The Stacks project

Proof. The morphism $Y \times _ X X' \to Y$ is proper and of finite presentation by Morphisms, Lemmas 29.41.5 and 29.21.4. The morphism $Y \times _ X Z \to Y$ is a closed immersion (Morphisms, Lemma 29.2.4) of finite presentation. The inverse image of $Y \times _ X Z$ in $Y \times _ X X'$ is equal to the inverse image of $E$ in $Y \times _ X X'$ and hence is locally principal (Divisors, Lemma 31.13.11). Let $X'' \subset X'$, resp. $Y'' \subset Y \times _ X X'$ be the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{X'}$, resp. $\mathcal{O}_{Y \times _ Y X'}$ supported on $E$, resp. $Y \times _ X E$. Clearly, $Y'' \subset Y \times _ X X''$ is the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{Y \times _ Y X''}$ supported on $Y \times _ X (E \cap X'')$. Thus $Y''$ is the strict transform of $Y$ relative to the blowing up $X'' \to X$, see Divisors, Definition 31.33.1. Thus by Divisors, Lemma 31.33.2 we see that $Y''$ is the blow up of $Y \times _ X Z$ on $Y$. $\square$

Proof. Assume (1). Then the surjection $\mathcal{O}_{X'} \to \mathcal{O}_ W$ is an isomorphism over the open $X' \subset E$. Since the ideal sheaf of $X'' \subset X'$ is the sections of $\mathcal{O}_{X'}$ supported on $E$ (by our definition of almost blow up squares) we conclude (2) is true. If (2) is true, then (3) holds. If (3) holds, then (1) holds because $X'' \cap (X' \setminus E)$ is isomorphic to $X \setminus Z$ which in turn is isomorphic to $X' \setminus E$. $\square$

Proof. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent sheaf of ideals corresponding to $X''$. By Properties, Lemma 28.22.3 we can write $\mathcal{I}$ as the filtered colimit $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits \mathcal{I}_ i$ of its quasi-coherent submodules of finite type. Since these modules correspond $1$-to-$1$ to the closed subschemes $X'_ i$ the proof is complete. $\square$

Proof. We may write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ 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 closed immersion $Z_ i \to X_ i$ whose base change to $X$ is the closed immersion $Z \to X$. See Limits, Lemmas 32.10.1 and 32.8.5. Let $b_ i : X'_ i \to X_ i$ be the blowing up with center $Z_ i$. This produces a blow up square

where all the morphisms are finite type morphisms of Noetherian schemes and hence of finite presentation. Thus this is an almost blow up square. By Lemma 38.37.1 the base change of this diagram to $X$ produces the desired almost blow up square. $\square$

Proof. Denote $X'' \to X$ the blowing up of $Z$ in $X$. We view $X''$ as a closed subscheme of both $X'_1$ and $X'_2$. Write $X'' = \mathop{\mathrm{lim}}\nolimits X'_{1, i}$ as in Lemma 38.37.3. By Limits, Proposition 32.6.1 there exists an $i$ and a morphism $h : X'_{1, i} \to X'_2$ agreeing with the inclusions $X'' \subset X'_{1, i}$ and $X'' \subset X'_2$. By Limits, Lemma 32.4.20 the restriction of $h$ to $X'_{1, i'}$ is a closed immersion for some $i' \geq i$. This finishes the proof. $\square$

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. $\square$

Proof. Since $Z \amalg X' \to X$ is a surjective proper morphism of finite presentation we see that $\{ Z \amalg X' \to X\} $ is an h covering (Lemma 38.34.7). We have

Since $\mathcal{F}$ is a Zariski sheaf we see that $\mathcal{F}$ sends disjoint unions to products. Thus the sheaf condition for the covering $\{ Z \amalg X' \to X\} $ says that $\mathcal{F}(X) \to \mathcal{F}(Z) \times \mathcal{F}(X')$ is injective with image the set of pairs $(t, s')$ such that (a) $t|_ E = s'|_ E$ and (b) $s'$ is in the equalizer of the two maps $\mathcal{F}(X') \to \mathcal{F}(X' \times _ X X')$. Next, observe that the obvious morphism

is a surjective proper morphism of finite presentation as $b$ induces an isomorphism $X' \setminus E \to X \setminus Z$. We conclude that $\mathcal{F}(X' \times _ X X') \to \mathcal{F}(E \times _ Z E) \times \mathcal{F}(X')$ is injective. It follows that (a) $\Rightarrow $ (b) which means that the lemma is true. $\square$

Proof. First proof. Observe that $X \to X'$ is a proper surjective morphism of finite presentation and $X \times _{X'} X = X$. By the sheaf property for the h covering $\{ X \to X'\} $ (Lemma 38.34.7) we conclude.

