Chapter IV (continued)

§2. Base change and flatness

This section (unlike §6) appeals only exceptionally to Noetherian techniques. Nos. 1 and 2 are scarcely more than translations of elementary properties of flatness from commutative algebra (cf. Bourbaki, Alg. comm., chap. I) and are included here for the convenience of references. The following numbers are devoted above all to "descent" properties along flat or faithfully flat morphisms: if $g:Y^{\prime }\to Y$ is such a morphism, the issue is to be able to assert that a part of $Y$, or an ${\mathcal{O}}_Y$-Module, or a morphism $X\to Y$, has a certain property, when one knows that its inverse image under $g$ has that property. We restrict ourselves here to properties that do not appeal to the general technique of "descent", which will be developed in Chapter V.

2.1. Flat modules on preschemes

(2.1.1)

Let $f:X\to Y$ be a morphism of preschemes and $\mathcal{F}$ an ${\mathcal{O}}_X$-Module; recall $(0_I,\mathrm{6.7.1})$ that $\mathcal{F}$ is said to be $f$-flat (or $Y$-flat) at a point $x\in X$ if ${\mathcal{F}}_x$ is a flat ${\mathcal{O}}_{f(x)}$-module, $f$-flat (or $Y$-flat) if it is $f$-flat at every $x\in X$, and finally that the morphism $f$ is said to be flat at the point $x\in X$ (resp. flat) if ${\mathcal{O}}_X$ is $f$-flat at $x$ (resp. $f$-flat). When $f=1_X$, one says simply that an ${\mathcal{O}}_X$-Module $\mathcal{F}$ is flat at the point $x$ (resp. flat) if it is $X$-flat at this point (resp. at every point $x\in X$), that is, if ${\mathcal{F}}_x$ is a flat ${\mathcal{O}}_x$-module (resp. if this holds for every $x\in X$). Recall that we have proved (III, 1.4.15.1) the following property:

Proposition (2.1.2).

Let $A$, $B$ be two rings, $\varphi :A\to B$ a ring homomorphism, $X=\mathrm{Spec}(B)$, $Y=\mathrm{Spec}(A)$, $f:X\to Y$ the morphism corresponding to $\varphi$, $M$ a $B$-module. For $\mathcal{F}=\tilde{M}$ to be $f$-flat, it is necessary and sufficient that $M$ be a flat $A$-module.

Proposition (2.1.3).

Let $f:X\to Y$ be a morphism of preschemes, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module. The following conditions are equivalent:

a) For every base change $g:Y^{\prime }\to Y$, if one sets $X^{\prime }=X{\times }_Y{Y}^{\prime }$, the functor ${\mathcal{F}}^{\prime }\mapsto \mathcal{F}{\otimes }_Y{\mathcal{F}}^{\prime }$ from the category of quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Modules to that of quasi-coherent ${\mathcal{O}}_{X^{\prime }}$-Modules is exact.

a') Condition a) is satisfied for all canonical morphisms $g:\mathrm{Spec}({\mathcal{O}}_y)\to Y$ (I, 2.4.1), where $y$ runs over $Y$.

b) $\mathcal{F}$ is $f$-flat.

The questions being local on $X$ and $Y$, one may restrict to the case where $Y=\mathrm{Spec}(B)$, $X=\mathrm{Spec}(A)$, $\mathcal{F}=M$, where $M$ is an $A$-module. It is clear that a) entails a'); condition a') entails that for every $x\in X$ the functor $N\mapsto M_{\mathfrak{n}}{\otimes }_{B_{\mathfrak{n}}}N$ is exact in the category of $B_{\mathfrak{n}}$-modules, $\mathfrak{n}$ being the ideal ${\mathfrak{j}}_{f(x)}$ of $B$; this means that $M_{\mathfrak{n}}$ is a flat $B_{\mathfrak{n}}$-module, and it results from $(0_I,\mathrm{6.3.3})$ and from (2.1.2) that $\mathcal{F}$ is $f$-flat. Finally, to see that b) entails a), one may also restrict to the case where $Y^{\prime }=\mathrm{Spec}(A^{\prime })$ is affine and ${\mathcal{F}}^{\prime }=N^{\prime }$, where $N^{\prime }$ is an $A^{\prime }$-module; the conclusion then again follows from (2.1.2) and the definition of flatness, since $(M{\otimes }_A{N}^{\prime })=\mathcal{F}{\otimes }_Y{\mathcal{F}}^{\prime }$.

Proposition (2.1.4).

Let $f:X\to Y$, $g:Y^{\prime }\to Y$ be two morphisms of preschemes, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module; set $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$, ${\mathcal{F}}^{\prime }=\mathcal{F}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$, and let $g^{\prime }$ be the canonical projection $X^{\prime }\to X$. Let $x^{\prime }$ be a point of $X^{\prime }$, $x=g^{\prime }(x^{\prime })$, $y^{\prime }=f^{\prime }(x^{\prime })$, $y=f(x)=g(y^{\prime })$. If $\mathcal{F}$ is $f$-flat at the point $x$, then ${\mathcal{F}}^{\prime }$ is $f^{\prime }$-flat at the point $x^{\prime }$; in particular if $\mathcal{F}$ is $f$-flat, ${\mathcal{F}}^{\prime }$ is $f^{\prime }$-flat; if $f$ is flat, $f^{\prime }$ is flat.

It suffices to prove the first assertion; applying (I, 3.6.5) three times, as well as (I, 2.4.4), one may reduce to the case where $Y=\mathrm{Spec}({\mathcal{O}}_y)$, $X=\mathrm{Spec}({\mathcal{O}}_x)$, $Y^{\prime }=\mathrm{Spec}({\mathcal{O}}_{y^{\prime }})$, $\mathcal{F}=\tilde{M}$, where $M={\mathcal{F}}_x$; the hypothesis and (2.1.2) then entail that $\mathcal{F}$ is $f$-flat — in other words one is reduced to proving a particular case of the second assertion, and this last follows at once from (2.1.3).

Proposition (2.1.5).

Consider a commutative diagram of morphisms of preschemes

  X  ←─g'──  X'
  │           │
  f│         │f'
  ↓           ↓
  Y  ←──g──  Y'
              │
              │h
              ↓
              Z

where $X^{\prime }=X{\times }_Y{Y}^{\prime }$ and $f^{\prime }=f_{(Y^{\prime })}$. Let $x^{\prime }$ be a point of $X^{\prime }$, and set $x=g^{\prime }(x^{\prime })$, $y^{\prime }=f^{\prime }(x^{\prime })$, $y=f(x)=g(y^{\prime })$, $z=h(y^{\prime })$. Let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-Module that is $f$-flat at the point $x$ (resp. $f$-flat), and ${\mathcal{G}}^{\prime }$ a quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Module that is $h$-flat at the point $y^{\prime }$ (resp. $h$-flat); then $\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{G}}^{\prime }$ is a quasi-coherent ${\mathcal{O}}_{X^{\prime }}$-Module that is $(h\circ f^{\prime })$-flat at the point $x^{\prime }$ (resp. $(h\circ f^{\prime })$-flat).

As in (2.1.4), one reduces to the case where $X=\mathrm{Spec}({\mathcal{O}}_x)$, $Y=\mathrm{Spec}({\mathcal{O}}_y)$,

$Y^{\prime }=\mathrm{Spec}({\mathcal{O}}_{y^{\prime }})$ and $Z=\mathrm{Spec}({\mathcal{O}}_z)$, and it then suffices to prove that $\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{G}}^{\prime }$ is $(h\circ f^{\prime })$-flat. Taking (2.1.2) into account, the proposition follows from Bourbaki, Alg. comm., chap. I, §2, n° 7, prop. 8.

Corollary (2.1.6).

Let $f:X\to Y$, $g:Y\to Z$ be two morphisms of preschemes, $\mathcal{F}$ an ${\mathcal{O}}_X$-Module. If $\mathcal{F}$ is $f$-flat at the point $x\in X$ and $g$ is flat at the point $f(x)$, then $\mathcal{F}$ is $(g\circ f)$-flat at the point $x$. In particular, if $f$ and $g$ are flat morphisms, so is $g\circ f$.

