-prescheme $Y^{\prime }$ finite over $Y$.

Proof. The hypothesis implies that there exists a projective $Y$-prescheme $Z$ such that $X$ is $Y$-isomorphic to a subprescheme induced on an open set of $Z$ (II, 5.3.2 and 5.5.1). We are thus reduced to proving the theorem when $f$ is a projective morphism, hence proper (II, 5.5.3), and it then follows at once from (4.4.1).

Remark (4.4.4).

If $X$ is reduced (resp. irreducible, and $X^{\prime }$ non-empty), one may suppose, in the statement of (4.4.3), that $Y^{\prime }$ is reduced (resp. irreducible). Indeed, one may always replace $Y^{\prime }$ by the subprescheme closure ${\bar{X}}^{\prime }$ of $X^{\prime }$ in $Y^{\prime }$ (I, 9.5.11 and II, 6.1.5, (i) and (ii)), and one knows that if $X^{\prime }$ is reduced, the same holds for ${\bar{X}}^{\prime }$ (I, 9.5.9, (i)); on the other hand, if $X^{\prime }$ is non-empty, it is irreducible if $X$ is, and ${\bar{X}}^{\prime }$ is then also irreducible.

Corollary (4.4.5).

Let $Y$ be a locally Noetherian scheme, $f:X\to Y$ a morphism of finite type, $x$ a point of $X$ isolated in its fibre $f^{-1}(f(x))$. Then there exists an open neighbourhood of $x$ in $X$ which is isomorphic to an open part of a $Y$-prescheme finite over $Y$.

Proof. Let $y=f(x)$, $U$ an affine open neighbourhood of $y$ in $Y$, $V$ an affine open neighbourhood of $x$ in $X$, contained in $f^{-1}(U)$. Since $Y$ is separated, the injection $U\to Y$ is affine (II, 1.6.3), and since $V$ is affine over $U$ (ibid.), the restriction of $f$ to $V$ is an affine morphism $V\to Y$ (II, 1.6.2, (ii)); a fortiori, this restriction is a quasi-projective morphism since it is of finite type (I, 6.3.5 and II, 5.3.4, (i)). It then suffices to apply (4.4.3) to this restriction.

Corollary (4.4.5) may be stated in the language of commutative algebra:

Corollary (4.4.6).

Let $A$ be a Noetherian ring, $B$ an $A$-algebra of finite type, $\mathfrak{q}$ a prime ideal of $B$, $\mathfrak{p}$ its inverse image in $A$. Suppose that $\mathfrak{q}$ is both maximal and minimal in the set of prime ideals of $B$ whose inverse image is $\mathfrak{p}$. Then there exist $g\in B-\mathfrak{q}$, a finite $A$-algebra $A^{\prime }$, and an element $f^{\prime }\in A^{\prime }$ such that the $A$-algebras $B_g$ and $A_{f^{\prime }}^{\prime }$ are isomorphic.

Proof. It suffices to apply (4.4.5) to $Y=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$, the hypothesis on $\mathfrak{q}$ meaning exactly that $\mathfrak{q}$ is isolated in its fibre (I, 1.1.7).

We deduce the following less general-looking result:

Corollary (4.4.7).

Let $A$ be a Noetherian local ring, $B$ an $A$-algebra of finite type, $\mathfrak{n}$ a prime ideal of $B$ whose inverse image in $A$ is the maximal ideal $\mathfrak{m}$. Suppose that $\mathfrak{n}$ is maximal in $B$ and is minimal in the set of prime ideals of $B$ whose inverse image is $\mathfrak{m}$ (which also means that $B_{\mathfrak{n}}$ is primary for $\mathfrak{n}$). Then there exists a finite $A$-algebra $A^{\prime }$ and a maximal ideal ${\mathfrak{m}}^{\prime }$ of $A^{\prime }$ (of which $\mathfrak{m}$ is the inverse image in $A$) such that $B_{\mathfrak{n}}$ is isomorphic to the $A$-algebra $A_{{\mathfrak{m}}^{\prime }}^{\prime }$.

The following particular case of (4.4.7) is also sometimes called the "Main Theorem":

Corollary (4.4.8).

Under the conditions of (4.4.7), suppose in addition that $A$ and $B$ are integral and have the same field of fractions $K$. Then, if $A$ is integrally closed, we have $B_{\mathfrak{n}}=A$.

Proof. Indeed, Remark (4.4.4) shows that we may suppose, in the application of (4.4.7), that $A^{\prime }$ is integral and has $K$ for field of fractions; the hypothesis on $A$ then implies $A^{\prime }=A$, hence $B_{\mathfrak{n}}=A$; since we have $A\subset B\subset B_{\mathfrak{n}}$, we indeed conclude $A=B$.

The statement (4.4.8) is the form given by Zariski to his "Main theorem" (extended to arbitrary Noetherian integral local rings).

The preceding corollaries were local-type variants of (4.4.3), which is a global result. Here is another consequence of global nature:

Corollary (4.4.9).

Let $Y$ be a locally Noetherian integral prescheme, $f:X\to Y$ a separated morphism, of finite type and birational. Suppose in addition $Y$ normal and all the fibres $f^{-1}(y)$ finite for $y\in Y$. Then $f$ is an open immersion; if in addition $f$ is closed (in particular if $f$ is proper), $f$ is an isomorphism.

Proof. Indeed, let $x\in X$, and set $y=f(x)$. Since $f^{-1}(y)$ is an algebraic scheme over $\kappa (y)$, the hypothesis that it is finite implies that it is discrete (I, 6.4.4); in addition, ${\mathcal{O}}_y$ is integrally closed and ${\mathcal{O}}_x$ and ${\mathcal{O}}_y$ have the same field of fractions (I, 7.1.5). We may thus apply (4.4.8), and if $f=(\psi ,\theta )$, the homomorphism ${\theta }_x^{♯}:{\mathcal{O}}_y\to {\mathcal{O}}_x$ is bijective; we conclude (I, 6.5.4) that $f$ is a local isomorphism. But since $f$ is separated and $X$ integral, $f$ is an open immersion (I, 8.2.8). The last assertion follows from the fact that $f$ is dominant.

Proposition (4.4.10).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type. The set $X^{\prime }$ of $x\in X$ isolated in their fibre $f^{-1}(f(x))$ is open in $X$.

Proof. The question being local on $X$ and $Y$, we may suppose $X$ and $Y$ affine Noetherian and $f$ of finite type; $f$ is then an affine morphism of finite type, hence quasi-projective (II, 5.3.4, (i)), and it suffices to apply (4.4.3).

Corollary (4.4.11).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a proper morphism. The set $U$ of points $y\in Y$ such that $f^{-1}(y)$ is discrete is open in $Y$, and the morphism $f^{-1}(U)\to U$ restriction of $f$ is finite. In particular, a proper and quasi-finite morphism $X\to Y$ is finite.