Éléments de Géométrie Algébrique — English

§11. Topological properties of flat morphisms of finite presentation; local criteria of flatness

While in §2 we considered the statements concerning flatness which do not depend on any finiteness hypothesis, and while §6 studies the notion of flatness in the framework of locally Noetherian preschemes (but without finiteness hypothesis on the morphisms), the present section is devoted to the notion of $f$-flatness in the case where the morphism $f:X\to Y$ is locally of finite presentation. The interest of the notion of flat morphism of finite presentation lies in the fact that it is the one which seems to express, in the most technically adequate manner, the intuitive notion of "family of algebraic preschemes parameterized by a scheme $Y$", whose study is one of the principal objects of Algebraic Geometry. Moreover, even if one were interested at the outset only in the case of a Noetherian base scheme, it is indispensable, for certain technical reasons (for example, for certain applications of the theory of "descent", which leads one to introduce schemes not necessarily Noetherian), not to confine oneself to that case, as soon as one deals with problems of essentially relative nature linked to morphisms locally of finite presentation. We shall systematically follow this principle, already supported by the results of §§8 and 9, in the entire continuation of this Chapter, and even in the continuation of our Treatise, even at the cost of sacrificing on occasion the simplicity of certain proofs, which Noetherian hypotheses sometimes permit one to lighten ^(*). In the present section, this leads us to take up again, in the context of "finite presentation" (notably in n° 3) certain flatness statements already obtained in the Noetherian context. The essential technical tool for making the reduction to the Noetherian case is the theorem of compatibility of flatness with projective limits of preschemes (11.2.6), completing the general results of §8. We also prove in passing (11.3.1) a result often used in the sequel, implying that the set of points of flatness of a morphism locally of finite presentation is open.

^(*) This principle is also inspired by the necessity of granting droit de cité, as "parameter spaces" for families of algebraic schemes, to arbitrary ringed spaces (and even arbitrary ringed "toposes"), for which there can no longer be any question in general of Noetherian hypotheses. It seems rather clear that one will no longer be able to elude for long this new extension of Algebraic Geometry, and it is fitting from the present moment to develop the "relative"-type notions and techniques of the theory of schemes in such a way that they can adapt practically as they stand to this more general framework.

In nos. 4 to 8, we study the question of the "descent" of flatness, consisting in finding useful conditions on a base-change morphism $Y^{\prime }\to Y$ (not flat in general) so as to be able to conclude that if $X{\times }_Y{Y}^{\prime }$ is flat over $Y^{\prime }$, then $X$ is flat over $Y$. These results, more technical than those of nos. 1 to 3, are of less frequent use in the sequel; they will however play an important role in the non-projective construction techniques in the following chapter. The only result of nos. 4 to 8 used in the sequel of Chap. IV is the valuative criterion of flatness (n° 8), which will be applied in (15.2).

Finally, nos. 9 and 10 are devoted to the study of a notion which makes precise, in the theory of schemes, that of density in the topological sense, namely the notion of family of sub-preschemes schematically dense in a given prescheme, and notably the study of the behaviour of this notion under base change (flat or arbitrary). This notion is used above all, for the moment, in the study of group schemes.

11.1. Flatness loci (Noetherian case)

Theorem (11.1.1).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module. Then the set $U$ of $x\in X$ such that $\mathcal{F}$ is $f$-flat at the point $x$ is open in $X$.

The question being evidently local on $X$ and $Y$, one may suppose $Y=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(B)$, with $A$ Noetherian and $B$ an $A$-algebra of finite type. One then has $\mathcal{F}=\tilde{M}$, where $M$ is a $B$-module of finite type. Let us apply the criterion $(0_{III},\mathrm{9.2.6})$: it therefore suffices to prove the following assertion:

  (11.1.1.1) Let A be a Noetherian ring, B an A-algebra of finite type, M a B-module of finite type, 𝔮 a prime
             ideal of B, 𝔭 its inverse image in A. Suppose that M_𝔮 is a flat A_𝔭-module. Then there exists
             g ∈ B − 𝔮 such that for every prime ideal 𝔮' ⊃ 𝔮 of B with g ∉ 𝔮', M_{𝔮'} is a flat A_{𝔭'}-module,
             where 𝔭' denotes the inverse image of 𝔮' in A (it amounts to the same thing (0_I, 6.3.1) to say
             that M_{𝔮'} is a flat A-module).

To this end, let us consider $B_{{\mathfrak{q}}^{\prime }}$ as an $A$-algebra; one evidently has $\mathfrak{p}{B}_{{\mathfrak{q}}^{\prime }}\subset {\mathfrak{q}}^{\prime }{B}_{{\mathfrak{q}}^{\prime }}$. One then knows $(0_{III},\mathrm{10.2.2})$ that for $M_{{\mathfrak{q}}^{\prime }}$ to be a flat $A$-module, it is necessary and sufficient that $M_{{\mathfrak{q}}^{\prime }}/\mathfrak{p}{M}_{{\mathfrak{q}}^{\prime }}$ be a flat $(A/\mathfrak{p})$-module and that one have $To{r}_1^A(M_{{\mathfrak{q}}^{\prime }},A/\mathfrak{p})=0$. Now, one has $M_{{\mathfrak{q}}^{\prime }}=M{\otimes }_B{B}_{{\mathfrak{q}}^{\prime }}$; since $B_{{\mathfrak{q}}^{\prime }}$ is flat over $B$, one has $M_{{\mathfrak{q}}^{\prime }}/\mathfrak{p}{M}_{{\mathfrak{q}}^{\prime }}=(M/\mathfrak{p}M{)}_{{\mathfrak{q}}^{\prime }}$ and $To{r}_1^A(M_{{\mathfrak{q}}^{\prime }},A/\mathfrak{p})=(To{r}_1^A(M,A/\mathfrak{p}){)}_{{\mathfrak{q}}^{\prime }}$ (defining Tor by means of a projective resolution of $A/\mathfrak{p}$); for the same reason, since one must have $g\notin {\mathfrak{q}}^{\prime }$, $B_{{\mathfrak{q}}^{\prime }}$ is flat over $B_g$, hence $(M/\mathfrak{p}M{)}_{{\mathfrak{q}}^{\prime }}=((M/\mathfrak{p}M{)}_g{)}_{{\mathfrak{q}}^{\prime }{B}_g}$ and $(To{r}_1^A(M,A/\mathfrak{p}){)}_{{\mathfrak{q}}^{\prime }}=((To{r}_1^A(M,A/\mathfrak{p}){)}_g{)}_{{\mathfrak{q}}^{\prime }{B}_g}$, where in these formulas $M/\mathfrak{p}M$ and $To{r}_1^A(M,A/\mathfrak{p})$ are considered as $B$-modules. Taking $(0_I,\mathrm{6.3.2})$ into account, one sees that one is reduced to proving the

Lemma (11.1.1.2).

Under the conditions of (11.1.1.1), there exists $g\in B-\mathfrak{q}$ such that: (i) $(M/\mathfrak{p}M{)}_g$ is a flat $(A/\mathfrak{p})$-module; (ii) $(To{r}_1^A(M,A/\mathfrak{p}){)}_g=0$.