This is the particular case of (2.1.5) with $Y^{\prime }=Y$, ${\mathcal{G}}^{\prime }={\mathcal{O}}_{Y^{\prime }}$.

Corollary (2.1.7).

If $f:X\to X^{\prime }$, $g:Y\to Y^{\prime }$ are two flat $S$-morphisms, then $f{\times }_{S}g:X{\times }_{S}Y\to X^{\prime }{\times }_S{Y}^{\prime }$ is a flat morphism.

This follows from (2.1.4) and (2.1.6) (cf. (I, 3.5.1)).

Proposition (2.1.8).

Let $f:X\to Y$ be a morphism of preschemes,

$$0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$$

an exact sequence of quasi-coherent ${\mathcal{O}}_X$-Modules such that ${\mathcal{F}}^{\prime \prime }$ is $Y$-flat.

(i) For every morphism $g:Y^{\prime }\to Y$ and every quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Module ${\mathcal{G}}^{\prime }$, the sequence

  0 → ℱ' ⊗_Y 𝒢' → ℱ ⊗_Y 𝒢' → ℱ'' ⊗_Y 𝒢' → 0

of ${\mathcal{O}}_{X^{\prime }}$-Modules (where $X^{\prime }=X{\times }_Y{Y}^{\prime }$) is exact.

(ii) For $\mathcal{F}$ to be $Y$-flat, it is necessary and sufficient that ${\mathcal{F}}^{\prime }$ be so.

One may obviously suppose $X$, $Y$, $Y^{\prime }$ affine; the conclusion then follows from (2.1.2) and $(0_I,\mathrm{6.1.2})$.

Corollary (2.1.9).

Let ${\mathcal{L}}^{\bullet }$ be a complex of quasi-coherent ${\mathcal{O}}_X$-Modules, $i$ an index such that if one denotes by $d^i:{\mathcal{L}}^i\to {\mathcal{L}}^{i+1}$ the differential, ${\mathcal{B}}^{i+1}({\mathcal{L}}^{\bullet })=Im(d^i)$ and ${\mathcal{Z}}^{i+1}({\mathcal{L}}^{\bullet })=Coker(d^i)$ are $Y$-flat. Then, with the notations of (2.1.8), the canonical homomorphism

  ℋ^i(ℒ^•) ⊗_Y 𝒢' → ℋ^i(ℒ^• ⊗_Y 𝒢')

is bijective.

Since the tensor product is right exact, one has

  𝒵^{i+1}(ℒ^•) ⊗_Y 𝒢' = Coker(d^i ⊗ 1) = 𝒵^{i+1}(ℒ^• ⊗_Y 𝒢')

and ${\mathcal{Z}}^{{}^{\prime }i}({\mathcal{L}}^{\bullet }){\otimes }_Y{\mathcal{G}}^{\prime }={\mathcal{Z}}^{{}^{\prime }i}({\mathcal{L}}^{\bullet }{\otimes }_Y{\mathcal{G}}^{\prime })$. Moreover, in the exact sequence

$$0\to {\mathcal{B}}^{i+1}({\mathcal{L}}^{\bullet })\to {\mathcal{L}}^{i+1}\to {\mathcal{Z}}^{i+1}({\mathcal{L}}^{\bullet })\to 0$$

${\mathcal{Z}}^{i+1}({\mathcal{L}}^{\bullet })$ is $Y$-flat, so it follows from (2.1.8, (i)) that one has the exact sequence

  0 → ℬ^{i+1}(ℒ^•) ⊗_Y 𝒢' → ℒ^{i+1} ⊗_Y 𝒢' → 𝒵^{i+1}(ℒ^• ⊗_Y 𝒢') → 0

whence ${\mathcal{B}}^{i+1}({\mathcal{L}}^{\bullet }){\otimes }_Y{\mathcal{G}}^{\prime }=Im(d^i\otimes 1)={\mathcal{B}}^{i+1}({\mathcal{L}}^{\bullet }{\otimes }_Y{\mathcal{G}}^{\prime })$. Then, in the exact sequence

$$0\to {\mathcal{H}}^i({\mathcal{L}}^{\bullet })\to {\mathcal{Z}}^i({\mathcal{L}}^{\bullet })\to {\mathcal{B}}^{i+1}({\mathcal{L}}^{\bullet })\to 0$$

${\mathcal{B}}^{i+1}({\mathcal{L}}^{\bullet })$ is $Y$-flat, so it follows from (2.1.8, (i)) and what precedes that one has the exact sequence

  0 → ℋ^i(ℒ^•) ⊗_Y 𝒢' → 𝒵^i(ℒ^• ⊗_Y 𝒢') → ℬ^{i+1}(ℒ^• ⊗_Y 𝒢') → 0

which proves the corollary.

Corollary (2.1.10).

Let $f:X\to Y$ be a morphism of preschemes, $\mathcal{F}$ a quasi-coherent and $Y$-flat ${\mathcal{O}}_X$-Module, ${\mathcal{L}}_{\bullet }=({\mathcal{L}}_i)$ a left resolution of $\mathcal{F}$ formed of quasi-coherent and $Y$-flat ${\mathcal{O}}_X$-Modules. Then, for every morphism $g:Y^{\prime }\to Y$ and every quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Module ${\mathcal{G}}^{\prime }$, the complex ${\mathcal{L}}_{\bullet }{\otimes }_Y{\mathcal{G}}^{\prime }=({\mathcal{L}}_i{\otimes }_Y{\mathcal{G}}^{\prime }{)}_{i\ge 0}$ is a left resolution of $\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$.

Moreover, if ${\mathcal{Z}}_i({\mathcal{L}}_{\bullet })=Ker({\mathcal{L}}_i\to {\mathcal{L}}_{i-1})$, the ${\mathcal{Z}}_i({\mathcal{L}}_{\bullet })$ are $Y$-flat, and one has

  𝒵_i(ℒ_•) ⊗_Y 𝒢' = 𝒵_i(ℒ_• ⊗_Y 𝒢') = Ker(ℒ_i ⊗_Y 𝒢' → ℒ_{i−1} ⊗_Y 𝒢').

Set ${\mathcal{R}}_i=Im({\mathcal{L}}_{i+1}\to {\mathcal{L}}_i)={\mathcal{Z}}_i({\mathcal{L}}_{\bullet })$; one then has the exact sequences

  0 ← ℱ ← ℒ_0 ← ℛ_0 ← 0
  ⋮
  0 ← ℛ_i ← ℒ_{i+1} ← ℛ_{i+1} ← 0
  ⋮

and since $\mathcal{F}$ and the ${\mathcal{L}}_i$ are $Y$-flat, one deduces from (2.1.8, (ii)) by induction that all the ${\mathcal{R}}_i$ are also $Y$-flat; using (2.1.8, (i)), one therefore has the exact sequences

  0 ← ℱ ⊗_Y 𝒢' ← ℒ_0 ⊗_Y 𝒢' ← ℛ_0 ⊗_Y 𝒢' ← 0
  0 ← ℛ_i ⊗_Y 𝒢' ← ℒ_{i+1} ⊗_Y 𝒢' ← ℛ_{i+1} ⊗_Y 𝒢' ← 0          (i ≥ 0)

which prove the corollary.

Proposition (2.1.11).

Let $f:X\to Y$ be a flat morphism, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_Y$-Module of finite presentation. If $\mathcal{J}$ is the Ideal of ${\mathcal{O}}_Y$ annihilator of $\mathcal{F}$, then $f\ast (\mathcal{J})$ is the Ideal of ${\mathcal{O}}_X$ annihilator of $f\ast (\mathcal{F})$.