Second proof (silly). The blow up of $X'$ in $X$ is the empty scheme. The reason is that the affine blowup algebra $A[\frac{I}{a}]$ (Algebra, Section 10.70) is zero if $a$ is a nilpotent element of $A$. Details omitted. Hence we get an almost blow up square of the form

Since $\mathcal{F}$ is a sheaf we have that $\mathcal{F}(\emptyset )$ is a singleton. Applying Lemma 38.37.7 we get the conclusion. $\square$

Proof. Assume $\mathcal{F}$ is a sheaf. Condition (1) holds because a Zariski covering is a h covering, see Lemma 38.34.6. Condition (2) holds because for $f$ as in (2) we have that $\{ X \to Y\} $ is an fppf covering (this is clear) and hence an h covering, see Lemma 38.34.6. Condition (3) holds by Lemma 38.37.7.

Conversely, assume $\mathcal{F}$ satisfies (1), (2), and (3). We will prove $\mathcal{F}$ is a sheaf by applying Lemma 38.34.17. Consider a surjective, finitely presented, proper morphism $f : X \to Y$ in $(\mathit{Sch}/S)_ h$ with $Y$ affine. It suffices to show that $\mathcal{F}(Y)$ is the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times _ Y X)$.

First, assume that $f : X \to Y$ is in addition a closed immersion (in other words, $f$ is a thickening). Then the blow up of $Y$ in $X$ is the empty scheme and this produces an almost blow up square consisting with $\emptyset , \emptyset , X, Y$ at the vertices (compare with the second proof of Lemma 38.37.8). Hence we see that condition (3) tells us that

is cartesian in the category of sets. Since $\mathcal{F}$ is a sheaf for the Zariski topology, we see that $\mathcal{F}(\emptyset )$ is a singleton. Hence we see that $\mathcal{F}(X) = \mathcal{F}(Y)$.

Interlude A: let $T \to T'$ be a morphism of $(\mathit{Sch}/S)_ h$ which is a thickening and of finite presentation. Then $\mathcal{F}(T') \to \mathcal{F}(T)$ is bijective. Namely, choose an affine open covering $T' = \bigcup T'_ i$ and let $T_ i = T \times _{T'} T'_ i$ be the corresponding affine opens of $T$. Then we have $\mathcal{F}(T'_ i) \to \mathcal{F}(T_ i)$ is bijective for all $i$ by the result of the previous paragraph. Using the Zariski sheaf property we see that $\mathcal{F}(T') \to \mathcal{F}(T)$ is injective. Repeating the argument we find that it is bijective. Minor details omitted.

Interlude B: consider an almost blow up square (38.37.0.1) in the category $(\mathit{Sch}/S)_ h$. Then we claim the diagram

is cartesian in the category of sets. This is a consequence of condition (3) as follows by choosing an affine open covering of $X$ and arguing as in Interlude A. We omit the details.

Next, let $f : X \to Y$ be a surjective, finitely presented, proper morphism in $(\mathit{Sch}/S)_ h$ with $Y$ affine. Choose a generic flatness stratification

as in More on Morphisms, Lemma 37.54.2 for $f : X \to Y$. We are going to use all the properties of the stratification without further mention. Set $X_0 = X \times _ Y Y_0$. By the Interlude B we have $\mathcal{F}(Y_0) = \mathcal{F}(Y)$, $\mathcal{F}(X_0) = \mathcal{F}(X)$, and $\mathcal{F}(X_0 \times _{Y_0} X_0) = \mathcal{F}(X \times _ Y X)$.

We are going to prove the result by induction on $t$. If $t = 1$ then $X_0 \to Y_0$ is surjective, proper, flat, and of finite presentation and we see that the result holds by property (2). For $t > 1$ we may replace $Y$ by $Y_0$ and $X$ by $X_0$ (see above) and assume $Y = Y_0$.