By virtue of the generic flatness theorem (6.9.1) applied to the integral ring $A/\mathfrak{p}$, to the $(A/\mathfrak{p})$-algebra of finite type $B/\mathfrak{p}B$, and to the $(B/\mathfrak{p}B)$-module of finite type $M/\mathfrak{p}M$, there exists $h\in A-\mathfrak{p}$ such that, if $\bar{h}$ is its canonical image in $A/\mathfrak{p}$, $(M/\mathfrak{p}M{)}_{\bar{h}}$ is a flat $(A/\mathfrak{p})$-module. On the other hand, since $M_{\mathfrak{q}}$ is a flat $A_{\mathfrak{p}}$-module, and consequently a flat $A$-module $(0_I,\mathrm{6.3.1})$, one has $To{r}_1^A(M_{\mathfrak{q}},A/\mathfrak{p})=0$, which one also writes $(To{r}_1^A(M,A/\mathfrak{p}){)}_{\mathfrak{q}}=0$. But since $A$ and $B$ are Noetherian, $To{r}_1^A(M,A/\mathfrak{p})$ is a $B$-module of finite type, hence $(0_I,\mathrm{5.2.2})$ there exists $g_1\in B-\mathfrak{q}$ such that $(To{r}_1^A(M,A/\mathfrak{p}){)}_{g_1}=0$. Moreover, one has $(M/\mathfrak{p}M{)}_{\bar{h}}=(M/\mathfrak{p}M{)}_h$ ($M/\mathfrak{p}M$ being considered in the second member as an $A$-module); in addition, $(M/\mathfrak{p}M{)}_h$ being a $B$-module, $(M/\mathfrak{p}M{)}_{h{g}_1}$ is again a flat $(A/\mathfrak{p})$-module, for it can be written $(M/\mathfrak{p}M{)}_{\bar{h}{\bar{g}}_1}$, where $\bar{h}{\bar{g}}_1$ is the canonical image of $h{g}_1$ in $B/\mathfrak{p}B$, and it suffices to apply $(0_I,\mathrm{6.3.2})$; finally, one has $(To{r}_1^A(M,A/\mathfrak{p}){)}_{h{g}_1}=0$ and $h{g}_1\in B-\mathfrak{q}$ since $h\in A-\mathfrak{p}$. Q.E.D.

Corollary (11.1.2).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $\mathcal{F}$, ${\mathcal{F}}^{\prime }$ two coherent ${\mathcal{O}}_X$-Modules, $u:{\mathcal{F}}^{\prime }\to \mathcal{F}$ a homomorphism of ${\mathcal{O}}_X$-Modules. Suppose that $\mathcal{F}$ is $f$-flat. Then the set $U$ of $x\in X$ such that, setting $y=f(x)$, the homomorphism $u_x\otimes 1:{\mathcal{F}}_x^{\prime }{\otimes }_{{\mathcal{O}}_y}k(y)\to {\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_y}k(y)$ is injective, is open in $X$.

Indeed, let $\mathcal{N}$ (resp. $\mathcal{Q}$) be the kernel (resp. the cokernel) of $u$; let us apply $(0_{III},\mathrm{10.2.4})$ ^(*) to the local rings ${\mathcal{O}}_y$ and ${\mathcal{O}}_x$ and to the ${\mathcal{O}}_y$-modules ${\mathcal{F}}_x^{\prime }$ and ${\mathcal{F}}_x$: to say that $u_x\otimes 1$ is injective amounts to saying that ${\mathcal{N}}_x=0$ and that $\mathcal{Q}$ is $f$-flat at the point $x$. Now since $\mathcal{N}$ and $\mathcal{Q}$ are coherent $(0_I,\mathrm{5.3.4})$, the set of $x$ where ${\mathcal{N}}_x=0$ is open $(0_I,\mathrm{5.2.2})$, and the set of $x$ where $\mathcal{Q}$ is $f$-flat is open by (11.1.1); whence the conclusion.

In particular:

Corollary (11.1.3).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a flat morphism locally of finite type, $g$ a section of ${\mathcal{O}}_X$ over $X$. The set of $x\in X$ such that $g_x\otimes 1$ is not a zero-divisor in ${\mathcal{O}}_x{\otimes }_{{\mathcal{O}}_{f(x)}}k(f(x))$ is open in $X$.

It suffices to apply (11.1.2) to the endomorphism of ${\mathcal{O}}_X$ defined by $g$.

Corollary (11.1.4).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module which is $f$-flat. Let $(g_i{)}_{1\le i\le n}$ be a sequence of sections of ${\mathcal{O}}_X$ over $X$. Then the set $U$ of $x\in X$ such that the sequence $((g_i{)}_x\otimes 1)$ is $({\mathcal{F}}_{f(x)}{\otimes }_{{\mathcal{O}}_{f(x)}}k(f(x)))$-regular is open in $X$.

Since $\mathcal{F}$ is $f$-flat, it follows from $(0_{IV},\mathrm{15.1.16})$ that $U$ is also the set of points

^(*) Here is a proof of $(0_{III},\mathrm{10.2.4})$ which was not published in N. Bourbaki's Algèbre commutative. Taking $(0_I,\mathrm{6.1.2})$ into account, it suffices to see that b) implies a). Set $P=Im(u)$, $Q=Coker(u)$, $R=Ker(u)$. The composite $M\otimes k\to P\otimes k\to N\otimes k$ is injective, and $M\otimes k\to P\otimes k$ is surjective, hence $P\otimes k\to N\otimes k$ is injective and $M\otimes k\to P\otimes k$ is bijective. The exact sequence $0\to P\to N\to Q\to 0$ gives the exact sequence $0\to P\otimes k\to N\otimes k\to Q\otimes k\to 0$ (since $P\otimes k\to N\otimes k$ is injective), and one has $To{r}_1^A(Q,k)=0$; since $Q$ is a $B$-module of finite type, $(0_{III},\mathrm{10.2.2})$ shows that $Q$ is a flat $A$-module; then $P$ is also a flat $A$-module by $(0_I,\mathrm{6.1.2})$. The sequence $0\to R\to M\to P\to 0$ being exact, so is $0\to R\otimes k\to M\otimes k\to P\otimes k\to 0$ by $(0_I,\mathrm{6.1.2})$; since $M\otimes k\to P\otimes k$ is bijective, one has $R\otimes k=0$; but since $B$ is Noetherian, $R$ is a $B$-module of finite type, hence one has $R=0$ by virtue of Nakayama's lemma.

$x\in X$ such that the sequence $(g_i{)}_x$ is ${\mathcal{F}}_x$-regular and the ${\mathcal{O}}_x$-module ${\mathcal{F}}_x/({\sum }_{i=1}^n(g_i{)}_x{\mathcal{F}}_x)$ is $f$-flat. But $\mathcal{F}$ and $\mathcal{F}/(\sum g_i\mathcal{F})$ are coherent, hence the corollary follows from (11.1.1) and $(0_{IV},\mathrm{15.2.4})$.

Corollary (11.1.5).