One has by definition an exact sequence $(0_I,\mathrm{5.3.7})$

  0 → 𝒥 → 𝒪_Y → ℋom_{𝒪_Y}(ℱ, ℱ)

whence, since $f$ is flat, an exact sequence

  0 → f*(𝒥) → 𝒪_X → f*(ℋom_{𝒪_Y}(ℱ, ℱ))

and since by hypothesis $\mathcal{F}$ is an ${\mathcal{O}}_Y$-Module of finite presentation, $f\ast (\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},\mathcal{F}))$ is canonically identified with $\mathcal{H}o{m}_{{\mathcal{O}}_X}(f\ast (\mathcal{F}),f\ast (\mathcal{F}))$ $(0_I,\mathrm{6.7.6})$, whence the conclusion.

Proposition (2.1.12).

Let $X$ be a prescheme, $\mathcal{F}$ an ${\mathcal{O}}_X$-Module of finite presentation, $x$ a point of $X$. The following conditions are equivalent:

a) ${\mathcal{F}}_x$ is a flat ${\mathcal{O}}_x$-module.

b) There exists an open neighbourhood $U$ of $x$ such that $\mathcal{F}|U$ is a locally free $({\mathcal{O}}_X|U)$-Module.

Indeed, ${\mathcal{F}}_x$ is an ${\mathcal{O}}_x$-module of finite presentation and ${\mathcal{O}}_x$ a local ring; it therefore amounts to the same to say that ${\mathcal{F}}_x$ is a flat ${\mathcal{O}}_x$-module or a free ${\mathcal{O}}_x$-module (Bourbaki, Alg. comm., chap. II, §3, n° 2, cor. 2 of prop. 5); whence the conclusion, taking account of $(0_I,\mathrm{5.2.7})$. We note that the proposition is valid for an arbitrary ringed space in local rings.

Proposition (2.1.13).

*Let $f:X\to Y$ be a morphism of preschemes. If $f$ is flat at a point $x\in X$ and the ring ${\mathcal{O}}_x$ is reduced (resp. integral and integrally closed), then the ring ${\mathcal{O}}_{f(x)}$

is reduced (resp. integral and integrally closed). If $f$ is faithfully flat and $X$ is reduced (resp. normal), then $Y$ is reduced (resp. normal).*

Set ${\mathcal{O}}_{f(x)}=A$, ${\mathcal{O}}_x=B$. If $B$ is a flat $A$-module, it is also a faithfully flat $A$-module $(0_I,\mathrm{6.6.2})$, so $A$ is identified with a subring of $B$; if $B$ is reduced, so therefore is $A$. Suppose now that $B$ is integral and integrally closed, and let $L$ be its field of fractions; then $A\subset B$ is integral; denote by $K\subset L$ its field of fractions. The hypothesis entails that $B\cap K=A$ (Bourbaki, Alg. comm., chap. I, §3, n° 5, prop. 10). If then $t\in K$ is integral over $A$, it is also integral over $B$, hence belongs to $B$ by hypothesis, and consequently $t\in A$, which proves that $A$ is integrally closed.

Proposition (2.1.14).

Let $f:X\to Y$ be a faithfully flat morphism. If $X$ is locally integral and the space underlying $Y$ is locally Noetherian, then $Y$ is locally integral.

Indeed, every $y\in Y$ is of the form $f(x)$ for some $x\in X$ and by hypothesis ${\mathcal{O}}_y$ is identified with a subring of ${\mathcal{O}}_x$ $(0_I,\mathrm{6.6.1})$; since ${\mathcal{O}}_x$ is integral, so is ${\mathcal{O}}_y$, and this proves the proposition (I, 5.1.4).

2.2. Faithfully flat modules on preschemes

Proposition (2.2.1).

Let $f:X\to Y$ be a morphism of preschemes, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module. The following conditions are equivalent:

a) For every base change $g:Y^{\prime }\to Y$, if one sets $X^{\prime }=X{\times }_Y{Y}^{\prime }$, the functor ${\mathcal{G}}^{\prime }\mapsto \mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ from the category of quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Modules to that of quasi-coherent ${\mathcal{O}}_{X^{\prime }}$-Modules is exact and faithful.

a') Condition a) is satisfied for all canonical morphisms $g:\mathrm{Spec}({\mathcal{O}}_y)\to Y$ (I, 2.4.1), where $y$ runs over $Y$.

a'') Condition a) is satisfied for all canonical immersions $Y^{\prime }\to Y$, where $Y^{\prime }$ runs over the set of affine open subschemes of $Y$.

b) $\mathcal{F}$ is $Y$-flat and, for every $y\in Y$, if one denotes by $X_y$ the fibre $f^{-1}(y)$, the ${\mathcal{O}}_{X_y}$-Module ${\mathcal{F}}_y=\mathcal{F}{\otimes }_{Y}k(y)$ is $\ne 0$.

It is clear that a) implies a') and a''); condition a') first implies that $\mathcal{F}$ is $Y$-flat (2.1.3); on the other hand it implies that for every $y\in Y$ the functor $N\mapsto {\mathcal{F}}_y{\otimes }_{{\mathcal{O}}_y}\tilde{N}$ is faithful in the category of ${\mathcal{O}}_y$-modules; taking in particular $N=k(y)$, one obtains the second assertion of b). To show that b) implies a), one may restrict to the case where $Y$ is affine, the question being local on $Y$. Similarly, to prove that a'') implies a), one is reduced to proving that when $Y$ is affine, the fact that ${\mathcal{G}}^{\prime }\mapsto \mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ is an exact and faithful functor entails condition a). In other words, one is reduced to proving the following more precise proposition:

Proposition (2.2.2).

Let $Y=\mathrm{Spec}(A)$ be an affine scheme, $f:X\to Y$ a morphism of preschemes, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module. Condition a) of (2.2.1) is equivalent to each of the following:

b') $\mathcal{F}$ is $Y$-flat and, for every closed point $y$ of $Y$, one has ${\mathcal{F}}_y\ne 0$.

c) The functor $\mathcal{G}\mapsto \mathcal{F}{\otimes }_Y\mathcal{G}$ from the category of quasi-coherent ${\mathcal{O}}_Y$-Modules to that of quasi-coherent ${\mathcal{O}}_X$-Modules is exact and faithful.

If b') is satisfied, there is at least one $x\in f^{-1}(y)$ such that $({\mathcal{F}}_y{)}_x\ne 0$; let $U=\mathrm{Spec}(B)$ be an affine open neighbourhood of $x$, and let $\mathcal{F}|U=M$, where $M$ is a $B$-module. Then b') implies that $M/{\mathfrak{j}}_{y}M\ne 0$, and consequently (since $M$ is a flat $A$-module by (2.1.2)) that $M{\otimes }_A{A}_y$ is a faithfully flat $A_y$-module $(0_I,\mathrm{6.4.5})$. The relation $(\mathcal{F}{\otimes }_Y\mathcal{G}){\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_y=0$ for a closed point $y$ of $Y$ implies $(\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_U{)}_x{\otimes }_{{\mathcal{O}}_y}{\mathcal{G}}_y=0$, hence ${\mathcal{G}}_y=0$. But if ${\mathcal{G}}_y=0$ for every closed point $y$ of $Y$, one concludes that $\mathcal{G}=0$; indeed, if $\mathcal{G}=N$, the annihilator of an element of $N$ is contained in no maximal ideal of $A$, so it equals all of $A$. The relation $\mathcal{F}{\otimes }_Y\mathcal{G}=0$ therefore implies $\mathcal{G}=0$; in other words, the functor $\mathcal{G}\mapsto \mathcal{F}{\otimes }_Y\mathcal{G}$ is faithful; we know moreover that this functor is exact (2.1.3), which shows that b') entails c).

