Chapter 0 — Preliminaries

§6. Flatness

(6.0) The notion of flatness is due to J.-P. Serre [16]; in what follows, we omit proofs that are given in N. Bourbaki's Algèbre commutative, to which we refer the reader. All rings are assumed commutative.¹

If $M$, $N$ are $A$-modules and $M^{\prime }$ (resp. $N^{\prime }$) is a submodule of $M$ (resp. $N$), we write $Im(M^{\prime }{\otimes }_A{N}^{\prime })$ for the submodule of $M{\otimes }_{A}N$ that is the image of the canonical map $M^{\prime }{\otimes }_A{N}^{\prime }\to M{\otimes }_{A}N$.

6.1. Flat modules

(6.1.1) Let $M$ be an $A$-module. The following are equivalent:

a) The functor $M{\otimes }_{A}N$ in $N$ is exact on $A$-modules; b) $To{r}_i^A(M,N)=0$ for every $i>0$ and every $A$-module $N$; c) $To{r}_1^A(M,N)=0$ for every $A$-module $N$.

When $M$ satisfies these conditions, $M$ is called a flat $A$-module. Every free $A$-module is plainly flat.

For $M$ to be flat, it suffices that for every finite-type ideal $\mathfrak{J}$ of $A$, the canonical map $M{\otimes }_A\mathfrak{J}\to M{\otimes }_{A}A=M$ be injective.

(6.1.2) Every inductive limit of flat $A$-modules is flat. A direct sum ${\oplus }_{\lambda \in L}{M}_{\lambda }$ is flat if and only if each $M_{\lambda }$ is flat. In particular, every projective $A$-module is flat.

Let $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ be an exact sequence with $M^{\prime \prime }$ flat. Then for every $A$-module $N$,

0 → M′ ⊗ N → M ⊗ N → M″ ⊗ N → 0

is exact. Moreover, for $M$ to be flat, it is necessary and sufficient that $M^{\prime }$ be flat (but $M$ and $M^{\prime }$ may both be flat without $M^{\prime \prime }=M/M^{\prime }$ being flat).

(6.1.3) Let $M$ be a flat $A$-module and $N$ an arbitrary $A$-module. For two submodules $N^{\prime },N^{\prime \prime }$ of $N$,

Im(M ⊗ (N′ + N″)) = Im(M ⊗ N′) + Im(M ⊗ N″)
Im(M ⊗ (N′ ∩ N″)) = Im(M ⊗ N′) ∩ Im(M ⊗ N″)

(images taken in $M\otimes N$).

(6.1.4) Let $M$, $N$ be $A$-modules, $M^{\prime }$ (resp. $N^{\prime }$) a submodule of $M$ (resp. $N$), and suppose one of $M/M^{\prime },N/N^{\prime }$ is flat. Then $Im(M^{\prime }\otimes N^{\prime })=Im(M^{\prime }\otimes N)\cap Im(M\otimes N^{\prime })$ (images in $M\otimes N$). In particular, if $\mathfrak{J}$ is an ideal of $A$ and $M/M^{\prime }$ is flat, then $\mathfrak{J}{M}^{\prime }=M^{\prime }\cap \mathfrak{J}M$.

6.2. Change of ring

When an abelian group $M$ is endowed with several module structures over rings $A$, $B$, …, instead of saying that $M$ is flat as an $A$-module, $B$-module, etc., we shall sometimes say $M$ is $A$-flat, $B$-flat, etc.

(6.2.1) Let A, B be rings, $M$ an $A$-module, and $N$ an $(A,B)$-bimodule. If $M$ is flat and $N$ is $B$-flat, then $M{\otimes }_{A}N$ is $B$-flat. In particular, if M, N are two flat $A$-modules, $M{\otimes }_{A}N$ is a flat $A$-module. If $B$ is an $A$-algebra and $M$ is a flat $A$-module, the $B$-module $M_{(B)}=M{\otimes }_{A}B$ is flat. Finally, if $B$ is an $A$-algebra which is flat as an $A$-module, and $N$ is a flat $B$-module, then $N$ is also $A$-flat.