Consider the quasi-compact open subscheme $V = Y \setminus Y_1 = Y_0 \setminus Y_1$. Choose a diagram

as in Lemma 38.37.6 for $f : X \to Y \supset V$. Then $f' : X' \to Y'$ is flat and of finite presentation. Also $f'$ is proper (use Morphisms, Lemmas 29.41.4 and 29.41.7 to see this). Thus the image $W = f'(X') \subset Y'$ is an open (Morphisms, Lemma 29.25.10) and closed subscheme of $Y'$. Observe that $Y' \setminus E$ is contained in $W$. By Lemma 38.37.2 this means we may replace $Y'$ by $W$ in the above diagram. In other words, we may and do assume $f'$ is surjective. At this point we know that

are cartesian by Interlude B. Note that $Z \cap Y_1 \to Z$ is a thickening of finite presentation (as $Z$ is set theoretically contained in $Y_1$ as a closed subscheme of $Y$ disjoint from $V$). Thus we obtain a filtration

as above for the restriction $T = Z \times _ Y X \to Z$ of $f$ to $T$. Thus by induction hypothesis we find that $\mathcal{F}(Z) \to \mathcal{F}(T)$ is an injective map of sets whose image is the equalizer of the two maps $\mathcal{F}(T) \to \mathcal{F}(T \times _ Z T)$.

Let $s \in \mathcal{F}(X)$ be in the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times _ Y X)$. By the above we see that the restriction $s|_ T$ comes from a unique element $t \in \mathcal{F}(Z)$ and similarly that the restriction $s|_{X'}$ comes from a unique element $t' \in \mathcal{F}(Y')$. Chasing sections using the restriction maps for $\mathcal{F}$ corresponding to the arrows in the huge commutative diagram above the reader finds that $t$ and $t'$ restrict to the same element of $\mathcal{F}(E)$ because they restrict to the same element of $\mathcal{F}(D)$ and we have (2); here we use that $D \to E$ is surjective, flat, proper, and of finite presentation as the restriction of $X' \to Y'$. Thus by the first of the two cartesian squares displayed above we get a unique section $u \in \mathcal{F}(Y)$ restricting to $t$ and $t'$ on $Z$ and $Y'$. To see that $u$ restrict to $s$ on $X$ use the second diagram. $\square$

Proof. By Proposition 38.37.9 it suffices to show that given an almost blow up square (38.37.0.1) with $X$ affine in the category $(\mathit{Sch}/S)_ h$ the diagram

is cartesian in the category of sets. The rough idea of the proof is to dominate the morphism by other almost blowup squares to which we can apply assumptions (3) and (4) locally.

Suppose we have an almost blow up square (38.37.0.1) in the category $(\mathit{Sch}/S)_ h$, an open covering $X = \bigcup U_ i$, and open coverings $U_ i \cap U_ j = \bigcup U_{ijk}$ such that the diagrams

are cartesian, then the same is true for

This follows as $\mathcal{F}$ is a sheaf in the Zariski topology.

In particular, if we have a blow up square (38.37.0.1) such that $b : X' \to X$ is a closed immersion and $Z$ is a locally principal closed subscheme, then we see that $\mathcal{F}(X) = \mathcal{F}(X') \times _{\mathcal{F}(E)} \mathcal{F}(Z)$. Namely, affine locally on $X$ we obtain an almost blow up square as in (3).

Let $Z \subset X$, $E_ k \subset X'_ k \to X$, $E \subset X' \to X$, and $i_ k : X' \to X'_ k$ be as in the statement of Lemma 38.37.5. Then

is an almost blow up square of the kind discussed in the previous paragraph. Thus

for $k = 1, 2$ by the result of the previous paragraph. It follows that

is bijective for $k = 1$ if and only if it is bijective for $k = 2$. Thus given a closed immersion $Z \to X$ of finite presentation with $X$ quasi-compact and quasi-separated, whether or not $\mathcal{F}(X) = \mathcal{F}(X') \times _{\mathcal{F}(E)} \mathcal{F}(Z)$ is independent of the choice of the almost blow up square (38.37.0.1) one chooses. (Moreover, by Lemma 38.37.4 there does indeed exist an almost blow up square for $Z \subset X$.)

Finally, consider an affine object $X$ of $(\mathit{Sch}/S)_ h$ and a closed immersion $Z \to X$ of finite presentation. We will prove the desired property for the pair $(X, Z)$ by induction on the number of generators $r$ for the ideal defining $Z$ in $X$. If the number of generators is $\leq 2$, then we can choose our almost blow up square as in Example 38.37.11 and we conclude by assumption (4).