Finally, to see that c) entails a), one may restrict to the case where $Y^{\prime }$ is also affine; as $g:Y^{\prime }\to Y$ is then an affine morphism, so is the projection $g^{\prime }:X^{\prime }\to X$ (II, 1.5.5); in addition, the functor $\mathcal{H}\mapsto g_{\ast }^{\prime }(\mathcal{H})$ is then exact in the category of quasi-coherent ${\mathcal{O}}_{X^{\prime }}$-Modules (I, 1.6.4), and if $g_{\ast }^{\prime }(\mathcal{H})=\mathcal{H}$, one has ${\mathcal{H}}^{\prime }=\tilde{\mathcal{H}}$, so the preceding functor is also faithful; to see that ${\mathcal{G}}^{\prime }\mapsto \mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ is exact and faithful, it therefore suffices to see that the functor ${\mathcal{G}}^{\prime }\mapsto g_{\ast }^{\prime }(\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime })$ is. Now, if $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$, one has $g_{\ast }^{\prime }(\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime })=g_{\ast }^{\prime }(g^{\prime }\ast (\mathcal{F}){\otimes }_{{\mathcal{O}}_{X^{\prime }}}{f}^{\prime }\ast ({\mathcal{G}}^{\prime }))$; the fact that $g$ is affine entails that one has a canonical isomorphism

  ℱ ⊗_{𝒪_X} f*(g_*(𝒢')) ⥲ g'_*(g'*(ℱ) ⊗_{𝒪_{X'}} f'*(𝒢')).            (2.2.2.1)

Indeed, one knows (II, 1.5.2) that one has a canonical isomorphism

$$f\ast (g_{\ast }({\mathcal{G}}^{\prime }))\stackrel{\sim }{\to }{g}_{\ast }^{\prime }(f^{\prime }\ast ({\mathcal{G}}^{\prime })),$$

and on the other hand one has a canonical homomorphism $\mathcal{F}\to g_{\ast }^{\prime }(g^{\prime }\ast (\mathcal{F}))$ $(0_I,\mathrm{4.4.3.2})$; composing the homomorphism $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}f\ast (g_{\ast }({\mathcal{G}}^{\prime }))\to g_{\ast }^{\prime }(g^{\prime }\ast (\mathcal{F})){\otimes }_{{\mathcal{O}}_X}{g}_{\ast }^{\prime }(f^{\prime }\ast ({\mathcal{G}}^{\prime }))$ with the canonical homomorphism $(0_I,\mathrm{4.2.2.1})$, one deduces the homomorphism (2.2.2.1); the verification that it is an isomorphism is immediate by reducing to the case where $X$ is affine. This being so, the functor ${\mathcal{G}}^{\prime }\mapsto g_{\ast }({\mathcal{G}}^{\prime })$ is exact and faithful and by hypothesis so is $\mathcal{G}\mapsto \mathcal{F}{\otimes }_Y\mathcal{G}=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}f\ast (\mathcal{G})$; their composite is therefore exact and faithful, which completes the proof of (2.2.1) and (2.2.2).

Corollary (2.2.3).

Let $X=\mathrm{Spec}(B)$, $Y=\mathrm{Spec}(A)$ be two affine schemes, $f:X\to Y$ a morphism, $\mathcal{F}=\tilde{M}$ a quasi-coherent ${\mathcal{O}}_X$-Module. For $\mathcal{F}$ to satisfy the equivalent conditions of (2.2.1) (or (2.2.2)), it is necessary and sufficient that the $A$-module $M$ be faithfully flat.

Indeed, condition c) of (2.2.2) then means that the functor $N\mapsto M{\otimes }_{A}N$ from the category of $A$-modules to that of $B$-modules is exact and faithful, and the conclusion follows from $(0_I,\mathrm{6.4.1})$.

Definition (2.2.4).

When the equivalent conditions of (2.2.1) are satisfied, one says that the quasi-coherent ${\mathcal{O}}_X$-Module $\mathcal{F}$ is faithfully flat relative to $f$ (or to $Y$).

One notes that this notion is local on $Y$, but not on $X$; in particular one can have ${\mathcal{F}}_x=0$ for certain $x\in X$, in other words $Supp(\mathcal{F})$ is not necessarily equal to $X$. Nevertheless, it follows at once from criterion (2.2.1, b) that for every $y\in Y$ there exists at least one $x\in f^{-1}(y)$ such that $({\mathcal{F}}_y{)}_x={\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_y}k(y)\ne 0$, and a fortiori ${\mathcal{F}}_x\ne 0$; in other words:

Corollary (2.2.5).

If $\mathcal{F}$ is a quasi-coherent ${\mathcal{O}}_X$-Module, faithfully flat relative to $f$, one has $f(Supp(\mathcal{F}))=Y$, and a fortiori $f$ is a surjective morphism.

This result admits a partial converse:

Corollary (2.2.6).

Let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-Module of finite type. For $\mathcal{F}$ to be faithfully flat relative to $f$, it is necessary and sufficient that $\mathcal{F}$ be $f$-flat and that $f(Supp(\mathcal{F}))=Y$.

Indeed (I, 9.1.13 and 3.6.1) one has $Supp({\mathcal{F}}_y)=f^{-1}(y)\cap Supp(\mathcal{F})$, so in this case criterion (2.2.1, b) is none other than the condition of the corollary.

In particular, the ${\mathcal{O}}_X$-Module ${\mathcal{O}}_X$ is faithfully flat relative to $f$ if and only if it is $f$-flat and $f$ is surjective, in other words if and only if the morphism $f$ is faithfully flat $(0_I,\mathrm{6.7.8})$.

Let us make explicit the properties involved in definition (2.2.4):

Proposition (2.2.7).

Let $f:X\to Y$ be a morphism of preschemes, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module faithfully flat relative to $f$. Then, for a sequence ${\mathcal{G}}^{\prime }\to \mathcal{G}\to {\mathcal{G}}^{\prime \prime }$ of quasi-coherent ${\mathcal{O}}_Y$-Modules to be exact, it is necessary and sufficient that the corresponding sequence $\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }\to \mathcal{F}{\otimes }_Y\mathcal{G}\to \mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime \prime }$ be so. In particular, for a homomorphism $u:\mathcal{G}\to {\mathcal{G}}^{\prime }$ of quasi-coherent ${\mathcal{O}}_Y$-Modules to be injective (resp. surjective, bijective, zero), it is necessary and sufficient that $1_{\mathcal{F}}\otimes f\ast (u):\mathcal{F}{\otimes }_Y\mathcal{G}\to \mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ be so. For a quasi-coherent ${\mathcal{O}}_Y$-Module $\mathcal{G}$ to be zero, it is necessary and sufficient that $\mathcal{F}{\otimes }_Y\mathcal{G}=0$. For every quasi-coherent ${\mathcal{O}}_Y$-Module $\mathcal{G}$, the map ${\mathcal{G}}^{\prime }\mapsto \mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ from the set of quasi-coherent sub-${\mathcal{O}}_Y$-Modules of $\mathcal{G}$ to the set of quasi-coherent sub-${\mathcal{O}}_X$-Modules of $\mathcal{F}{\otimes }_Y\mathcal{G}$ is injective.

To prove the last assertion — that is, that for two quasi-coherent sub-${\mathcal{O}}_Y$-Modules ${\mathcal{G}}^{\prime }$, ${\mathcal{G}}^{\prime \prime }$ of $\mathcal{G}$, the relation $\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }=\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime \prime }$ entails ${\mathcal{G}}^{\prime }={\mathcal{G}}^{\prime \prime }$ — one may (replacing ${\mathcal{G}}^{\prime \prime }$ by ${\mathcal{G}}^{\prime }+{\mathcal{G}}^{\prime \prime }$) reduce to the case where ${\mathcal{G}}^{\prime }\subset {\mathcal{G}}^{\prime \prime }$, and it then suffices to apply the second assertion of the statement to the injection $u:{\mathcal{G}}^{\prime }\to {\mathcal{G}}^{\prime \prime }$.

Corollary (2.2.8).

Let $f:X\to Y$ be a faithfully flat morphism. For every quasi-coherent ${\mathcal{O}}_Y$-Module $\mathcal{G}$, the canonical map

  Γ(Y, 𝒢) → Γ(X, f*(𝒢))                                       (2.2.8.1)

is injective.

