Lemma 35.14.8. Let X \to Y \to Z be morphism of schemes. Let P be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that X \to Z has P and that \{ X \to Y\} can be refined by an fppf covering of Y. Then Y \to Z is P.
Proof. Let \mathop{\mathrm{Spec}}(C) \subset Z be an affine open and let \mathop{\mathrm{Spec}}(B) \subset Y be an affine open which maps into \mathop{\mathrm{Spec}}(C). The assumption on X \to Y implies we can find a standard affine fppf covering \{ \mathop{\mathrm{Spec}}(B_ j) \to \mathop{\mathrm{Spec}}(B)\} and lifts x_ j : \mathop{\mathrm{Spec}}(B_ j) \to X. Since \mathop{\mathrm{Spec}}(B_ j) is quasi-compact we can find finitely many affine opens \mathop{\mathrm{Spec}}(A_ i) \subset X lying over \mathop{\mathrm{Spec}}(B) such that the image of each x_ j is contained in the union \bigcup \mathop{\mathrm{Spec}}(A_ i). Hence after replacing each \mathop{\mathrm{Spec}}(B_ j) by a standard affine Zariski coverings of itself we may assume we have a standard affine fppf covering \{ \mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(B)\} such that each \mathop{\mathrm{Spec}}(B_ i) \to Y factors through an affine open \mathop{\mathrm{Spec}}(A_ i) \subset X lying over \mathop{\mathrm{Spec}}(B). In other words, we have ring maps C \to B \to A_ i \to B_ i for each i. Note that we can also consider
and that the ring map B \to \prod B_ i is faithfully flat and of finite presentation.
The case P = flat. In this case we know that C \to A is flat and we have to prove that C \to B is flat. Suppose that N \to N' \to N'' is an exact sequence of C-modules. We want to show that N \otimes _ C B \to N' \otimes _ C B \to N'' \otimes _ C B is exact. Let H be its cohomology and let H' be the cohomology of N \otimes _ C B' \to N' \otimes _ C B' \to N'' \otimes _ C B'. As B \to B' is flat we know that H' = H \otimes _ B B'. On the other hand N \otimes _ C A \to N' \otimes _ C A \to N'' \otimes _ C A is exact hence has zero cohomology. Hence the map H \to H' is zero (as it factors through the zero module). Thus H' = 0. As B \to B' is faithfully flat we conclude that H = 0 as desired.
The case P = locally\ finite\ type. In this case we know that C \to A is of finite type and we have to prove that C \to B is of finite type. Because B \to B' is of finite presentation (hence of finite type) we see that A \to B' is of finite type, see Algebra, Lemma 10.6.2. Therefore C \to B' is of finite type and we conclude by Lemma 35.14.2.
The case P = locally\ finite\ presentation. In this case we know that C \to A is of finite presentation and we have to prove that C \to B is of finite presentation. Because B \to B' is of finite presentation and B \to A of finite type we see that A \to B' is of finite presentation, see Algebra, Lemma 10.6.2. Therefore C \to B' is of finite presentation and we conclude by Lemma 35.14.1. \square
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