## 29.27 Generic flatness

A scheme of finite type over an integral base is flat over a dense open of the base. In Algebra, Section 10.118 we proved a Noetherian version, a version for morphisms of finite presentation, and a general version. We only state and prove the general version here. However, it turns out that this will be superseded by Proposition 29.27.2 which shows the result holds if we only assume the base is reduced.

Proposition 29.27.1 (Generic flatness). Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. Assume

1. $S$ is integral,

2. $f$ is of finite type, and

3. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.

Then there exists an open dense subscheme $U \subset S$ such that $X_ U \to U$ is flat and of finite presentation and such that $\mathcal{F}|_{X_ U}$ is flat over $U$ and of finite presentation over $\mathcal{O}_{X_ U}$.

Proof. As $S$ is integral it is irreducible (see Properties, Lemma 28.3.4) and any nonempty open is dense. Hence we may replace $S$ by an affine open of $S$ and assume that $S = \mathop{\mathrm{Spec}}(A)$ is affine. As $S$ is integral we see that $A$ is a domain. As $f$ is of finite type, it is quasi-compact, so $X$ is quasi-compact. Hence we can find a finite affine open cover $X = \bigcup _{i = 1, \ldots , n} X_ i$. Write $X_ i = \mathop{\mathrm{Spec}}(B_ i)$. Then $B_ i$ is a finite type $A$-algebra, see Lemma 29.15.2. Moreover there are finite type $B_ i$-modules $M_ i$ such that $\mathcal{F}|_{X_ i}$ is the quasi-coherent sheaf associated to the $B_ i$-module $M_ i$, see Properties, Lemma 28.16.1. Next, for each pair of indices $i, j$ choose an ideal $I_{ij} \subset B_ i$ such that $X_ i \setminus X_ i \cap X_ j = V(I_{ij})$ inside $X_ i = \mathop{\mathrm{Spec}}(B_ i)$. Set $M_{ij} = B_ i/I_{ij}$ and think of it as a $B_ i$-module. Then $V(I_{ij}) = \text{Supp}(M_{ij})$ and $M_{ij}$ is a finite $B_ i$-module.

At this point we apply Algebra, Lemma 10.118.3 the pairs $(A \to B_ i, M_{ij})$ and to the pairs $(A \to B_ i, M_ i)$. Thus we obtain nonzero $f_{ij}, f_ i \in A$ such that (a) $A_{f_{ij}} \to B_{i, f_{ij}}$ is flat and of finite presentation and $M_{ij, f_{ij}}$ is flat over $A_{f_{ij}}$ and of finite presentation over $B_{i, f_{ij}}$, and (b) $B_{i, f_ i}$ is flat and of finite presentation over $A_ f$ and $M_{i, f_ i}$ is flat and of finite presentation over $B_{i, f_ i}$. Set $f = (\prod f_ i) (\prod f_{ij})$. We claim that taking $U = D(f)$ works.

To prove our claim we may replace $A$ by $A_ f$, i.e., perform the base change by $U = \mathop{\mathrm{Spec}}(A_ f) \to S$. After this base change we see that each of $A \to B_ i$ is flat and of finite presentation and that $M_ i$, $M_{ij}$ are flat over $A$ and of finite presentation over $B_ i$. This already proves that $X \to S$ is quasi-compact, locally of finite presentation, flat, and that $\mathcal{F}$ is flat over $S$ and of finite presentation over $\mathcal{O}_ X$, see Lemma 29.21.2 and Properties, Lemma 28.16.2. Since $M_{ij}$ is of finite presentation over $B_ i$ we see that $X_ i \cap X_ j = X_ i \setminus \text{Supp}(M_{ij})$ is a quasi-compact open of $X_ i$, see Algebra, Lemma 10.40.8. Hence we see that $X \to S$ is quasi-separated by Schemes, Lemma 26.21.6. This proves the proposition. $\square$

It actually turns out that there is also a version of generic flatness over an arbitrary reduced base. Here it is.

Proposition 29.27.2 (Generic flatness, reduced case). Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. Assume

1. $S$ is reduced,

2. $f$ is of finite type, and

3. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.

Then there exists an open dense subscheme $U \subset S$ such that $X_ U \to U$ is flat and of finite presentation and such that $\mathcal{F}|_{X_ U}$ is flat over $U$ and of finite presentation over $\mathcal{O}_{X_ U}$.