Let $X$, $Y$, $Z$ be three locally Noetherian preschemes, $f:X\to Y$, $g:Y\to Z$ two morphisms of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module. Then the set $U$ of $y\in Y$ such that, for every generization $y^{\prime }$ of $y$, $\mathcal{F}$ is $(g\circ f)$-flat at all points of $f^{-1}(y^{\prime })$ (i.e. such that $\mathcal{F}{\otimes }_Y\mathrm{Spec}({\mathcal{O}}_{Y,y^{\prime }})$ is flat relative to the morphism $X{\times }_Y\mathrm{Spec}({\mathcal{O}}_{Y,y^{\prime }})\to Z$) is open in $Y$.

If $V$ is the set of $x\in X$ where $\mathcal{F}$ is $(g\circ f)$-flat, $U$ is the set of $y\in Y$ such that for every generization $y^{\prime }$ of $y$, one has $f^{-1}(y^{\prime })\subset V$. Now $V$ is open (11.1.1), hence locally constructible in $X$, and the set $W$ of $y\in Y$ such that $f^{-1}(y)\subset V$ is equal to $Y-f(X-V)$, hence is also locally constructible in $Y$ by virtue of Chevalley's theorem (1.8.4). It then follows from $(0_{III},\mathrm{9.2.5})$ that the points of $U$ are the points interior to $W$, whence the conclusion.

Corollary (11.1.6).

Let $A$ be a Noetherian ring, $B$ an $A$-algebra of finite type, $C$ a $B$-algebra of finite type, $M$ a $C$-module of finite type. Then the set of $\mathfrak{q}\in \mathrm{Spec}(B)$ such that $M_{\mathfrak{q}}$ is a flat $A$-module is open in $\mathrm{Spec}(B)$.

Taking (2.1.2) into account, this is a consequence of (11.1.5) applied to $Z=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(B)$, $X=\mathrm{Spec}(C)$, $\mathcal{F}=\tilde{M}$.

The results of this number will be freed of the Noetherian hypotheses in (11.3).

11.2. Flatness of a projective limit of preschemes

(11.2.1) Let $A$ be a ring, $M$, $N$ two $A$-modules, $A^{\prime }$ an $A$-algebra; set $M^{\prime }=M{\otimes }_A{A}^{\prime }$, $N^{\prime }=N{\otimes }_A{A}^{\prime }$. Recall (III, 6.3.8) that for every $i$ one defines a canonical homomorphism of $A$-modules

  (11.2.1.1)    φ_i : Tor_i^A(M, N) → Tor_i^{A'}(M', N')

in the following manner: one considers a left resolution of $M$ by free $A$-modules

  (11.2.1.2)    … → L_{i+1} →^{f_{i+1}} L_i → … → L_0 →^ε M → 0

whence one deduces by tensoring with $A^{\prime }$ a complex of $A^{\prime }$-modules

  (11.2.1.3)    … → L'_{i+1} →^{f'_{i+1}} L'_i → … → L'_0 →^{ε'} M' → 0

where one has set $L_i^{\prime }=L_i{\otimes }_A{A}^{\prime }$, $f_i^{\prime }=f_i\otimes 1_{A^{\prime }}$, ${ϵ}^{\prime }=ϵ\otimes 1_{A^{\prime }}$. Let us consider on the other hand a left resolution of $M^{\prime }$ by free $A^{\prime }$-modules

  (11.2.1.4)    … → L''_{i+1} →^{f''_{i+1}} L''_i → … → L''_0 →^{ε''} M' → 0

Since the $L_i^{\prime \prime }$ are free $A^{\prime }$-modules, one knows (M, V, 1.1) that there are $A^{\prime }$-homomorphisms $u_i$ forming a commutative diagram

  L'_i  ─f'_i─→  L'_{i-1}  ─→  ⋯  ─→  L'_0  ─ε'─→  M'
   │              │                    │            ‖
   u_i           u_{i-1}                u_0          1_{M'}
   ↓              ↓                    ↓            ↓
  L''_i ─f''_i─→ L''_{i-1} ─→  ⋯  ─→  L''_0 ─ε''─→  M'

If one composes the homomorphism $u_{\bullet }:L_{\bullet }^{\prime }\to L_{\bullet }^{\prime \prime }$ of complexes thus defined with the canonical homomorphism $L_{\bullet }\to L_{\bullet }^{\prime }$, one obtains a homomorphism of complexes of $A$-modules $L_{\bullet }\to L_{\bullet }^{\prime \prime }$; noting that one has $(L_i{\otimes }_{A}N){\otimes }_A{A}^{\prime }=L_i^{\prime }{\otimes }_{A^{\prime }}{N}^{\prime }$, one deduces from this a homomorphism of complexes of $A$-modules $L_{\bullet }{\otimes }_{A}N\to L_{\bullet }^{\prime \prime }{\otimes }_{A^{\prime }}{N}^{\prime }$, whence, on passing to homology, the canonical homomorphisms (11.2.1.1). Since the $u_i$ are well determined up to homotopy (M, V, 1.1), the homomorphisms (11.2.1.1) do not depend on the choice of the $u_i$ nor on the choice of the free resolutions $L_{\bullet }$ and $L_{\bullet }^{\prime \prime }$.

Since the $To{r}_i^{A^{\prime }}(M^{\prime },N^{\prime })$ are $A^{\prime }$-modules, one canonically deduces from (11.2.1.1) $A^{\prime }$-homomorphisms

  (11.2.1.5)    ψ_i : Tor_i^A(M, N) ⊗_A A' → Tor_i^{A'}(M', N').

Let us now consider two ring homomorphisms $\alpha :A\to A^{(1)}$, $\beta :A^{(1)}\to A^{(2)}$ and their composite $\beta \circ \alpha :A\to A^{(2)}$; set $M^{(j)}=M{\otimes }_A{A}^{(j)}$, $N^{(j)}=N{\otimes }_A{A}^{(j)}$ for $j=1,2$. Then the canonical composite homomorphism

  Tor_i^A(M, N) → Tor_i^{A^{(1)}}(M^{(1)}, N^{(1)}) → Tor_i^{A^{(2)}}(M^{(2)}, N^{(2)})

is the same as the canonical homomorphism deduced from $\beta \circ \alpha$; this results from the fact that, if $L_{\bullet }^{(j)}$ is a free resolution of $M^{(j)}$, the diagram

  L_•^{(1)}  ─→  L_•^{(2)}
     ↑              ↑
     L_•   ──────────

is commutative.