(6.2.2) Let $A$ be a ring and $B$ an $A$-algebra which is flat as an $A$-module. Let M, N be $A$-modules with $M$ admitting a finite presentation. The canonical homomorphism

(6.2.2.1)    Hom_A(M, N) ⊗_A B → Hom_B(M ⊗_A B, N ⊗_A B)

sending $u\otimes b$ to the homomorphism $m\otimes b^{\prime }\mapsto u(m)\otimes b^{\prime }b$ is an isomorphism.

(6.2.3) Let $(A_{\lambda },{\varphi }_{\mu \lambda })$ be a filtered inductive system of rings with $A=\underrightarrow{\lim }{A}_{\lambda }$. For each $\lambda$, let $M_{\lambda }$ be an $A_{\lambda }$-module, and for $\lambda \le \mu$ let ${\theta }_{\mu \lambda }:M_{\lambda }\to M_{\mu }$ be a ${\varphi }_{\mu \lambda }$-homomorphism, with $(M_{\lambda },{\theta }_{\mu \lambda })$ an inductive system; set $M=\underrightarrow{\lim }{M}_{\lambda }$, an $A$-module. If each $M_{\lambda }$ is $A_{\lambda }$-flat, then $M$ is $A$-flat. Indeed, let $\mathfrak{J}$ be a finite-type ideal of $A$; by the definition of the inductive limit, there is an index $\lambda$ and an ideal ${\mathfrak{J}}_{\lambda }\subset A_{\lambda }$ with $\mathfrak{J}={\mathfrak{J}}_{\lambda }A$. Setting ${\mathfrak{J}}_{\mu }^{\prime }={\mathfrak{J}}_{\lambda }{A}_{\mu }$ for $\mu \ge \lambda$, also $\mathfrak{J}=\underrightarrow{\lim }{\mathfrak{J}}_{\mu }^{\prime }$ (over $\mu \ge \lambda$). Since $\underrightarrow{\lim }$ is exact and commutes with tensor products,

M ⊗_A 𝔍 = lim⃗ (M_μ ⊗_{A_μ} 𝔍′_μ) = lim⃗ 𝔍′_μ M_μ = 𝔍 M.

6.3. Localization of flatness

(6.3.1) Let $A$ be a ring and $S$ a multiplicative subset. Then $S^{-1}A$ is a flat $A$-module: for every $A$-module $N$, $N{\otimes }_A{S}^{-1}A$ is identified with $S^{-1}N$ (1.2.5), and $S^{-1}N$ is exact in $N$ (1.3.2).

If $M$ is a flat $A$-module, then $S^{-1}M=M{\otimes }_A{S}^{-1}A$ is $S^{-1}A$-flat (6.2.1) — hence also $A$-flat by (6.2.1). In particular, if $P$ is an $S^{-1}A$-module viewed as an $A$-module (isomorphic to $S^{-1}P$), then $P$ is $A$-flat if and only if it is $S^{-1}A$-flat.

(6.3.2) Let $A$ be a ring, $B$ an $A$-algebra, and $T$ a multiplicative subset of $B$. If $P$ is a $B$-module which is $A$-flat, then $T^{-1}P$ is $A$-flat. Indeed, for every $A$-module $N$,

(T⁻¹P) ⊗_A N = (T⁻¹B ⊗_B P) ⊗_A N = T⁻¹B ⊗_B (P ⊗_A N) = T⁻¹(P ⊗_A N);

$T^{-1}(P{\otimes }_{A}N)$ is exact in $N$ as the composite of two exact functors $P{\otimes }_{A}N$ (in $N$) and $T^{-1}Q$ (in $Q$). If $S\subset A$ is multiplicative with its image in $B$ contained in $T$, then $T^{-1}P=S^{-1}(T^{-1}P)$, so also $S^{-1}A$-flat by (6.3.1).