Indeed, $\Gamma (Y,\mathcal{G})$ is canonically identified with ${\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{O}}_Y,\mathcal{G})$ $(0_I,\mathrm{5.1.1})$ and likewise $\Gamma (X,f\ast (\mathcal{G}))$ with ${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{O}}_X,f\ast (\mathcal{G}))$. By virtue of (2.2.1) and (2.2.4), the hypothesis entails that the functor $\mathcal{G}\mapsto f\ast (\mathcal{G})$ is exact and faithful in the category of quasi-coherent ${\mathcal{O}}_Y$-Modules, and consequently a homomorphism $u:{\mathcal{O}}_Y\to \mathcal{G}$ is zero if and only if $f\ast (u):f\ast ({\mathcal{O}}_Y)={\mathcal{O}}_X\to f\ast (\mathcal{G})$ is zero.

Remark (2.2.9).

The results of (2.2.7) and (2.2.8) still hold when the ${\mathcal{O}}_Y$-Modules ${\mathcal{G}}^{\prime }$, $\mathcal{G}$, ${\mathcal{G}}^{\prime \prime }$ appearing there are arbitrary (not necessarily quasi-coherent). Indeed, for every $y\in Y$, there exists $x\in f^{-1}(y)$ such that ${\mathcal{F}}_x$ is a ${\mathcal{O}}_y$-module

faithfully flat, and consequently the functor ${\mathcal{G}}_y\mapsto {\mathcal{G}}_y{\otimes }_{{\mathcal{O}}_y}{\mathcal{F}}_x$ is faithful; since moreover for every $x\in f^{-1}(y)$ the functor ${\mathcal{G}}_y\mapsto {\mathcal{G}}_y{\otimes }_{{\mathcal{O}}_y}{\mathcal{F}}_x$ is exact, one deduces the conclusion at once.

Proposition (2.2.10).

Let $f:X\to Y$, $g:Y^{\prime }\to Y$, $h:Y^{\prime }\to Z$ be three morphisms of preschemes, $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module, faithfully flat relative to $Y$, ${\mathcal{G}}^{\prime }$ a quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Module. For ${\mathcal{G}}^{\prime }$ to be flat (resp. faithfully flat) relative to $Z$, it is necessary and sufficient that $\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ be a flat (resp. faithfully flat) ${\mathcal{O}}_{X^{\prime }}$-Module relative to $Z$.

One knows already that if ${\mathcal{G}}^{\prime }$ is $Z$-flat, then so is $\mathcal{F}{\otimes }_Y{\mathcal{G}}^{\prime }$ (2.1.5). Consider an arbitrary base change $Z^{\prime \prime }\to Z$ and set $X^{\prime \prime }=X^{\prime }{\times }_Z{Z}^{\prime \prime }$; if ${\mathcal{G}}^{\prime }$ is faithfully flat relative to $Z$, the functor

  ℋ'' ↦ ℋ'' ⊗_Z 𝒢' → (ℋ'' ⊗_Z 𝒢') ⊗_Y ℱ = ℋ'' ⊗_Z (𝒢' ⊗_Y ℱ)        (2.2.10.1)

from the category of quasi-coherent ${\mathcal{O}}_{Z^{\prime \prime }}$-Modules to that of ${\mathcal{O}}_{X^{\prime \prime }}$-Modules is the composite of two exact and faithful functors, hence is exact and faithful. Conversely, if this composite functor is exact (resp. exact and faithful), so is ${\mathcal{M}}^{\prime \prime }\mapsto {\mathcal{M}}^{\prime \prime }{\otimes }_Z{\mathcal{G}}^{\prime }$, since the functor ${\mathcal{M}}^{\prime }\mapsto {\mathcal{M}}^{\prime }{\otimes }_Y\mathcal{F}$ (from the category of quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-Modules to that of ${\mathcal{O}}_{X^{\prime }}$-Modules) is exact and faithful by hypothesis.

Corollary (2.2.11).

(i) Let $f:X\to Y$, $g:Y^{\prime }\to Y$ be two morphisms, $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module. If $\mathcal{F}$ is faithfully flat relative to $Y$, then ${\mathcal{F}}^{\prime }=\mathcal{F}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$ is faithfully flat relative to $Y^{\prime }$.

(ii) Let $f:X\to Y$, $g:Y\to Z$ be two morphisms, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module, faithfully flat relative to $Y$. For $g$ to be a faithfully flat morphism, it is necessary and sufficient that $\mathcal{F}$ be faithfully flat relative to $Z$.

(iii) Let $f:X\to Y$, $g:Y\to Z$ be two morphisms, $\mathcal{G}$ a quasi-coherent ${\mathcal{O}}_Y$-Module. Suppose the morphism $f$ is faithfully flat. For $\mathcal{G}$ to be flat (resp. faithfully flat) relative to $Z$, it is necessary and sufficient that $f\ast (\mathcal{G})$ be flat (resp. faithfully flat) relative to $Z$.

(iv) Let $f:X\to Y$, $g:Y\to Z$ be two morphisms, $x$ a point of $X$. Suppose $f$ is flat at the point $x$. For $g\circ f$ to be flat at the point $x$, it is necessary and sufficient that $g$ be flat at the point $f(x)$.

To prove (i), one applies (2.2.10) replacing $Z$ by $Y^{\prime }$ and ${\mathcal{G}}^{\prime }$ by ${\mathcal{O}}_{Y^{\prime }}$. To prove (ii), one applies (2.2.10) taking the identity for the morphism $Y^{\prime }\to Y$ and replacing ${\mathcal{G}}^{\prime }$ by ${\mathcal{O}}_Y$. To prove (iii), one applies (2.2.10) again taking the identity for $Y^{\prime }\to Y$ and replacing $\mathcal{F}$ by ${\mathcal{O}}_X$ and ${\mathcal{G}}^{\prime }$ by $\mathcal{G}$. Finally (iv) is deduced from (ii) by replacing $X$ by $\mathrm{Spec}({\mathcal{O}}_x)$, $\mathcal{F}$ by ${\mathcal{O}}_X$, $Y$ by $\mathrm{Spec}({\mathcal{O}}_{f(x)})$, and $Z$ by $\mathrm{Spec}({\mathcal{O}}_{g(f(x))})$, taking account of $(0_I,\mathrm{6.6.2})$.

Corollary (2.2.12).

Let $Y$ be an affine scheme, $f:X\to Y$ a quasi-compact morphism, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module. If $\mathcal{F}$ is faithfully flat relative to $f$, there exists an affine scheme $X^{\prime }$ and a surjective local isomorphism $g:X^{\prime }\to X$ such that $g\ast (\mathcal{F})$ is faithfully flat relative to $f\circ g$.

Indeed, $X$ is quasi-compact, hence a finite union of affine open sets $X_i$; it suffices to take for $X^{\prime }$ the affine scheme sum of the preschemes induced on the open sets $X_i$, and for $g:X^{\prime }\to X$ the canonical morphism; it is clear that $g$ is faithfully flat, and

the hypothesis entails that $g\ast (\mathcal{F})$ is faithfully flat relative to $f\circ g$, by virtue of (2.2.11, (iii)).

Corollary (2.2.13).

(i) Let $f:X\to Y$, $g:Y^{\prime }\to Y$ be two morphisms, $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$. If $f$ is a faithfully flat morphism, so is $f^{\prime }$.

(ii) If $f:X\to X^{\prime }$, $g:Y\to Y^{\prime }$ are two faithfully flat $S$-morphisms, then

  f ×_S g : X ×_S Y → X' ×_S Y'

is faithfully flat.

(iii) Let $f:X\to Y$, $g:Y\to Z$ be two morphisms such that $f$ is faithfully flat. For $g$ to be a flat (resp. faithfully flat) morphism, it is necessary and sufficient that $g\circ f$ be a flat (resp. faithfully flat) morphism.

Proposition (2.2.14).

Let $f:X\to Y$ be a faithfully flat and quasi-compact morphism. If $X$ is locally Noetherian, so is $Y$.