Induction step. Suppose $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/(f_1, \ldots , f_ r))$ with $r > 2$. Choose a blow up square (38.37.0.1) for the pair $(X, Z)$. Set $Z_1 = \mathop{\mathrm{Spec}}(A/(f_1, f_2))$ and let

be the almost blow up square constructed in Example 38.37.11. By Lemma 38.37.1 the base changes

are almost blow up squares. The ideal of $Z$ in $Z_1$ is generated by $r - 2$ elements. The ideal of $Y \times _ X Z$ is generated by the pullbacks of $f_1, \ldots , f_ r$ to $Y$. Locally on $Y$ the ideal generated by $f_1, f_2$ can be generated by one element, thus $Y \times _ X Z$ is affine locally on $Y$ cut out by at most $r - 1$ elements. By induction hypotheses and the discussion above

and

By assumption (4) we have

Now suppose we have a pair $(s', t)$ with $s' \in \mathcal{F}(X')$ and $t \in \mathcal{F}(Z)$ with same restriction in $\mathcal{F}(E)$. Then $(s'|{Z_1 \times _ X X'}, t)$ are the image of a unique element $t_1 \in \mathcal{F}(Z_1)$. Similarly, $(s'|_{Y \times _ X X'}, t|_{Y \times _ X Z})$ are the image of a unique element $s_ Y \in \mathcal{F}(Y)$. We claim that $s_ Y$ and $t_1$ restrict to the same element of $\mathcal{F}(E_1)$. This is true because the almost blow up square

is the base change of almost blow up square (I) via $E_1 \to Y$ and the base change of almost blow up square (II) via $E_1 \to Z_1$ and because the pairs of sections used to construct $s_ Y$ and $t_1$ match. Thus by the third fibre product equality we see that there is a unique $s \in \mathcal{F}(X)$ mapping to $s_ Y$ in $\mathcal{F}(Y)$ and to $t_1$ in $\mathcal{F}(Z)$. We omit the verification that $s$ maps to $s'$ in $\mathcal{F}(X')$ and to $t$ in $\mathcal{F}(Z)$; hint: use uniqueness of $s$ just constructed and work affine locally. $\square$

Proof. This lemma is a formal consequence of Lemma 38.37.12 and our defnition of stacks in groupoids. For example, assume (1), (2), (3). To show that $\mathcal{S}$ is a stack, we have to prove descent for morphisms and objects, see Stacks, Definition 8.5.1.

If $x, y$ are objects of $\mathcal{S}$ over an object $U$ of $(\mathit{Sch}/S)_ h$, then our assumptions imply $\mathit{Isom}(x, y)$ is a presheaf on $(\mathit{Sch}/U)_ h$ which satisfies (1), (2), (3), and (4) of Lemma 38.37.12 and therefore is a sheaf. Some details omitted.

Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $(\mathit{Sch}/S)_ h$. Let $(x_ i, \varphi _{ij})$ be a descent datum in $\mathcal{S}$ relative to the family $\{ U_ i \to U\} _{i \in I}$, see Stacks, Definition 8.3.1. Consider the rule $F$ which to $V/U$ in $(\mathit{Sch}/U)_ h$ associates the set of pairs $(y, \psi _ i)$ where $y$ is an object of $\mathcal{S}_ V$ and $\psi _ i : y|_{U_ i \times _ U V} \to x_ i|_{U_ i \times _ U V}$ is a morphism of $\mathcal{S}$ over $U_ i \times _ U V$ such that

up to isomorphism. Since we already have descent for morphisms, it is clear that $F(V/U)$ is either empty or a singleton set. On the other hand, we have $F(U_{i_0}/U)$ is nonempty because it contains $(x_{i_0}, \varphi _{i_0i})$. Since our goal is to prove that $F(U/U)$ is nonempty, it suffices to show that $F$ is a sheaf on $(\mathit{Sch}/U)_ h$. To do this we may use the criterion of Lemma 38.37.12. However, our assumptions (1), (2), (3) imply (by drawing some commutative diagrams which we omit), that properties (1), (2), (3), and (4) of Lemma 38.37.12 hold for $F$.

We omit the verification that if $\mathcal{S}$ is a stack in groupoids, then (1), (2), and (3) are satisfied. $\square$