(6.3.3) Let $\varphi :A\to B$ be a ring homomorphism and $M$ a $B$-module. The following are equivalent:

a) $M$ is $A$-flat; b) For every maximal ideal $\mathfrak{n}$ of $B$, $M_{\mathfrak{n}}$ is $A$-flat; c) For every maximal ideal $\mathfrak{n}$ of $B$, with $\mathfrak{m}={\varphi }^{-1}(\mathfrak{n})$, $M_{\mathfrak{n}}$ is $A_{\mathfrak{m}}$-flat.

Indeed, since $M_{\mathfrak{n}}=(M_{\mathfrak{n}}{)}_{\mathfrak{m}}$, the equivalence of b) and c) follows from (6.3.1), and a) ⟹ b) is (6.3.2). For b) ⟹ a): given an injective $u:N^{\prime }\to N$ of $A$-modules, we must show $v=1\otimes u:M{\otimes }_A{N}^{\prime }\to M{\otimes }_{A}N$ is injective. Since $v$ is also a $B$-module homomorphism, it suffices that $v_{\mathfrak{n}}$ be injective for every maximal $\mathfrak{n}$. But

(M ⊗_A N)_𝔫 = B_𝔫 ⊗_B (M ⊗_A N) = M_𝔫 ⊗_A N,

so $v_{\mathfrak{n}}$ is $1\otimes u:M_{\mathfrak{n}}{\otimes }_A{N}^{\prime }\to M_{\mathfrak{n}}{\otimes }_{A}N$, injective since $M_{\mathfrak{n}}$ is $A$-flat.

In particular (taking $B=A$), an $A$-module $M$ is flat iff $M_{\mathfrak{m}}$ is $A_{\mathfrak{m}}$-flat for every maximal ideal $\mathfrak{m}$ of $A$.

(6.3.4) Let $M$ be an $A$-module. If $M$ is flat and $f\in A$ is not a zero divisor of $A$, then $f$ annihilates no nonzero element of $M$: the homomorphism $m\mapsto f\cdot m$ is $1\otimes u$, where $u:a\mapsto f\cdot a$ on $A$ and $M\cong M{\otimes }_{A}A$; $u$ is injective, so $1\otimes u$ is injective since $M$ is flat. In particular, if $A$ is an integral ring, $M$ is torsion-free.

Conversely, suppose $A$ is integral and $M$ is torsion-free, and that for every maximal $\mathfrak{m}$ of $A$, $A_{\mathfrak{m}}$ is a discrete valuation ring. Then $M$ is $A$-flat. By (6.3.3), it suffices to show $M_{\mathfrak{m}}$ is $A_{\mathfrak{m}}$-flat, so assume $A$ is already a DVR. Since $M=\underrightarrow{\lim }{M}_i$ over its finite-type submodules (each torsion-free), one may assume $M$ is of finite type (6.1.2); then $M$ is a free $A$-module, whence the assertion.

In particular, if $A$ is integral and $\varphi :A\to B$ is a homomorphism making $B$ a flat $A$-module with $B\ne 0$, then $\varphi$ is injective. Conversely, if $B$ is integral, $A$ is a subring of $B$, and $A_{\mathfrak{m}}$ is a DVR for every maximal $\mathfrak{m}$, then $B$ is $A$-flat.

6.4. Faithfully flat modules

(6.4.1) For an $A$-module $M$, the following are equivalent:

a) A sequence $N^{\prime }\to N\to N^{\prime \prime }$ of $A$-modules is exact iff $M\otimes N^{\prime }\to M\otimes N\to M\otimes N^{\prime \prime }$ is exact; b) $M$ is flat, and for every $A$-module $N$, $M\otimes N=0$ implies $N=0$; c) $M$ is flat, and for every homomorphism $v:N\to N^{\prime }$ of $A$-modules, $1_M\otimes v=0$ implies $v=0$; d) $M$ is flat, and $\mathfrak{m}M>\ne M$ for every maximal ideal $\mathfrak{m}$ of $A$.

