The Stacks project

41.9 Flat morphisms

This section simply exists to summarize the properties of flatness that will be useful to us. Thus, we will be content with stating the theorems precisely and giving references for the proofs.

After briefly recalling the necessary facts about flat modules over Noetherian rings, we state a theorem of Grothendieck which gives sufficient conditions for “hyperplane sections” of certain modules to be flat.

Definition 41.9.1. Flatness of modules and rings.

1. A module $N$ over a ring $A$ is said to be flat if the functor $M \mapsto M \otimes _ A N$ is exact.

2. If this functor is also faithful, we say that $N$ is faithfully flat over $A$.

3. A morphism of rings $f : A \to B$ is said to be flat (resp. faithfully flat) if the functor $M \mapsto M \otimes _ A B$ is exact (resp. faithful and exact).

Here is a list of facts with references to the algebra chapter.

1. Free and projective modules are flat. This is clear for free modules and follows for projective modules as they are direct summands of free modules and $\otimes$ commutes with direct sums.

2. Flatness is a local property, that is, $M$ is flat over $A$ if and only if $M_{\mathfrak p}$ is flat over $A_{\mathfrak p}$ for all $\mathfrak p \in \mathop{\mathrm{Spec}}(A)$. See Algebra, Lemma 10.39.18.

3. If $M$ is a flat $A$-module and $A \to B$ is a ring map, then $M \otimes _ A B$ is a flat $B$-module. See Algebra, Lemma 10.39.7.

4. Finite flat modules over local rings are free. See Algebra, Lemma 10.78.5.

5. If $f : A \to B$ is a morphism of arbitrary rings, $f$ is flat if and only if the induced maps $A_{f^{-1}(\mathfrak q)} \to B_{\mathfrak q}$ are flat for all $\mathfrak q \in \mathop{\mathrm{Spec}}(B)$. See Algebra, Lemma 10.39.18

6. If $f : A \to B$ is a local homomorphism of local rings, $f$ is flat if and only if it is faithfully flat. See Algebra, Lemma 10.39.17.

7. A map $A \to B$ of rings is faithfully flat if and only if it is flat and the induced map on spectra is surjective. See Algebra, Lemma 10.39.16.

8. If $A$ is a Noetherian local ring, the completion $A^\wedge$ is faithfully flat over $A$. See Algebra, Lemma 10.97.3.

9. Let $A$ be a Noetherian local ring and $M$ an $A$-module. Then $M$ is flat over $A$ if and only if $M \otimes _ A A^\wedge$ is flat over $A^\wedge$. (Combine the previous statement with Algebra, Lemma 10.39.8.)

Before we move on to the geometric category, we present Grothendieck's theorem, which provides a convenient recipe for producing flat modules.

Theorem 41.9.2. Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be a local homomorphism. If $M$ is a finite $B$-module that is flat as an $A$-module, and $t \in \mathfrak m_ B$ is an element such that multiplication by $t$ is injective on $M/\mathfrak m_ AM$, then $M/tM$ is also $A$-flat.

Proof. See Algebra, Lemma 10.99.1. See also [Section 20, MatCA]. $\square$

Definition 41.9.3. (See Morphisms, Definition 29.25.1). Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

1. Let $x \in X$. We say $\mathcal{F}$ is flat over $Y$ at $x \in X$ if $\mathcal{F}_ x$ is a flat $\mathcal{O}_{Y, f(x)}$-module. This uses the map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ to think of $\mathcal{F}_ x$ as a $\mathcal{O}_{Y, f(x)}$-module.

2. Let $x \in X$. We say $f$ is flat at $x \in X$ if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is flat.

3. We say $f$ is flat if it is flat at all points of $X$.

4. A morphism $f : X \to Y$ that is flat and surjective is sometimes said to be faithfully flat.

Once again, here is a list of results:

1. The property (of a morphism) of being flat is, by fiat, local in the Zariski topology on the source and the target.

2. Open immersions are flat. (This is clear because it induces isomorphisms on local rings.)

3. Flat morphisms are stable under base change and composition. Morphisms, Lemmas 29.25.8 and 29.25.6.

4. If $f : X \to Y$ is flat, then the pullback functor $\mathit{QCoh}(\mathcal{O}_ Y) \to \mathit{QCoh}(\mathcal{O}_ X)$ is exact. This is immediate by looking at stalks.

5. Let $f : X \to Y$ be a morphism of schemes, and assume $Y$ is quasi-compact and quasi-separated. In this case if the functor $f^*$ is exact then $f$ is flat. (Proof omitted. Hint: Use Properties, Lemma 28.22.1 to see that $Y$ has “enough” ideal sheaves and use the characterization of flatness in Algebra, Lemma 10.39.5.)

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