(11.2.2) The notation being that of (11.2.1), let us now consider a filtered inductive system of $A$-algebras $(A_{\alpha },{\xi }_{\beta \alpha })$, and for every index $\alpha$, set $M_{\alpha }=M{\otimes }_A{A}_{\alpha }$, $N_{\alpha }=N{\otimes }_A{A}_{\alpha }$; it then follows from (11.2.1) that for each $i$, $(To{r}_i^{A_{\alpha }}(M_{\alpha },N_{\alpha }),{\psi }_{\beta \alpha })$, where ${\psi }_{\beta \alpha }$ is the canonical homomorphism (11.2.1.1) corresponding to ${\xi }_{\beta \alpha }$, is an inductive system

of $A_{\alpha }$-modules. Set $A^{\prime }=\lim A_{\alpha }$, $M^{\prime }=\lim M_{\alpha }=M{\otimes }_A{A}^{\prime }$, $N^{\prime }=\lim N_{\alpha }=N{\otimes }_A{A}^{\prime }$; if one denotes by ${\xi }_{\alpha }:A_{\alpha }\to A^{\prime }$ the canonical homomorphism, one deduces from this canonical homomorphisms (11.2.1.1) ${\psi }_{\alpha }:To{r}_i^{A_{\alpha }}(M_{\alpha },N_{\alpha })\to To{r}_i^{A^{\prime }}(M^{\prime },N^{\prime })$ which (still by virtue of (11.2.1)) form an inductive system of homomorphisms; we propose to complete the result of (M, V, 9.4*) by showing that the

  (11.2.2.1)    ψ = lim ψ_α : lim_α Tor_i^{A_α}(M_α, N_α) → Tor_i^{A'}(M', N')

are isomorphisms of $A^{\prime }$-modules. For this, we proceed as in (M, V, 9.5*), associating to each $M_{\alpha }$ its canonical free resolution. Everything boils down (taking into account the exactness of the functor lim) to proving the

Lemma (11.2.2.2).

Let $(A_{\alpha },{\xi }_{\beta \alpha })$ be a filtered inductive system of rings, $(M_{\alpha },h_{\beta \alpha })$ an inductive system of sets, $A^{\prime }=\lim A_{\alpha }$, $M^{\prime }=\lim M_{\alpha }$, ${\xi }_{\alpha }:A_{\alpha }\to A^{\prime }$, $h_{\alpha }:M_{\alpha }\to M^{\prime }$ the canonical maps. For every $\alpha$, let $F(M_{\alpha })$ be the $A_{\alpha }$-module of formal linear combinations of elements of $M_{\alpha }$; let similarly $F(M^{\prime })$ be the $A^{\prime }$-module of formal linear combinations of elements of $M^{\prime }$; if $h_{\beta \alpha }^{\prime }:F(M_{\alpha })\to F(M_{\beta })$ (for $\alpha \le \beta$) and $h_{\alpha }^{\prime }:F(M_{\alpha })\to F(M^{\prime })$ are the $A_{\alpha }$-homomorphisms deduced from $h_{\beta \alpha }$ and $h_{\alpha }$ respectively, $(F(M_{\alpha }),h_{\beta \alpha }^{\prime })$ is an inductive system of $A_{\alpha }$-modules and $(h_{\alpha }^{\prime })$ an inductive system of homomorphisms. Then

  h' = lim h'_α : lim F(M_α) → F(M') = F(lim M_α)

is an isomorphism.

For the proof, see Bourbaki, Alg., chap. II, 3rd ed., §6, n° 6, cor. of prop. 10.

(11.2.3) Let us resume the notation of (11.2.1) and consider particularly the case $i=1$; set $T=To{r}_1^A(M,N)$, $T^{\prime }=T{\otimes }_A{A}^{\prime }$, $T^{\prime \prime }=To{r}_1^{A^{\prime }}(M^{\prime },N^{\prime })$; then ${\psi }_1$ is the homomorphism $Ker(f_0\otimes 1_N)/Im(f_1\otimes 1_N)\to Ker(f_0^{\prime \prime }\otimes 1_{N^{\prime }})/Im(f_1^{\prime \prime }\otimes 1_{N^{\prime }})$ which is deduced by passage to quotients from the restriction $Ker(f_0\otimes 1_N)\to Ker(f_0^{\prime \prime }\otimes 1_{N^{\prime }})$ of

  L_1 ⊗_A N → L''_1 ⊗_{A'} N'.

Set $R=Ker(ϵ)$, $R^{\prime \prime }=Ker({ϵ}^{\prime \prime })$, so that one has the exact sequences

  0 → R → L_0 →^ε M → 0    and    0 → R'' → L''_0 →^{ε''} M' → 0,

whence one deduces the exact sequences of homology

  (11.2.3.1)    0 = Tor_1^A(L_0, N) → T →^∂ R ⊗_A N → L_0 ⊗_A N → M ⊗_A N → 0
  (11.2.3.2)    0 = Tor_1^{A'}(L''_0, N') → T'' →^{∂''} R'' ⊗_{A'} N' → L''_0 ⊗_{A'} N' → M' ⊗_{A'} N' → 0

One has on the other hand a homomorphism of $A^{\prime }$-modules

  v : R' = Ker(ε) ⊗_A A' → Ker(ε') →^∼ Ker(ε'') = R''.

Let us show that the diagram

  (11.2.3.3)
       T'  ─∂⊗1─→  R' ⊗_{A'} N'  ─→  L'_0 ⊗_{A'} N'  ─→  M' ⊗_{A'} N'
       │              │                  ‖                   ‖
       ψ_1            v ⊗ 1              u_0 ⊗ 1             1
       ↓              ↓                  ↓                   ↓
       T''  ─∂''─→  R'' ⊗_{A'} N'  ─→  L''_0 ⊗_{A'} N'  ─→  M' ⊗_{A'} N'

is commutative. For this, one verifies at once (M, IV, 1) that the homomorphism $\partial :T\to R{\otimes }_{A}N$ comes (in the present case) by passage to the quotient from the homomorphism $w:Ker(f_0\otimes 1_N)\to R{\otimes }_{A}N$, restriction of the homomorphism $g_0\otimes 1_N:L_1{\otimes }_{A}N\to R{\otimes }_{A}N$, where $g_0:L_1\to R$ is such that $f_0=j\circ g_0$; similarly for ${\partial }^{\prime \prime }$. It therefore suffices to see that the diagram

  Ker(f_0 ⊗ 1_N)  ─→  R ⊗_A N  ─→  R' ⊗_{A'} N' = (R ⊗_A N) ⊗_A A'
       │                                            │
       ↓                                            ↓
  Ker(f''_0 ⊗ 1_{N'}) ──────────────────────────→  R'' ⊗_{A'} N'

is commutative, which results from the commutativity of the diagram

  L_1  ─→  R
   │       │
   ↓       ↓
  L''_1 ─→ R''.

Lemma (11.2.4).

Let $A$ be a ring, $C$ an $A$-algebra, $M$ an $A$-module, $N$ a $C$-module, $A^{\prime }$ an $A$-algebra. Set $C^{\prime }=C{\otimes }_A{A}^{\prime }$, $M^{\prime }=M{\otimes }_A{A}^{\prime }$, $N^{\prime }=N{\otimes }_A{A}^{\prime }=N{\otimes }_C{C}^{\prime }$. Suppose that $M{\otimes }_{A}N$ is a flat $C$-module. Then the canonical homomorphism

  ψ_1 : Tor_1^A(M, N) ⊗_A A' → Tor_1^{A'}(M', N')

(cf. (11.2.1.5)) is surjective.

