The Stacks project

Proof. To obtain $\alpha $ set $r = \dim _{\kappa (\mathfrak p)} N \otimes _ S \kappa (\mathfrak p)$ and pick $x_1, \ldots , x_ r \in N$ which form a basis of $N \otimes _ S \kappa (\mathfrak p)$. Define $\alpha (s_1, \ldots , s_ r) = \sum s_ i x_ i$. This proves the existence.

Fix a choice of $\alpha $. We may apply Lemma 38.10.1 to the map $\alpha _{\mathfrak r} : S_{\mathfrak r}^{\oplus r} \to N_{\mathfrak r}$. Hence we see that (1), (3), (4), (5), and (6) are all equivalent. Since it is also clear that (2) implies (3) we see that all we have to do is show that (1) implies (2).

Assume (1). By openness of flatness, see Algebra, Theorem 10.129.4, the set

is open in $\mathop{\mathrm{Spec}}(S)$. It contains $\mathfrak q$ by assumption and hence $\mathfrak p$. Because $S^{\oplus r}$ and $N$ are finitely presented $S$-modules the set

is open in $\mathop{\mathrm{Spec}}(S)$, see Algebra, Lemma 10.79.2. It contains $\mathfrak p$ by (5). As $R \to S$ is finitely presented and flat the map $\Phi : \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is open, see Algebra, Proposition 10.41.8. For any prime $\mathfrak r' \in \Phi (U_1 \cap U_2)$ we see that there exists a prime $\mathfrak q'$ lying over $\mathfrak r'$ such that $N_{\mathfrak q'}$ is flat and such that $\alpha _{\mathfrak q'}$ is an isomorphism, which implies that $\alpha \otimes \kappa (\mathfrak p')$ is an isomorphism where $\mathfrak p' = \mathfrak r' S$. Thus $\alpha _{\mathfrak r'}$ is $R_{\mathfrak r'}$-universally injective by the implication (1) $\Rightarrow $ (3). Hence if we pick $f \in R$, $f \not\in \mathfrak r$ such that $D(f) \subset \Phi (U_1 \cap U_2)$ then we conclude that $\alpha _ f$ is $R_ f$-universally injective, see Algebra, Lemma 10.82.12. The same reasoning also shows that for any $\mathfrak q' \in U_1 \cap \Phi ^{-1}(\Phi (U_1 \cap U_2))$ the module $\mathop{\mathrm{Coker}}(\alpha )_{\mathfrak q'}$ is $R$-flat. Note that $\mathfrak q \in U_1 \cap \Phi ^{-1}(\Phi (U_1 \cap U_2))$. Hence we can find a $g \in S$, $g \not\in \mathfrak q$ such that $D(g) \subset U_1 \cap \Phi ^{-1}(\Phi (U_1 \cap U_2))$ and we win. $\square$

Proof. Let $(A_ i, B_ i, M_ i, \alpha _ i)_{i = 1, \ldots , n}$ be the given complete dévissage. We prove the lemma by induction on $n$. Note that the assertions of the lemma are entirely about the structure of $N$ as an $R$-module. Hence we may replace $N$ by $M_1$, and we may think of $M_1$ as a $B_1$-module. See Remark 38.6.3 in order to see why $M_1$ is of finite presentation as a $B_1$-module. By Lemma 38.12.1 we may, after replacing $R$ by $R_ f$ for some $f \in R$, $f \not\in \mathfrak r$, assume the map $\alpha _1 : B_1^{\oplus r_1} \to M_1$ is $R$-universally injective. Since $M_1$ and $B_1^{\oplus r_1}$ are $R$-flat and finitely presented as $B_1$-modules we see that $\mathop{\mathrm{Coker}}(\alpha _1)$ is $R$-flat (Algebra, Lemma 10.82.7) and finitely presented as a $B_1$-module. Note that $(A_ i, B_ i, M_ i, \alpha _ i)_{i = 2, \ldots , n}$ is a complete dévissage of $\mathop{\mathrm{Coker}}(\alpha _1)$. Hence the induction hypothesis implies that, after replacing $R$ by $R_ f$ for some $f \in R$, $f \not\in \mathfrak r$, we may assume that $\mathop{\mathrm{Coker}}(\alpha _1)$ has a decomposition as in the lemma and is projective. In particular $M_1 = B_1^{\oplus r_1} \oplus \mathop{\mathrm{Coker}}(\alpha _1)$. This proves the statement regarding the decomposition. The statement on projectivity follows as $B_1$ is projective as an $R$-module by Lemma 38.9.3. $\square$

Proof. By openness of flatness, see More on Morphisms, Theorem 37.15.1 we may replace $X$ by an open neighbourhood of $x$ and assume that $\mathcal{F}$ is flat over $S$. Next, we apply Proposition 38.5.7 to find a diagram as in the statement of the proposition such that $g^*\mathcal{F}/X'/S'$ has a complete dévissage over $s'$. (In particular $S'$ and $X'$ are affine.) By Morphisms, Lemma 29.25.13 we see that $g^*\mathcal{F}$ is flat over $S$ and by Lemma 38.2.3 we see that it is flat over $S'$. Via Remark 38.6.5 we deduce that

has a complete dévissage over the prime of $\Gamma (S', \mathcal{O}_{S'})$ corresponding to $s'$. Thus Lemma 38.12.2 implies that the result of the proposition holds after replacing $S'$ by a standard open neighbourhood of $s'$. $\square$