When $M$ satisfies these conditions, $M$ is faithfully flat; $M$ is then necessarily a faithful module. Moreover, for $u:N\to N^{\prime }$, $u$ is injective (resp. surjective, bijective) iff $1\otimes u:M\otimes N\to M\otimes N^{\prime }$ is.

(6.4.2) A nonzero free module is faithfully flat. So is the direct sum of a flat module and a faithfully flat one. If $S\subset A$ is multiplicative, $S^{-1}A$ is faithfully flat as an $A$-module only if $S$ consists of invertible elements (so $S^{-1}A=A$).

(6.4.3) Let $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ be exact with $M^{\prime }$ and $M^{\prime \prime }$ flat. If either is faithfully flat, then $M$ is faithfully flat.

(6.4.4) Let A, B be rings, $M$ an $A$-module, and $N$ an $(A,B)$-bimodule. If $M$ is faithfully flat and $N$ is a faithfully flat $B$-module, then $M{\otimes }_{A}N$ is a faithfully flat $B$-module. In particular, if M, N are two faithfully flat $A$-modules, so is $M{\otimes }_{A}N$. If $B$ is an $A$-algebra and $M$ is a faithfully flat $A$-module, then $M_{(B)}$ is a faithfully flat $B$-module.

(6.4.5) If $M$ is a faithfully flat $A$-module and $S\subset A$ is multiplicative, then $S^{-1}M$ is a faithfully flat $S^{-1}A$-module (since $S^{-1}M=M{\otimes }_A{S}^{-1}A$, by (6.4.4)). Conversely, if $M_{\mathfrak{m}}$ is faithfully flat over $A_{\mathfrak{m}}$ for every maximal $\mathfrak{m}$, then $M$ is faithfully flat over $A$: it is $A$-flat by (6.3.3), and

M_𝔪 / 𝔪 M_𝔪 = (M ⊗_A A_𝔪) ⊗_{A_𝔪} (A_𝔪 / 𝔪 A_𝔪) = M ⊗_A (A/𝔪) = M / 𝔪 M,

so the hypothesis gives $M/\mathfrak{m}M\ne 0$ for every maximal $\mathfrak{m}$, whence (6.4.1).

6.5. Restriction of scalars

(6.5.1) Let $A$ be a ring and $\varphi :A\to B$ a homomorphism making $B$ an $A$-algebra. Suppose there is a $B$-module $N$ that is a faithfully flat $A$-module. Then for every $A$-module $M$, the map $x\mapsto 1\otimes x$ from $M$ to $B{\otimes }_{A}M=M_{(B)}$ is injective. In particular, $\varphi$ is injective; for every ideal $\mathfrak{a}\subset A$, ${\varphi }^{-1}(\mathfrak{a}B)=\mathfrak{a}$; for every maximal (resp. prime) ideal $\mathfrak{m}\subset A$, there is a maximal (resp. prime) ideal $\mathfrak{n}\subset B$ with ${\varphi }^{-1}(\mathfrak{n})=\mathfrak{m}$.

(6.5.2) Under the conditions of (6.5.1), one identifies $A$ with a subring of $B$ via $\varphi$; more generally, $M$ is identified with an $A$-submodule of $M_{(B)}$. Note that if $B$ is Noetherian, so is $A$: the map $\mathfrak{a}\mapsto \mathfrak{a}B$ is an increasing injection from ideals of $A$ into ideals of $B$, so a strictly increasing infinite chain of ideals of $A$ would give one of $B$.