The question being local on $Y$, one may suppose $Y=\mathrm{Spec}(A)$ affine; since $f$ is quasi-compact, it follows from (2.2.12) that one may also restrict to the case where $X=\mathrm{Spec}(B)$ is affine. Then $B$ is a Noetherian ring by hypothesis, and a faithfully flat $A$-module (2.2.3); hence $A$ is Noetherian $(0_I,\mathrm{6.5.2})$.

Proposition (2.2.15).

Let $f:X\to Y$ be a faithfully flat morphism. If $\mathfrak{S}(X)$ and $\mathfrak{S}(Y)$ denote respectively the set of sub-preschemes of $X$ and $Y$, the map $Z\mapsto f^{-1}(Z)$ from $\mathfrak{S}(Y)$ to $\mathfrak{S}(X)$ is injective.

Since $f$ is surjective, one has, for the underlying set of a sub-prescheme $Z$ of $Y$, $f(f^{-1}(Z))=Z$. On the other hand, if $U$ is an open set of $Y$ containing $Z$ and in which $Z$ is closed, $f^{-1}(U)$ is open in $X$, $f^{-1}(Z)$ is closed in $f^{-1}(U)$, and the restriction $f^{-1}(U)\to U$ of $f$ is a faithfully flat morphism. One may therefore restrict to considering only closed sub-preschemes of $Y$. Now, if $Z$ is a closed sub-prescheme of $Y$ corresponding to a quasi-coherent Ideal $\mathcal{J}$ of ${\mathcal{O}}_Y$, one knows that $f^{-1}(Z)$ corresponds to the quasi-coherent Ideal $f\ast (\mathcal{J}){\mathcal{O}}_X$ of ${\mathcal{O}}_X$ (I, 4.4.5), and since $f$ is flat, $f\ast (\mathcal{J}){\mathcal{O}}_X$ is identified with $f\ast (\mathcal{J})$. But the map $\mathcal{J}\mapsto f\ast (\mathcal{J})$ from the set of quasi-coherent Ideals of ${\mathcal{O}}_Y$ to the set of quasi-coherent Ideals of ${\mathcal{O}}_X$ is injective (2.2.7), whence the conclusion.

Corollary (2.2.16).

Let $X$, $Y$ be two $S$-preschemes; if $S^{\prime }\to S$ is a faithfully flat morphism, the map $f\mapsto f_{(S^{\prime })}$ from ${\mathrm{Hom}}_S(X,Y)$ to ${\mathrm{Hom}}_{S^{\prime }}(X_{(S^{\prime })},Y_{(S^{\prime })})$ is injective.

One has $X_{(S^{\prime })}{\times }_{S^{\prime }}{Y}_{(S^{\prime })}=(X{\times }_{S}Y{)}_{(S^{\prime })}$ (I, 3.3.10), so the projection morphism $X_{(S^{\prime })}{\times }_{S^{\prime }}{Y}_{(S^{\prime })}\to X{\times }_{S}Y$ is faithfully flat (2.2.13). The elements of ${\mathrm{Hom}}_S(X,Y)$ correspond bijectively to the sub-preschemes of $X{\times }_{S}Y$ that are the graphs of these $S$-morphisms (I, 5.3.11), and if $Z_f$ is the graph of $f\in {\mathrm{Hom}}_S(X,Y)$, one has $Z_{f_{(S^{\prime })}}=g^{-1}(Z_f)$ (I, 5.3.12). It therefore suffices to apply proposition (2.2.15) to $g$.

Proposition (2.2.17).

Let $A$ be a ring, $B$ an $A$-algebra such that $B$ is a faithfully flat and finitely presented $A$-module. Then the structure homomorphism $\varphi :A\to B$ is an isomorphism of the $A$-module $A$ onto a direct factor of the $A$-module $B$. If $A$ is a local ring, $B$ is a free $A$-module and there exists a basis of this module containing the unit element of $B$.

By virtue of Bourbaki, Alg. comm., chap. II, §3, n° 3, prop. 12, it suffices to prove the proposition when $A$ is a local ring; one then knows (loc. cit., n° 2, cor. 2 of prop. 5) that $B$ is a free $A$-module of finite type, and the conclusion follows from loc. cit., prop. 5.

2.3. Topological properties of flat morphisms

Lemma (2.3.1).

Let $f:X\to Y$ be a quasi-compact and quasi-separated morphism, $g:Y^{\prime }\to Y$ a flat morphism; set $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$. Then, for every quasi-coherent ${\mathcal{O}}_X$-Module $\mathcal{F}$, the canonical homomorphism

  g*(f_*(ℱ)) → f'_*(ℱ ⊗_{𝒪_X} 𝒪_{X'})                            (2.3.1.1)

is bijective.

This is a particular case of (III, 1.4.15) (improved in (1.7.21)).

Proposition (2.3.2).

Let $S$ be a prescheme, $f:X\to Y$ a quasi-compact and quasi-separated $S$-morphism; let $Z$ be the sub-prescheme of $Y$, closed image of $X$ under $f$ ((I, 9.5.3) and (1.7.8)), and let $j:Z\to Y$ be the canonical injection, so that one has $f=j\circ g$, where $g:X\to Z$ is a morphism (loc. cit.). Let $h:S^{\prime }\to S$ be a flat morphism, and set $f^{\prime }=f_{(S^{\prime })}:X_{(S^{\prime })}\to Y_{(S^{\prime })}$; then $j^{\prime }=j_{(S^{\prime })}:Z_{(S^{\prime })}\to Y_{(S^{\prime })}$ is identified with the canonical injection of the sub-prescheme of $Y_{(S^{\prime })}$ closed image of $X_{(S^{\prime })}$ under $f^{\prime }$.

Since the morphism $Y_{(S^{\prime })}\to Y$ is flat (2.1.4), one may restrict to the case where $S=Y$ (I, 3.3.11); if $f=(\psi ,\theta )$, one knows that $Z$ is the closed sub-prescheme of $S$ defined by the (quasi-coherent) Ideal $\mathcal{J}$ of ${\mathcal{O}}_S$, kernel of the homomorphism $\theta :{\mathcal{O}}_S\to f_{\ast }({\mathcal{O}}_X)$ (I, 9.5.2). Since $h$ is a flat morphism, the quasi-coherent Ideal $\mathcal{J}{\mathcal{O}}_{S^{\prime }}$ of ${\mathcal{O}}_{S^{\prime }}$ is identified with the kernel of $h\ast (\theta ):{\mathcal{O}}_{S^{\prime }}\to h\ast (f_{\ast }({\mathcal{O}}_X))$. Now, if $f^{\prime }=({\psi }^{\prime },{\theta }^{\prime })$, one verifies easily (for example by reducing to the case where $Y$ and $Y^{\prime }$ are affine and using (I, 2.2.4)) that ${\theta }^{\prime }:{\mathcal{O}}_{S^{\prime }}\to f_{\ast }^{\prime }({\mathcal{O}}_{X^{\prime }})$ is the composite of the canonical homomorphism (2.3.1.1) $h\ast (f_{\ast }({\mathcal{O}}_X))\to f_{\ast }^{\prime }({\mathcal{O}}_{X^{\prime }})$ and of $h\ast (\theta )$; the conclusion therefore follows from (2.3.1) and (I, 9.5.2), since $f$ is quasi-compact and quasi-separated (1.1.2 and 1.2.2).

(2.3.3)

We shall say that a morphism $f:X\to Y$ is quasi-flat if there exists a quasi-coherent ${\mathcal{O}}_X$-Module $\mathcal{F}$ of finite type that is $f$-flat and whose support is equal to $X$. We shall say that $f$ is quasi-faithfully flat if it is quasi-flat and surjective. Every flat (resp. faithfully flat) morphism is quasi-flat (resp. quasi-faithfully flat), since then $\mathcal{F}={\mathcal{O}}_X$ satisfies the preceding conditions.

