The Stacks project

Lemma 38.37.6. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme of finite presentation over $Y$. Let $V \subset Y$ be a quasi-compact open such that $X_ V \to V$ is flat. Then there exist a commutative diagram

\[ \xymatrix{ E \ar[ddd] \ar[rd] & & & D \ar[lll] \ar[ddd] \ar[ld] \\ & Y' \ar[d] & X' \ar[l] \ar[d] \\ & Y & X \ar[l] \\ Z \ar[ru] & & & T \ar[lll] \ar[lu] } \]

whose right and left hand squares are almost blow up squares, whose lower and top squares are cartesian, such that $Z \cap V = \emptyset $, and such that $X' \to Y'$ is flat (and of finite presentation).

Proof. If $Y$ is a Noetherian scheme, then this lemma follows immediately from Lemma 38.31.1 because in this case blow up squares are almost blow up squares (we also use that strict transforms are blow ups). The general case is reduced to the Noetherian case by absolute Noetherian approximation.

We may write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ as a directed limit of an inverse system of Noetherian schemes with affine transition morphisms, see Limits, Proposition 32.5.4. We can find an index $i$ and a morphism $X_ i \to Y_ i$ of finite presentation whose base change to $Y$ is $X \to Y$. See Limits, Lemmas 32.10.1. After increasing $i$ we may assume $V$ is the inverse image of an open subscheme $V_ i \subset Y_ i$, see Limits, Lemma 32.4.11. Finally, after increasing $i$ we may assume that $X_{i, V_ i} \to V_ i$ is flat, see Limits, Lemma 32.8.7. By the Noetherian case, we may construct a diagram as in the lemma for $X_ i \to Y_ i \supset V_ i$. The base change of this diagram by $Y \to Y_ i$ provides the solution. Use that base change preserves properties of morphisms, see Morphisms, Lemmas 29.41.5, 29.21.4, 29.2.4, and 29.25.8 and that base change of an almost blow up square is an almost blow up square, see Lemma 38.37.1. $\square$

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