Proof. For every point $x \in X_ s$ we can use Proposition 38.12.4 to find a commutative diagram

whose horizontal arrows are elementary étale neighbourhoods such that $Y_ x$, $S_ x$ are affine and such that $\Gamma (Y_ x, g_ x^*\mathcal{F})$ is a projective $\Gamma (S_ x, \mathcal{O}_{S_ x})$-module. In particular $g_ x(Y_ x) \cap X_ s$ is an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $g_{x_ i}(Y_{x_ i}) \cap X_ s$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. We may also assume that $S'$ is affine. Set $X' = \coprod Y_{x_ i} \times _{S_{x_ i}} S'$ and endow it with the obvious morphism $g : X' \to X$. By construction $g(X')$ contains $X_ s$ and

This is a projective $\Gamma (S', \mathcal{O}_{S'})$-module, see Algebra, Lemma 10.94.1. $\square$

Proof. This is a special case of Proposition 38.12.4. $\square$

Proof. This is a special case of Lemma 38.12.5. $\square$

Proof. To prove the lemma we may replace $(S, s)$ by any elementary étale neighbourhood, and we may also replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$. Hence by Proposition 38.10.3 we may assume that $\mathcal{F}$ is finitely presented and flat over $S$ in a neighbourhood of $x$. In this case the result follows from Proposition 38.12.4 because Algebra, Theorem 10.85.4 assures us that projective $=$ free over a local ring. $\square$

Proof. (The only difference with Lemma 38.12.8 is that we do not assume $f$ is of finite presentation.) The problem is local on $X$ and $S$. Hence we may assume $X$ and $S$ are affine, say $X = \mathop{\mathrm{Spec}}(B)$ and $S = \mathop{\mathrm{Spec}}(A)$. Since $B$ is a finite type $A$-algebra we can find a surjection $A[x_1, \ldots , x_ n] \to B$. In other words, we can choose a closed immersion $i : X \to \mathbf{A}^ n_ S$. Set $t = i(x)$ and $\mathcal{G} = i_*\mathcal{F}$. Note that $\mathcal{G}_ t \cong \mathcal{F}_ x$ are $\mathcal{O}_{S, s}$-modules. Hence $\mathcal{G}$ is flat over $S$ at $t$. We apply Lemma 38.12.8 to the morphism $\mathbf{A}^ n_ S \to S$, the point $t$, and the sheaf $\mathcal{G}$. Thus we can find an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of pointed schemes

such that $Y \to \mathbf{A}^ n_{\mathcal{O}_{S', s'}}$ is étale, $\kappa (t) = \kappa (y)$, the scheme $Y$ is affine, and such that $\Gamma (Y, h^*\mathcal{G})$ is a projective $\mathcal{O}_{S', s'}$-module. Then a solution to the original problem is given by the closed subscheme $X' = Y \times _{\mathbf{A}^ n_ S} X$ of $Y$. $\square$

Proof. For every point $x \in X_ s$ we can use Lemma 38.12.8 to find an elementary étale neighbourhood $(S_ x , s_ x) \to (S, s)$ and a commutative diagram

such that $Y_ x \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x})$ is étale, $\kappa (x) = \kappa (y_ x)$, the scheme $Y_ x$ is affine of finite presentation over $\mathcal{O}_{S_ x, s_ x}$, the sheaf $g_ x^*\mathcal{F}$ is of finite presentation over $\mathcal{O}_{Y_ x}$, and such that $\Gamma (Y_ x, g_ x^*\mathcal{F})$ is a free $\mathcal{O}_{S_ x, s_ x}$-module. In particular $g_ x((Y_ x)_{s_ x})$ is an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $g_{x_ i}((Y_{x_ i})_{s_{x_ i}})$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. Set

and endow it with the obvious morphism $g : X' \to X$. By construction $X_ s = g(X'_{s'})$ and

This is a free $\mathcal{O}_{S', s'}$-module as a direct sum of base changes of free modules. Some minor details omitted. $\square$

Proof. (The only difference with Lemma 38.12.10 is that we do not assume $f$ is of finite presentation.) For every point $x \in X_ s$ we can use Lemma 38.12.9 to find an elementary étale neighbourhood $(S_ x , s_ x) \to (S, s)$ and a commutative diagram

such that $Y_ x \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x})$ is étale, $\kappa (x) = \kappa (y_ x)$, the scheme $Y_ x$ is affine, and such that $\Gamma (Y_ x, g_ x^*\mathcal{F})$ is a free $\mathcal{O}_{S_ x, s_ x}$-module. In particular $g_ x((Y_ x)_{s_ x})$ is an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $g_{x_ i}((Y_{x_ i})_{s_{x_ i}})$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. Set

and endow it with the obvious morphism $g : X' \to X$. By construction $X_ s = g(X'_{s'})$ and

This is a free $\mathcal{O}_{S', s'}$-module as a direct sum of base changes of free modules. $\square$