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The Stacks project

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


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