Proof. For the impatient reader: This proof is a repeat of the proof of Proposition 29.27.1 using Algebra, Lemma 10.118.7 instead of Algebra, Lemma 10.118.3.

Since being flat and being of finite presentation is local on the base, see Lemmas 29.25.2 and 29.21.2, we may work affine locally on $S$. Thus we may assume that $S = \mathop{\mathrm{Spec}}(A)$, where $A$ is a reduced ring (see Properties, Lemma 28.3.2). As $f$ is of finite type, it is quasi-compact, so $X$ is quasi-compact. Hence we can find a finite affine open cover $X = \bigcup _{i = 1, \ldots , n} X_ i$. Write $X_ i = \mathop{\mathrm{Spec}}(B_ i)$. Then $B_ i$ is a finite type $A$-algebra, see Lemma 29.15.2. Moreover there are finite type $B_ i$-modules $M_ i$ such that $\mathcal{F}|_{X_ i}$ is the quasi-coherent sheaf associated to the $B_ i$-module $M_ i$, see Properties, Lemma 28.16.1. Next, for each pair of indices $i, j$ choose an ideal $I_{ij} \subset B_ i$ such that $X_ i \setminus X_ i \cap X_ j = V(I_{ij})$ inside $X_ i = \mathop{\mathrm{Spec}}(B_ i)$. Set $M_{ij} = B_ i/I_{ij}$ and think of it as a $B_ i$-module. Then $V(I_{ij}) = \text{Supp}(M_{ij})$ and $M_{ij}$ is a finite $B_ i$-module.

At this point we apply Algebra, Lemma 10.118.7 the pairs $(A \to B_ i, M_{ij})$ and to the pairs $(A \to B_ i, M_ i)$. Thus we obtain dense opens $U(A \to B_ i, M_{ij}) \subset S$ and dense opens $U(A \to B_ i, M_ i) \subset S$ with notation as in Algebra, Equation (10.118.3.2). Since a finite intersection of dense opens is dense open, we see that

$U = \bigcap \nolimits _{i, j} U(A \to B_ i, M_{ij}) \quad \cap \quad \bigcap \nolimits _ i U(A \to B_ i, M_ i)$

is open and dense in $S$. We claim that $U$ is the desired open.

Pick $u \in U$. By definition of the loci $U(A \to B_ i, M_{ij})$ and $U(A \to B, M_ i)$ there exist $f_{ij}, f_ i \in A$ such that (a) $u \in D(f_ i)$ and $u \in D(f_{ij})$, (b) $A_{f_{ij}} \to B_{i, f_{ij}}$ is flat and of finite presentation and $M_{ij, f_{ij}}$ is flat over $A_{f_{ij}}$ and of finite presentation over $B_{i, f_{ij}}$, and (c) $B_{i, f_ i}$ is flat and of finite presentation over $A_ f$ and $M_{i, f_ i}$ is flat and of finite presentation over $B_{i, f_ i}$. Set $f = (\prod f_ i) (\prod f_{ij})$. Now it suffices to prove that $X \to S$ is flat and of finite presentation over $D(f)$ and that $\mathcal{F}$ restricted to $X_{D(f)}$ is flat over $D(f)$ and of finite presentation over the structure sheaf of $X_{D(f)}$.

Hence we may replace $A$ by $A_ f$, i.e., perform the base change by $\mathop{\mathrm{Spec}}(A_ f) \to S$. After this base change we see that each of $A \to B_ i$ is flat and of finite presentation and that $M_ i$, $M_{ij}$ are flat over $A$ and of finite presentation over $B_ i$. This already proves that $X \to S$ is quasi-compact, locally of finite presentation, flat, and that $\mathcal{F}$ is flat over $S$ and of finite presentation over $\mathcal{O}_ X$, see Lemma 29.21.2 and Properties, Lemma 28.16.2. Since $M_{ij}$ is of finite presentation over $B_ i$ we see that $X_ i \cap X_ j = X_ i \setminus \text{Supp}(M_{ij})$ is a quasi-compact open of $X_ i$, see Algebra, Lemma 10.40.8. Hence we see that $X \to S$ is quasi-separated by Schemes, Lemma 26.21.6. This proves the proposition. $\square$

Remark 29.27.3. The results above are a first step towards more refined flattening techniques for morphisms of schemes. The article [GruRay] by Raynaud and Gruson contains many wonderful results in this direction.

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