It follows at once from (2.1.4) and (I, 9.1.13) that if $f$ is quasi-flat, then for every morphism $Y^{\prime }\to Y$ the morphism $f_{(Y^{\prime })}:X{\times }_Y{Y}^{\prime }\to Y^{\prime }$ is quasi-flat. Similarly (taking (I, 3.5.2) into account), if $f$ is quasi-faithfully flat, so is $f_{(Y^{\prime })}$.

Proposition (2.3.4).

Let $f:X\to Y$ be a quasi-flat morphism (2.3.3). Then $f$ possesses the following properties (which are equivalent by virtue of (1.10.4)):

(i) For every $x\in X$ and every generization $y^{\prime }$ of $y=f(x)$, there exists a generization $x^{\prime }$ of $x$ such that $f(x^{\prime })=y^{\prime }$.

(ii) For every $x\in X$, the image under $f$ of $\mathrm{Spec}({\mathcal{O}}_{X,x})$ is $\mathrm{Spec}({\mathcal{O}}_{Y,y})$.

(iii) For every irreducible closed part $Y^{\prime }$ of $Y$, every irreducible component of $f^{-1}(Y^{\prime })$ dominates $Y^{\prime }$.

It suffices, for example, to prove (ii). By hypothesis, there is a quasi-coherent ${\mathcal{O}}_X$-Module of finite type $\mathcal{F}$, which is $f$-flat and such that $Supp(\mathcal{F})=X$. For every $x\in X$, ${\mathcal{F}}_x$ is then an ${\mathcal{O}}_x$-module of finite type, not reduced to 0, and moreover ${\mathcal{F}}_x$ is a flat ${\mathcal{O}}_y$-module, for the homomorphism $\rho :{\mathcal{O}}_y\to {\mathcal{O}}_x$. Since this latter is local and ${\mathcal{F}}_x\ne 0$, Nakayama's lemma shows that ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_y}k(y)\ne 0$, hence ${\mathcal{F}}_x$ is a faithfully flat ${\mathcal{O}}_y$-module $(0_I,\mathrm{6.4.1})$. It results that for every prime ideal $\mathfrak{q}$ of ${\mathcal{O}}_y$, there exists a prime ideal $\mathfrak{p}$ of ${\mathcal{O}}_x$ such that $\mathfrak{q}={\rho }^{-1}(\mathfrak{p})$ $(0_I,\mathrm{6.5.1})$, which proves (ii).

Corollary (2.3.5).

Let $f:X\to Y$ be a morphism satisfying the equivalent conditions (i), (ii), (iii) of (2.3.4) (in particular a quasi-flat morphism, or an open morphism (1.10.4)).

(i) Let $Z$, $Z^{\prime }$ be two irreducible closed parts of $Y$ such that $Z\subset Z^{\prime }$, and let $T$ be an irreducible component of $f^{-1}(Z)$; then there exists an irreducible component $T^{\prime }$ of $f^{-1}(Z^{\prime })$ containing $T$ (and dominating $Z^{\prime }$).

(ii) For every irreducible component $T$ of $X$, $\overline{f(T)}$ is an irreducible component of $Y$.

(iii) Suppose $Y$ is irreducible, and, if $y$ is its generic point, suppose that $f^{-1}(y)$ is irreducible. Then $X$ is irreducible.

(i) It suffices to apply (2.3.4, (i)) taking for $x$ the generic point of $T$ ($y=f(x)$ then being the generic point of $Z$) and for $y^{\prime }$ the generic point of $Z^{\prime }$.

(ii) It is clear that $f(T)=Z$ is irreducible, and by virtue of (i), $Z$ cannot be strictly contained in an irreducible closed part $Z^{\prime }$ of $Y$.

(iii) By virtue of (ii), every irreducible component of $X$ dominates $Y$, hence meets $f^{-1}(y)$; the conclusion then follows from $(0_I,\mathrm{2.1.8})$.

Proposition (2.3.6).

Let $Y$ be a prescheme whose set of irreducible components is locally finite (which is the case if the space underlying $Y$ is locally Noetherian (cf. (I, 6.1.9))).

(i) Every closed part $W$ of $Y$ stable under generization $(0_I,\mathrm{2.1.2})$ is open. In particular, every connected component of $Y$ is open.

(ii) Let $f:X\to Y$ be a closed morphism satisfying moreover the equivalent conditions (i), (ii), (iii) of (2.3.4) (which will be the case if $f$ is quasi-flat or open). Then the image under $f$ of every connected component $C$ of $X$ is a connected component of $Y$.

(iii) Let $f:X\to Y$ be a morphism satisfying the equivalent conditions (i), (ii), (iii) of (2.3.4) and such moreover that for every $y\in Y$ the set $f^{-1}(y)$ is finite (which will be the case if $f$ is quasi-finite or radicial). Then the set of irreducible components of $X$ is locally finite.

(i) If $y\in W$, the generic points ${\eta }_i$ ($1\le i\le m$) of the irreducible components $Y_i$ of $Y$ containing $y$ belong by hypothesis to $W$; hence, since $W$ is closed, these components themselves are contained in $W$; since by hypothesis there is a neighbourhood $U$ of $y$ such that $U$ is the union of the $U\cap Y_i$ $(0_I,\mathrm{2.1.6})$, one has $U\subset W$, so $W$

is open. Since for every $y\in Y$, $\overline{y}$ is connected, a connected component of $Y$ is stable under generization, so the second assertion follows at once from the first.

(ii) Since $C$ is closed in $X$, $f(C)$ is closed in $Y$ by hypothesis. Furthermore, since $C$ is stable under generization, the hypothesis on $f$ entails that $f(C)$ is stable under generization; hence, by virtue of (i), $f(C)$ is open and closed, and since it is connected it is a connected component of $Y$.

(iii) Let $x\in X$; by hypothesis, there is an open neighbourhood $U$ of $y=f(x)$ meeting only a finite number of irreducible components $Y_i$ of $Y$ ($1\le i\le n$); let $y_i$ be the generic point of $Y_i$ ($1\le i\le n$). For every irreducible component $Z$ of $X$ meeting $f^{-1}(U)$, the generic point $z$ of $Z$ is necessarily in one of the sets $f^{-1}(y_i)$ (2.3.4). Since each of these sets is finite by hypothesis, this proves our assertion.

Proposition (2.3.7).

Let $f:X\to Y$ be a morphism, $g:Y^{\prime }\to Y$ a quasi-flat morphism, $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$.

(i) If $f$ is quasi-compact and dominant, so is $f^{\prime }$.

(ii) If every irreducible component of $X$ dominates an irreducible component of $Y$, then every irreducible component of $X^{\prime }$ dominates an irreducible component of $Y^{\prime }$.

Denote by $g^{\prime }:X^{\prime }\to X$ the canonical projection, which is a quasi-flat morphism (2.3.3).

(i) One already knows (1.1.2) that $f^{\prime }$ is quasi-compact; furthermore, if $y^{\prime }$ is a maximal point (1.1.4) of $Y^{\prime }$, $y=g(y^{\prime })$ is a maximal point of $Y$, as results from (2.3.4, (iii)). By hypothesis (1.1.5) there exists $x\in X$ such that $f(x)=y$; hence (I, 3.4.7) there exists $x^{\prime }\in X^{\prime }$ such that $f^{\prime }(x^{\prime })=y^{\prime }$.

(ii) Let $x^{\prime }$ be a maximal point of $X^{\prime }$, and let $x=g^{\prime }(x^{\prime })$; it follows from (2.3.4, (ii)) that $x$ is a maximal point of $X$, and by hypothesis $y=f(x)$ is a maximal point of $Y$. Set $y^{\prime }=f^{\prime }(x^{\prime })$, so that $g(y^{\prime })=y$; the issue is to show that $y^{\prime }$ is a maximal point of $Y^{\prime }$. By virtue of (I, 5.1.7) and (2.3.3), one may restrict to the case where $X$ and $Y$ are reduced, so ${\mathcal{O}}_x$ and ${\mathcal{O}}_y$ are fields; moreover, by virtue of (I, 3.6.5) applied twice and of (I, 2.4.4), one may restrict to the case where $Y=\mathrm{Spec}({\mathcal{O}}_y)$ and $Y^{\prime }=\mathrm{Spec}({\mathcal{O}}_{y^{\prime }})$. Then $f$ is a flat morphism since ${\mathcal{O}}_y$ is a field (2.1.2), and the same is true of $f^{\prime }$ (2.1.4); hence it follows from (2.3.4, (ii)) that $y^{\prime }=f^{\prime }(x^{\prime })$ is a maximal point of $Y^{\prime }$.