6.6. Faithfully flat rings

(6.6.1) Let $\varphi :A\to B$ be a ring homomorphism. The following are equivalent:

a) $B$ is a faithfully flat $A$-module (equivalently, $M_{(B)}$ is exact and faithful in $M$); b) $\varphi$ is injective and $B/\varphi (A)$ is $A$-flat; c) $B$ is $A$-flat (so $M_{(B)}$ is exact) and $x\mapsto 1\otimes x$ from $M$ to $M_{(B)}$ is injective for every $M$; d) $B$ is $A$-flat and ${\varphi }^{-1}(\mathfrak{a}B)=\mathfrak{a}$ for every ideal $\mathfrak{a}\subset A$; e) $B$ is $A$-flat and for every maximal $\mathfrak{m}\subset A$ there is a maximal $\mathfrak{n}\subset B$ with ${\varphi }^{-1}(\mathfrak{n})=\mathfrak{m}$.

Under these conditions, $A$ is identified with a subring of $B$.

(6.6.2) Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and $B$ an $A$-algebra with $\mathfrak{m}B\ne B$ (this holds, for example, when $B$ is local and $A\to B$ is local). If $B$ is $A$-flat, then $B$ is $A$-faithfully flat. Indeed, $\mathfrak{m}B\ne B$ gives a maximal ideal $\mathfrak{n}\subset B$ containing $\mathfrak{m}B$; ${\varphi }^{-1}(\mathfrak{n})\cap A$ contains $\mathfrak{m}$ and not 1, so ${\varphi }^{-1}(\mathfrak{n})=\mathfrak{m}$, and criterion e) of (6.6.1) applies. Consequently, if $B$ is Noetherian, so is $A$ (6.5.2).

(6.6.3) Let $B$ be an $A$-algebra which is faithfully flat as an $A$-module. For every $A$-module $M$ and submodule $M^{\prime }\subset M$, identifying $M$ with a submodule of $M_{(B)}$, one has $M^{\prime }=M\cap M_{(B)}^{\prime }$. For $M$ to be $A$-flat (resp. faithfully flat), it is necessary and sufficient that $M_{(B)}$ be $B$-flat (resp. faithfully flat).

(6.6.4) Let $B$ be an $A$-algebra and $N$ a faithfully flat $B$-module. For $B$ to be $A$-flat (resp. faithfully flat), it is necessary and sufficient that $N$ be.

In particular, let $C$ be a $B$-algebra. If $C$ is faithfully flat over $B$ and $B$ is faithfully flat over $A$, then $C$ is faithfully flat over $A$. If $C$ is faithfully flat over $B$ and over $A$, then $B$ is faithfully flat over $A$.

6.7. Flat morphisms of ringed spaces

(6.7.1) Let $f:X\to Y$ be a morphism of ringed spaces and $\mathcal{F}$ an ${\mathcal{O}}_X$-Module. $\mathcal{F}$ is $f$-flat (or $Y$-flat when no confusion about $f$ can arise) at a point $x\in X$ if ${\mathcal{F}}_x$ is a flat ${\mathcal{O}}_{f(x)}$-module; $\mathcal{F}$ is $f$-flat over $y\in Y$ if it is $f$-flat at every $x\in f^{-1}(y)$; $\mathcal{F}$ is $f$-flat if it is $f$-flat at every point of $X$. The morphism $f$ is flat at $x\in X$ (resp. flat over $y\in Y$, resp. flat) if ${\mathcal{O}}_X$ is $f$-flat at $x$ (resp. over $y$, resp. $f$-flat).

(6.7.2) With the notation of (6.7.1), if $\mathcal{F}$ is $f$-flat at $x$, then for every open neighborhood $U$ of $y=f(x)$, the functor $(f\ast (\mathcal{G}){\otimes }_{{\mathcal{O}}_X}\mathcal{F}{)}_x$ in $\mathcal{G}$ is exact on $({\mathcal{O}}_Y|U)$-Modules: this stalk is canonically ${\mathcal{G}}_y{\otimes }_{{\mathcal{O}}_y}{\mathcal{F}}_x$, whence the assertion. In particular, if $f$ is a flat morphism, $f\ast$ is exact on ${\mathcal{O}}_Y$-Modules.