Let us keep the notation of (11.2.3); right exactness of the tensor product shows that the sequence $L_1^{\prime }\to L_0^{\prime }\to M^{\prime }\to 0$ is exact; since $L_0^{\prime }$ and $L_1^{\prime }$ are free $A^{\prime }$-modules, one may suppose that one has taken $L_0^{\prime \prime }=L_0^{\prime }$, $L_1^{\prime \prime }=L_1^{\prime }$, with $u_0$ and $u_1$ being the identity maps and $f_0^{\prime \prime }=f_0^{\prime }$. Since $R=Im(f_1)$ and $R^{\prime \prime }=Im(f_1^{\prime \prime })=Im(f_1^{\prime })$, the homomorphism $v$ is surjective, and so therefore is $v\otimes 1$. The proof will be complete if one proves that the first row of (11.2.3.3) is exact, $ϵ\otimes 1$ being surjective and $u_0\otimes 1$ injective (Bourbaki, Alg. comm., chap. I, §1, n° 4, cor. 2 of prop. 2). Let us use for this

the hypothesis that $M{\otimes }_{A}N$ is a flat $C$-module. Setting $Q=Ker(ϵ\otimes 1)=Im(j\otimes 1)$ in the exact sequence (11.2.3.1), one has the two exact sequences $0\to Q\to L_0{\otimes }_{A}N\to M{\otimes }_{A}N\to 0$ and $T\to R{\otimes }_{A}N\to Q\to 0$, where the homomorphisms are $C$-module homomorphisms; using the flatness hypothesis, one deduces from this $(0_I,\mathrm{6.1.2})$ the exact sequence

  0 → Q ⊗_C C' → (L_0 ⊗_A N) ⊗_C C' → (M ⊗_A N) ⊗_C C' → 0

and on the other hand, the tensor product being right exact, one has the exact sequence

  T ⊗_C C' → (R ⊗_A N) ⊗_C C' → Q ⊗_C C' → 0

whence finally the exact sequence

  T ⊗_C C' → (R ⊗_A N) ⊗_C C' → (L_0 ⊗_A N) ⊗_C C' → (M ⊗_A N) ⊗_C C' → 0.

But by definition, for every $C$-module $P$, one has $P{\otimes }_C{C}^{\prime }=P{\otimes }_A{A}^{\prime }$, whence the conclusion.

Lemma (11.2.5).

Let $A$ be a ring, $\mathfrak{J}$ an ideal of $A$, $B$ an $A$-algebra, $M$ a $B$-module, $A\to A^{\prime }$ a ring homomorphism. Set ${\mathfrak{J}}^{\prime }=\mathfrak{J}{A}^{\prime }$, $B^{\prime }=B{\otimes }_A{A}^{\prime }$, $M^{\prime }=M{\otimes }_A{A}^{\prime }=M{\otimes }_B{B}^{\prime }$. Let ${\mathfrak{p}}^{\prime }$ be a prime ideal of $B^{\prime }$ containing $\mathfrak{J}{B}^{\prime }$. Suppose one of the following hypotheses is verified:

a) $\mathfrak{J}$ is nilpotent.

b) $A^{\prime }$ is Noetherian, $B^{\prime }$ is an $A^{\prime }$-algebra of finite type, $M^{\prime }$ an $B^{\prime }$-module of finite type.

Under these conditions, suppose verified the following two properties:

(i) $M{\otimes }_A(A/\mathfrak{J})$ is a flat $(A/\mathfrak{J})$-module.

(ii) The canonical composite homomorphism

  Tor_1^A(M, A/𝔍) → Tor_1^{A'}(M', A'/𝔍') → (Tor_1^{A'}(M', A'/𝔍'))_{𝔭'}

(where ${\psi }_1$ is the homomorphism (11.2.1.1) and $\theta$ the canonical homomorphism from a $B^{\prime }$-module to its localization at ${\mathfrak{p}}^{\prime }$) is zero.

Then $M_{{\mathfrak{p}}^{\prime }}^{\prime }$ is a flat $A^{\prime }$-module.

Note that in hypothesis b), $M_{{\mathfrak{p}}^{\prime }}^{\prime }$ is a $B_{{\mathfrak{p}}^{\prime }}^{\prime }$-module of finite type, $B_{{\mathfrak{p}}^{\prime }}^{\prime }$ a Noetherian $A^{\prime }$-algebra, and ${\mathfrak{J}}^{\prime }{B}_{{\mathfrak{p}}^{\prime }}^{\prime }$ is contained in the radical ${\mathfrak{p}}^{\prime }{B}_{{\mathfrak{p}}^{\prime }}^{\prime }$ of $B_{{\mathfrak{p}}^{\prime }}^{\prime }$; in hypothesis a), ${\mathfrak{J}}^{\prime }$ is nilpotent; one will therefore be able to apply the flatness criterion $(0_{III},\mathrm{10.2.1})$ or $(0_{III},\mathrm{10.2.2})$ according as a) or b) is verified. In the first place, one has $M_{{\mathfrak{p}}^{\prime }}^{\prime }{\otimes }_{A^{\prime }}(A^{\prime }/{\mathfrak{J}}^{\prime })=(M^{\prime }{\otimes }_{A^{\prime }}(A^{\prime }/{\mathfrak{J}}^{\prime }){)}_{{\mathfrak{p}}^{\prime }}=((M{\otimes }_A(A/\mathfrak{J})){\otimes }_{A/\mathfrak{J}}(A^{\prime }/{\mathfrak{J}}^{\prime }){)}_{{\mathfrak{p}}^{\prime }}$; hypothesis (i) therefore entails that $M_{{\mathfrak{p}}^{\prime }}^{\prime }{\otimes }_{A^{\prime }}(A^{\prime }/{\mathfrak{J}}^{\prime })$ is a flat $(A^{\prime }/{\mathfrak{J}}^{\prime })$-module, taking $(0_I,\mathrm{6.2.1}and\mathrm{6.3.2})$ into account. It remains therefore to see that $To{r}_1^{A^{\prime }}(M_{{\mathfrak{p}}^{\prime }}^{\prime },A^{\prime }/{\mathfrak{J}}^{\prime })=0$; but this $B_{{\mathfrak{p}}^{\prime }}^{\prime }$-module is equal to $(To{r}_1^{A^{\prime }}(M^{\prime },A^{\prime }/{\mathfrak{J}}^{\prime }){)}_{{\mathfrak{p}}^{\prime }}$ by virtue of the flatness of $B_{{\mathfrak{p}}^{\prime }}^{\prime }$ over $B^{\prime }$. But by virtue of hypothesis (ii), the composite homomorphism $\theta \circ {\psi }_1:To{r}_1^A(M,A/\mathfrak{J}){\otimes }_A{A}^{\prime }\to (To{r}_1^{A^{\prime }}(M^{\prime },A^{\prime }/{\mathfrak{J}}^{\prime }){)}_{{\mathfrak{p}}^{\prime }}$ is zero; moreover, (11.2.4) applied to $C=A/\mathfrak{J}$ and $N=C$ shows (taking hypothesis (i) into account) that ${\psi }_1$ is surjective (for $A^{\prime }/{\mathfrak{J}}^{\prime }=C{\otimes }_A{A}^{\prime }$); hence the homomorphism $\theta$ is zero, and since the image under $\theta$ of $To{r}_1^{A^{\prime }}(M^{\prime },A^{\prime }/{\mathfrak{J}}^{\prime })$ generates the $B_{{\mathfrak{p}}^{\prime }}^{\prime }$-module $(To{r}_1^{A^{\prime }}(M^{\prime },A^{\prime }/{\mathfrak{J}}^{\prime }){)}_{{\mathfrak{p}}^{\prime }}$, the latter is zero. Q.E.D.