Corollary (2.3.8) (Zariski).

Let $A$, $B$ be two Noetherian local rings, $\mathfrak{m}$, $\mathfrak{n}$ their respective maximal ideals, $\varphi :A\to B$ a local homomorphism. Suppose the following hypotheses are satisfied:

1° $B$ is an $A$-algebra essentially of finite type (1.3.8).

2° The completion Â of $A$ for the $\mathfrak{m}$-adic topology is integral.

3° $\varphi$ is injective.

Then the $\mathfrak{m}$-adic topology of $A$ is induced by the $\mathfrak{n}$-adic topology of $B$.

Set $B^{\prime }=B{\otimes }_A\hat{A}$; by virtue of 1°, $B^{\prime }$ is of the form $S^{-1}(C{\otimes }_A\hat{A})$, where $C$ is an $A$-algebra of finite type and $S$ a multiplicative subset of $C$, so $B^{\prime }$ is a Noetherian ring. Since $A$ is identified with a subring of Â $(0_I,\mathrm{7.3.5})$, $A$ is integral by 2°. Hypothesis 3° then entails that there exists a prime ideal $\mathfrak{q}$ of $B$ inducing the ideal 0 of $A$ $(0_I,\mathrm{1.5.8})$, and consequently the local homomorphism $A\to B/\mathfrak{q}$ is injective. One may therefore restrict to proving the conclusion of (2.3.8) by adding the hypothesis that $B$ is an integral local ring. Apply (2.3.7, (ii)) to $Y=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(B)$, $Y^{\prime }=\mathrm{Spec}(\hat{A})$ and $X^{\prime }=\mathrm{Spec}(B^{\prime })$; since the morphism $Y^{\prime }\to Y$ is flat and $X$

(which is integral) dominates the integral scheme $Y$, every irreducible component of $X^{\prime }$ dominates the scheme (integral by hypothesis) $Y^{\prime }$. If $y$, $x$, $y^{\prime }$ are the closed points of $Y$, $X$, $Y^{\prime }$ respectively, there is a point $x^{\prime }\in X^{\prime }$ (in fact unique) above $x$ and $y^{\prime }$ (I, 3.4.9) and $\mathrm{Spec}({\mathcal{O}}_{x^{\prime }})$ therefore dominates $\mathrm{Spec}({\mathcal{O}}_{y^{\prime }})$; consequently one has a commutative diagram of local homomorphisms of Noetherian local rings

  B = 𝒪_x ────────→ 𝒪_{x'}
    ↑                  ↑
   φ│                 │v
    │                  │
  A = 𝒪_y ────u────→ 𝒪_{y'} = Â

such that $u$ and $v$ are injective (I, 1.2.7); identifying $A$ and Â with subrings of ${\mathcal{O}}_{x^{\prime }}$, and denoting by $\mathfrak{r}$ the maximal ideal of ${\mathcal{O}}_{x^{\prime }}$, the intersection of the ideals ${\mathfrak{r}}^k\cap \hat{A}$ is therefore zero $(0_I,\mathrm{7.3.5})$; since Â is complete and these ideals are open in Â, this entails (Bourbaki, Alg. comm., chap. III, §2, n° 7, prop. 8) that the topology of Â is induced by the $\mathfrak{r}$-preadic topology of ${\mathcal{O}}_{x^{\prime }}$; a fortiori the same is true of the topology of $A$ $(0_I,\mathrm{7.3.5})$. Moreover one has ${\mathfrak{n}}^k\cap A\subset {\mathfrak{r}}^k\cap A$, so the $\mathfrak{n}$-preadic topology of $B$ induces on $A$ a topology finer than the $\mathfrak{m}$-preadic topology; but since ${\mathfrak{m}}^k\subset {\mathfrak{n}}^k\cap A$, these two topologies are identical. Q.E.D.

Remark (2.3.9).

We shall see further on (7.8.3, (vii)) that for the Noetherian local rings $A$ most usual in algebraic geometry, the hypothesis that $A$ is integral and integrally closed implies that the same holds for Â. That is why, in algebraic geometry over a base field, one generally states (2.3.8) under the hypothesis that $A$ is integral and integrally closed.

Theorem (2.3.10).

Let $f:X\to Y$ be a quasi-flat morphism (2.3.3). Then, for every proconstructible part (1.9.4) $Z$ of $Y$, one has $\overline{f^{-1}(Z)}=f^{-1}(\overline{Z})$.

Since $f$ is continuous, one has $f^{-1}(Z)\subset f^{-1}(\overline{Z})$ and everything reduces to proving that for every $x\in X$ such that $f(x)\in \overline{Z}$, $x$ is adherent to $f^{-1}(Z)$; it is clear that the question is local on $Y$, so one may suppose $Y$ affine. By virtue of the hypothesis, there exists an affine scheme $Y^{\prime }$ and a morphism $g:Y^{\prime }\to Y$ such that $g(Y^{\prime })=Z$ (1.9.5, (ix)). Let Y_1 be the closed image of $Y^{\prime }$ under $g$ (I, 9.5.3), and let X_1 be the closed sub-prescheme $f^{-1}(Y_1)$ of $X$; if $f_1:X_1\to Y_1$ is the restriction of $f$ to X_1, one knows that $f_1$ is quasi-flat (2.3.3); one may therefore replace $X$, $Y$ by X_1, Y_1 respectively, in other words suppose that $g$ is dominant. Set then $X^{\prime }=X{\times }_Y{Y}^{\prime }$, and let $f^{\prime }$ and $g^{\prime }$ be the projections of $X^{\prime }$ onto $Y^{\prime }$ and $X$ respectively, so that one has the commutative diagram

  X  ←─g'──  X'
  │           │
  f│         │f'
  ↓           ↓
  Y  ←──g──  Y'

Since $f$ is quasi-flat, $g$ quasi-compact and dominant, it follows from (2.3.7) (where the roles of $f$ and $g$ are interchanged) that $g^{\prime }$ is a dominant morphism, which proves the theorem.

Corollary (2.3.11).

Let $f$ be a quasi-flat and quasi-compact morphism, $F$ a closed part of $X$ such that $F=f^{-1}(f(F))$; then one has $F=f^{-1}(\overline{f(F)})$.

Let $Y^{\prime }$ be the reduced sub-prescheme of $X$ having $F$ as underlying space (I, 5.2.1) and let $j:Y^{\prime }\to X$ be the canonical injection; then $f\circ j$ is quasi-compact (1.1.2),

so $Z=f(F)$ is pro-constructible in $Y$ (1.9.5, (vii)), and the corollary follows from the fact that $F$ is closed.

One may further write the result of (2.3.11) in the form $F=f^{-1}(f(X)\cap \overline{f(F)})$, in other words, the closed sets of the subspace $f(X)$ of $Y$ are the parts $Z\subset f(X)$ such that $f^{-1}(Z)$ is closed in $X$; this also means that the topology induced by $Y$ on $f(X)$ is the quotient of that of $X$ by the equivalence relation defined by $f$. In particular:

Corollary (2.3.12).

Let $f:X\to Y$ be a quasi-faithfully flat (2.3.3) and quasi-compact morphism. Then the topology of $Y$ is the quotient of that of $X$ by the equivalence relation defined by $f$ (in other words, for $Z\subset Y$ to be open (resp. closed) in $Y$, it is necessary and sufficient that $f^{-1}(Z)$ be open (resp. closed) in $X$).

Indeed, one then has $f(X)=Y$.