(6.7.3) Conversely, suppose ${\mathcal{O}}_Y$ is coherent, and that for every open neighborhood $U$ of $y$, the functor $(f\ast (\mathcal{G}){\otimes }_{{\mathcal{O}}_X}\mathcal{F}{)}_x$ is exact in $\mathcal{G}$ on coherent $({\mathcal{O}}_Y|U)$-Modules. Then $\mathcal{F}$ is $f$-flat at $x$. It suffices to show that for every finite-type ideal $\mathfrak{J}\subset {\mathcal{O}}_y$, the canonical map $\mathfrak{J}{\otimes }_{{\mathcal{O}}_y}{\mathcal{F}}_x\to {\mathcal{F}}_x$ is injective (6.1.1). By (5.3.8) there is an open $U\ni y$ and a coherent ideal sheaf $\mathcal{J}\subset {\mathcal{O}}_Y|U$ with ${\mathcal{J}}_y=\mathfrak{J}$, whence the conclusion.

(6.7.4) The results of (6.1) on flat modules carry over at once to propositions on $f$-flat sheaves at a point:

If $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ is exact and ${\mathcal{F}}^{\prime \prime }$ is $f$-flat at $x$, then for every open $U\ni y=f(x)$ and every $({\mathcal{O}}_Y|U)$-Module $\mathcal{G}$, the sequence

0 → (f*(𝒢) ⊗_{𝒪_X} ℱ′)_x → (f*(𝒢) ⊗_{𝒪_X} ℱ)_x → (f*(𝒢) ⊗_{𝒪_X} ℱ″)_x → 0

is exact. For $\mathcal{F}$ to be $f$-flat at $x$, it is necessary and sufficient that ${\mathcal{F}}^{\prime }$ be. Analogous statements hold for $f$-flatness over $y\in Y$ and $f$-flatness on $X$.

(6.7.5) Let $f:X\to Y$ and $g:Y\to Z$ be morphisms of ringed spaces, $x\in X$, $y=f(x)$, and $\mathcal{F}$ an ${\mathcal{O}}_X$-Module. If $\mathcal{F}$ is $f$-flat at $x$ and $g$ is flat at $y$, then $\mathcal{F}$ is $(g\circ f)$-flat at $x$ (6.2.1). In particular, if $f$ and $g$ are flat, so is $g\circ f$.

(6.7.6) Let X, Y be ringed spaces and $f:X\to Y$ a flat morphism. The canonical homomorphism of bifunctors (4.4.6)

(6.7.6.1)    f*(ℋom_{𝒪_Y}(ℱ, 𝒢)) → ℋom_{𝒪_X}(f*(ℱ), f*(𝒢))

is an isomorphism when $\mathcal{F}$ admits a finite presentation (5.2.5).

Indeed, the question being local, one may assume an exact sequence ${\mathcal{O}}_Y^m\to {\mathcal{O}}_Y^n\to \mathcal{F}\to 0$. Both sides of (6.7.6.1) are left exact in $\mathcal{F}$ (using flatness of $f$), so one reduces to $\mathcal{F}={\mathcal{O}}_Y$, where the assertion is trivial.

(6.7.8) A morphism $f:X\to Y$ of ringed spaces is faithfully flat if $f$ is surjective and, for every $x\in X$, ${\mathcal{O}}_x$ is a faithfully flat ${\mathcal{O}}_{f(x)}$-module. When X, Y are spaces ringed in local rings (5.5.1), this amounts to $f$ being surjective and flat (6.6.2). When $f$ is faithfully flat, $f\ast$ is exact and faithful on ${\mathcal{O}}_Y$-Modules (6.6.1, a)), and an ${\mathcal{O}}_Y$-Module $\mathcal{G}$ is $Y$-flat iff $f\ast (\mathcal{G})$ is (6.6.3).