Theorem (11.2.6).

The notation being that of (8.5.1) and (8.8.1), suppose $S_{\alpha }$ quasi-compact, $X_{\alpha }$ and $Y_{\alpha }$ of finite presentation over $S_{\alpha }$; let $f_{\alpha }:X_{\alpha }\to Y_{\alpha }$ be an $S_{\alpha }$-morphism, ${\mathcal{F}}_{\alpha }$ a quasi-coherent ${\mathcal{O}}_{X_{\alpha }}$-Module of finite presentation.

(i) Let $x$ be a point of $X$, $x_{\lambda }$ its canonical projection in $X_{\lambda }$. For $\mathcal{F}$ to be $f$-flat at the point $x$, it is necessary and sufficient that there exist $\lambda \ge \alpha$ such that ${\mathcal{F}}_{\lambda }$ is $f_{\lambda }$-flat at the point $x_{\lambda }$.

(ii) For $\mathcal{F}$ to be $f$-flat, it is necessary and sufficient that there exist $\lambda \ge \alpha$ such that ${\mathcal{F}}_{\lambda }$ is $f_{\lambda }$-flat.

One may suppose that $S_{\alpha }=S_0$; since Y_0 is of finite presentation over S_0, Y_0 is quasi-compact, and $f_0:X_0\to Y_0$ is a morphism of finite presentation (1.6.2, (v)), hence one may also confine oneself to the case where $S_0=Y_0$. Moreover, by virtue of the quasi-compactness of S_0 and of the fact that the index set $L$ is filtered, one may confine oneself to the case where $S_0=\mathrm{Spec}(A_0)$ is affine. In addition X_0 is quasi-compact, hence the same reasoning shows that one may also suppose $X_0=\mathrm{Spec}(B_0)$ affine; one then has ${\mathcal{F}}_0={\tilde{M}}_0$, where M_0 is a B_0-module of finite presentation, and the statement (11.2.6) is in this case equivalent to the following (taking $(0_I,\mathrm{6.3.1})$ into account):

Corollary (11.2.6.1).

Let A_0 be a ring, $(A_{\lambda }{)}_{\lambda \in L}$ a filtered inductive system of A_0-algebras, B_0 an A_0-algebra of finite presentation, M_0 a B_0-module of finite presentation. Set $B_{\lambda }=B_0{\otimes }_{A_0}{A}_{\lambda }$, $M_{\lambda }=M_0{\otimes }_{A_0}{A}_{\lambda }=M_0{\otimes }_{B_0}{B}_{\lambda }$, $A=\lim A_{\lambda }$, $B=\lim B_{\lambda }=B_0{\otimes }_{A_0}A$, $M=\lim M_{\lambda }=M_0{\otimes }_{A_0}A=M_0{\otimes }_{B_0}B$.

(i) Let $\mathfrak{p}$ be a prime ideal of $B$, and for every $\lambda$, let ${\mathfrak{p}}_{\lambda }$ be its inverse image in $B_{\lambda }$. For $M_{\mathfrak{p}}$ to be a flat $A$-module, it is necessary and sufficient that there exist $\lambda$ such that $(M_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}$ is a flat $A_{\lambda }$-module.

(ii) For $M$ to be a flat $A$-module, it is necessary and sufficient that there exist $\lambda$ such that $M_{\lambda }$ is a flat $A_{\lambda }$-module.

One has only to prove that the conditions are necessary (2.1.4). We shall proceed in several steps.

I) Reduction to the case where the $A_{\lambda }$ are Noetherian. By virtue of (8.9.1), there exists a sub-ring $A_0^{\prime }$ of A_0 which is a $\mathbb{Z}$-algebra of finite type, an $A_0^{\prime }$-algebra of finite type $B_0^{\prime }$ and a $B_0^{\prime }$-module of finite type $M_0^{\prime }$ such that one has $B_0=B_0^{\prime }{\otimes }_{A_0^{\prime }}{A}_0$ and $M_0=M_0^{\prime }{\otimes }_{A_0^{\prime }}{A}_0$; since one has $B_{\lambda }=B_0^{\prime }{\otimes }_{A_0^{\prime }}{A}_{\lambda }$ and $M_{\lambda }=M_0^{\prime }{\otimes }_{A_0^{\prime }}{A}_{\lambda }$, one may replace A_0, B_0, M_0 by $A_0^{\prime }$, $B_0^{\prime }$, $M_0^{\prime }$ in the statement of (11.2.6.1), considering the $A_{\lambda }$ as $A_0^{\prime }$-algebras; one may therefore already suppose that A_0 is Noetherian. Let $H$ be the set of pairs $(\lambda ,C_{\lambda })$, where $C_{\lambda }$ is a sub-A_0-algebra of finite type of $A_{\lambda }$; order $H$ by setting $(\lambda ,C_{\lambda })\le (\mu ,D_{\mu })$ if $\lambda \le \mu$ and the homomorphism ${\varphi }_{\mu \lambda }:A_{\lambda }\to A_{\mu }$ is such that ${\varphi }_{\mu \lambda }(C_{\lambda })\subset D_{\mu }$; for this order $H$ is filtered increasing, for if $(\lambda ,C_{\lambda })$ and $(\mu ,D_{\mu })$ are two arbitrary elements of $H$, one majorizes them by an $(\nu ,E_{\nu })$ by taking $\nu \ge \lambda$, $\nu \ge \mu$ in $L$, then $E_{\nu }$ equal to the sub-A_0-algebra of $A_{\nu }$ generated by ${\varphi }_{\nu \lambda }(C_{\lambda })$ and ${\varphi }_{\nu \mu }(D_{\mu })$. For an element $\xi =(\lambda ,C_{\lambda })$ of $H$, one will set $A_{\xi }=C_{\lambda }$, and for $\xi \le \eta =(\mu ,D_{\mu })$ (hence $\lambda \le \mu$ and ${\varphi }_{\mu \lambda }(C_{\lambda })\subset D_{\mu }$), ${\varphi }_{\eta \xi }:A_{\xi }\to A_{\eta }$ will be the restriction to $C_{\lambda }$ of ${\varphi }_{\mu \lambda }$, considered as a homomorphism into $D_{\mu }$; it is clear that one thus obtains a filtered inductive system of A_0-algebras. One sets $B_{\xi }=B_0{\otimes }_{A_0}{A}_{\xi }$, $M_{\xi }=M_0{\otimes }_{A_0}{A}_{\xi }$; this time the $A_{\xi }$ are Noetherian; moreover the double-inductive-limit formula (Bourbaki, Alg., chap. II, 3rd ed., §6, n° 4, prop. 7) proves that one again has ${\lim }_H{A}_{\xi }=A$, ${\lim }_H{B}_{\xi }=B$, ${\lim }_H{M}_{\xi }=M$. Suppose

(11.2.6.1) proved for the inductive system $(A_{\xi }{)}_{\xi \in H}$; let $\mathfrak{p}$ be a prime ideal of $B$, such that $M_{\mathfrak{p}}$ is a flat $A$-module; there then exists $\xi \in H$ such that, if ${\mathfrak{p}}_{\xi }$ is the inverse image of $\mathfrak{p}$ in $B_{\xi }$, $(M_{\xi }{)}_{{\mathfrak{p}}_{\xi }}$ is a flat $A_{\xi }$-module. Let $\xi =(\lambda ,C_{\lambda })$, so that the injection $A_{\xi }=C_{\lambda }\to A_{\lambda }$ gives a homomorphism $B_{\xi }=B_0{\otimes }_{A_0}{C}_{\lambda }\to B_{\lambda }=B_0{\otimes }_{A_0}{A}_{\lambda }$, and if ${\mathfrak{p}}_{\lambda }$ is the inverse image of $\mathfrak{p}$ in $B_{\lambda }$, ${\mathfrak{p}}_{\xi }$ is the inverse image of ${\mathfrak{p}}_{\lambda }$ in $B_{\xi }$; consequently $(M_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}=(M_{\xi }{\otimes }_{C_{\lambda }}{A}_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}=((M_{\xi }{)}_{{\mathfrak{p}}_{\xi }}{\otimes }_{C_{\lambda }}{A}_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}$, hence $(M_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}$ is a flat $A_{\lambda }$-module $(0_I,\mathrm{6.2.1}and\mathrm{6.3.2})$. One treats similarly case (ii) of the statement. We may therefore in the sequel suppose the $A_{\lambda }$ Noetherian for $\lambda \in L$ (but not necessarily $A$ itself).

II) Reduction of the global statement (ii) to the pointwise statement (i). Suppose that $\mathcal{F}$ is $f$-flat. For every $\lambda$, let $U_{\lambda }$ be the set of $x_{\lambda }\in X_{\lambda }$ such that ${\mathcal{F}}_{\lambda }$ is $f_{\lambda }$-flat at the point $x_{\lambda }$; one knows that $U_{\lambda }$ is open in $X_{\lambda }$ since $S_{\lambda }$ is Noetherian and $f_{\lambda }$ of finite type (11.1.1); let $V_{\lambda }=v_{\lambda }^{-1}(U_{\lambda })$ be its inverse image in $X$. Since by hypothesis, for every $x\in X$, there is a $\lambda$ such that ${\mathcal{F}}_{\lambda }$ is $f_{\lambda }$-flat at the point $x_{\lambda }$, projection of $x$ in $X_{\lambda }$, this signifies that $x\in V_{\lambda }$ for some $\lambda$; in other words, $X$ is the union of the $V_{\lambda }$. Moreover (2.1.4), for $\lambda \le \mu$, one has $V_{\lambda }\subset V_{\mu }$, hence, since $X$ is quasi-compact, there exists an index $\mu$ such that $X=V_{\mu }$. Since the $X_{\lambda }$ are quasi-compact, it follows from (8.3.4) that there exists an index $\nu \ge \mu$ such that $v_{\nu \mu }^{-1}(U_{\mu })=X_{\nu }$; but by (2.1.4), this entails that ${\mathcal{F}}_{\nu }$ is $f_{\nu }$-flat.

III) End of the proof. It remains to prove (i), supposing S_0 affine and Noetherian; if $y_0$ is the projection of $x$ in S_0, one may in addition, by virtue of (2.1.4) and (I, 3.6.5), replace S_0 by $\mathrm{Spec}({\mathcal{O}}_{S_0,y_0})$, in other words one may confine oneself to the case where A_0 is a Noetherian local ring, of maximal ideal ${\mathfrak{m}}_0=y_0$; by definition, ${\mathfrak{m}}_0$ is the inverse image of the prime ideal $\mathfrak{p}=x$ of $B$, and $M_{\mathfrak{p}}$ is supposed to be a flat $A$-module; one therefore has in particular $To{r}_1^A(A/{\mathfrak{m}}_{0}A,M_{\mathfrak{p}})=0$, which also writes, since the $To{r}_i^A(A/{\mathfrak{m}}_{0}A,M)$ are $B$-modules and $B_{\mathfrak{p}}$ is flat over $B$, $(To{r}_1^A(A/{\mathfrak{m}}_{0}A,M){)}_{\mathfrak{p}}=0$. Let us note now that $To{r}_1^{A_0}(A_0/{\mathfrak{m}}_0,M_0)$ is a B_0-module of finite type, for one may define it by taking a resolution of $A_0/{\mathfrak{m}}_0$ by free A_0-modules of finite type (since A_0 is Noetherian) and tensoring with M_0, which gives B_0-modules of finite type; since B_0 is Noetherian, the homology of the complex thus obtained is indeed formed of B_0-modules of finite type. Let $(t_i^0{)}_{1\le i\le m}$ be a system of generators of the B_0-module $To{r}_1^{A_0}(A_0/{\mathfrak{m}}_0,M_0)$ and let $t_i$ be the canonical image (11.2.1.1) of $t_i^0$ in $To{r}_1^A(A/{\mathfrak{m}}_{0}A,M)$. The hypothesis entails that there exists $h\in B-\mathfrak{p}$ such that $h{t}_i=0$ for $1\le i\le m$. Now one has (11.2.2.1) Tor_1^A(A/𝔪_0 A, M) = lim_λ Tor_1^{A_λ}(A_λ/𝔪_0 A_λ, M_λ); there exists therefore a $\lambda$ such that, if the $t_i^{\lambda }$ are the images of the $t_i^0$ in $To{r}_1^{A_{\lambda }}(A_{\lambda }/{\mathfrak{m}}_0{A}_{\lambda },M_{\lambda })$, there exists $h_{\lambda }\in B_{\lambda }$ of image $h$, such that $h_{\lambda }{t}_i^{\lambda }=0$ for $1\le i\le m$. Let ${\mathfrak{p}}_{\lambda }=v_{\lambda }(\mathfrak{p})$ be the prime ideal of $B_{\lambda }$ inverse image of $\mathfrak{p}$; one has $h_{\lambda }\in B_{\lambda }-{\mathfrak{p}}_{\lambda }$, hence the canonical images of the $t_i^{\lambda }$ in $(To{r}_1^{A_{\lambda }}(A_{\lambda }/{\mathfrak{m}}_0{A}_{\lambda },M_{\lambda }){)}_{{\mathfrak{p}}_{\lambda }}$ are zero, and consequently the homomorphism $To{r}_1^{A_0}(A_0/{\mathfrak{m}}_0,M_0)\to (To{r}_1^{A_{\lambda }}(A_{\lambda }/{\mathfrak{m}}_0{A}_{\lambda },M_{\lambda }){)}_{{\mathfrak{p}}_{\lambda }}$ is zero. The conditions of lemma (11.2.5) are therefore satisfied ($A_0/{\mathfrak{m}}_0$ being a field), and $(M_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}$ is a flat $A_{\lambda }$-module, which finishes the proof of the theorem.

Corollary (11.2.7).

Let $S=\mathrm{Spec}(A)$ be an affine scheme, $f:X\to S$ a morphism, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module, $x$ a point of $X$. The following conditions are equivalent:

a) $f$ is a morphism of finite presentation, $\mathcal{F}$ is an ${\mathcal{O}}_X$-Module of finite presentation and $\mathcal{F}$ is $f$-flat at the point $x$ (resp. $f$-flat).

b) There exist a Noetherian affine scheme $S_0=\mathrm{Spec}(A_0)$, a morphism of finite type $f_0:X_0\to S_0$, a coherent ${\mathcal{O}}_{X_0}$-Module ${\mathcal{F}}_0$, a morphism $S\to S_0$ such that the $S$-prescheme $X_0{\otimes }_{S_0}S$ is $S$-isomorphic to $X$ and that, if one identifies $X$ with $X_0{\otimes }_{S_0}S$, $\mathcal{F}$ is isomorphic to ${\mathcal{F}}_0{\otimes }_{{\mathcal{O}}_{S_0}}{\mathcal{O}}_S$, and ${\mathcal{F}}_0$ is $f_0$-flat at the point $x_0$ projection of $x$ in X_0 (resp. $f_0$-flat).

c) The conditions of b) are verified and in addition A_0 is a sub-$\mathbb{Z}$-algebra of finite type of $A$, the morphism $S\to S_0$ corresponding to the canonical injection $A_0\to A$.

It is clear that c) implies b) and b) implies a) by virtue of (2.1.4). On the other hand, one may consider $A$ as the inductive limit of its sub-$\mathbb{Z}$-algebras of finite type, and one knows by (8.9.1) that there is such a sub-algebra A_0 and a morphism of finite type $f_0:X_0\to S_0$ such that $X$ is $S$-isomorphic to $X_0{\times }_{S_0}S$ and $\mathcal{F}$ isomorphic to ${\mathcal{F}}_0{\otimes }_{{\mathcal{O}}_{S_0}}{\mathcal{O}}_S$; one may therefore write $X=\lim X_{\lambda }$, where $X_{\lambda }=X_0{\otimes }_{A_0}{A}_{\lambda }$, $A_{\lambda }$ running through the set of sub-$\mathbb{Z}$-algebras of finite type of $A$ containing A_0, and $\mathcal{F}=v_{\lambda }^{\ast }({\mathcal{F}}_{\lambda })$, where $v_{\lambda }:X\to X_{\lambda }$ is the canonical projection and ${\mathcal{F}}_{\lambda }={\mathcal{F}}_0{\otimes }_{A_0}{A}_{\lambda }$. It follows from (11.2.6) that there exists $\lambda$ such that ${\mathcal{F}}_{\lambda }$ is $f_{\lambda }$-flat at the point $x_{\lambda }=v_{\lambda }(x)$ (resp. $f_{\lambda }$-flat); then the sub-ring $A_{\lambda }$ of $A$ verifies the conditions of c).

Proposition (11.2.8).

Let $g:X\to S$, $h:Y\to S$ be two morphisms of finite presentation, $f:X\to Y$ an $S$-morphism, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module of finite presentation; for every $s\in S$, let $X_s=g^{-1}(s)=X{\times }_S\mathrm{Spec}(k(s))$, $Y_s=h^{-1}(s)=Y{\times }_S\mathrm{Spec}(k(s))$, $f_s$ the morphism $f{\times }_{S}1:X_s\to Y_s$, ${\mathcal{F}}_s=\mathcal{F}{\otimes }_{{\mathcal{O}}_S}k(s)$. Then the set $E$ of $s\in S$ such that ${\mathcal{F}}_s$ is $f_s$-flat is locally constructible.

The property that we wish to show constructible verifies condition (9.2.1, (i)), by virtue of (2.2.13) and (2.5.1). Taking (9.2.3) into account, one may therefore confine oneself to the case where $S$ is affine, Noetherian, and integral, with generic point $\eta$, and to prove that $E$ or $S-E$ is a neighbourhood of $\eta$ in $S$. If $\eta \in S-E$, this follows at once from lemma (9.4.7.1). If on the contrary $\eta \in E$, it follows first of all from (11.2.6), applied according to the method of (8.1.2, a)), that there is an open neighbourhood $U$ of $\eta$ in $S$ such that $\mathcal{F}|g^{-1}(U)$ is $h^{-1}(U)$-flat; a fortiori (2.1.4) ${\mathcal{F}}_s$ is $f_s$-flat for every $s\in U$, which finishes the proof.

The following theorem generalizes (11.2.6.1, (ii)):

Theorem (11.2.9) (Raynaud).

Let A_0 be a ring, $(A_{\lambda }{)}_{\lambda \in L}$ a filtered inductive system of A_0-algebras, B_0 an A_0-algebra of finite presentation, ${\mathfrak{J}}_0$ an ideal of finite type of B_0, M_0 a B_0-module of finite presentation. Set $B_{\lambda }=B_0{\otimes }_{A_0}{A}_{\lambda }$, ${\mathfrak{J}}_{\lambda }={\mathfrak{J}}_0{B}_{\lambda }$, $M_{\lambda }=M_0{\otimes }_{A_0}{A}_{\lambda }=M_0{\otimes }_{B_0}{B}_{\lambda }$, $A=\lim A_{\lambda }$, $B=\lim B_{\lambda }=B_0{\otimes }_{A_0}A$, $\mathfrak{J}={\mathfrak{J}}_{0}B$, $M=\lim M_{\lambda }=M_0{\otimes }_{A_0}A=M_0{\otimes }_{B_0}B$. For $g{r}_{\mathfrak{J}}^{\bullet }(M)$ to be a flat $A$-module, it is necessary and sufficient that there exist $\lambda$ such that $g{r}_{{\mathfrak{J}}_{\lambda }}^{\bullet }(M_{\lambda })$ is a flat $A_{\lambda }$-module; the canonical homomorphisms

  (11.2.9.1)    gr_{𝔍_λ}^•(M_λ) ⊗_{A_λ} A_μ → gr_{𝔍_μ}^•(M_μ)    for μ ≥ λ

are then bijective and $g{r}_{{\mathfrak{J}}_{\mu }}^{\bullet }(M_{\mu })$ is a flat $A_{\mu }$-module for $\mu \ge \lambda$.

Let us first show that the conditions are sufficient, which amounts to proving the bijectivity of (11.2.9.1). This follows from the following lemma:

Lemma (11.2.9.2).

Let $A$ be a ring, $B$ an $A$-algebra, $M$ a $B$-module, $\mathfrak{J}$ an ideal of $B$. Let $A^{\prime }$ be an $A$-algebra; set $B^{\prime }=B{\otimes }_A{A}^{\prime }$, $M^{\prime }=M{\otimes }_A{A}^{\prime }=M{\otimes }_B{B}^{\prime }$, ${\mathfrak{J}}^{\prime }=\mathfrak{J}{B}^{\prime }$. If $g{r}_{\mathfrak{J}}^{\bullet }(M)$ is a flat $A$-module, the canonical homomorphism

  gr_𝔍^•(M) ⊗_A A' → gr_{𝔍'}^•(M')

is bijective.

Indeed, by induction on $k$, the hypothesis that the ${\mathfrak{J}}^{k}M/{\mathfrak{J}}^{k+1}M$ are flat $A$-modules for $k\ge 0$ first entails, by $(0_I,\mathrm{6.1.2})$, that $M/{\mathfrak{J}}^{k+1}M$ is a flat $A$-module; moreover $(0_I,\mathrm{6.1.2})$, the sequence