The Stacks project

Lemma 38.37.12. Let $\mathcal{F}$ be a presheaf on one of the sites $(\mathit{Sch}/S)_ h$ constructed in Definition 38.34.12. Then $\mathcal{F}$ is a sheaf if and only if the following conditions are satisfied

  1. $\mathcal{F}$ is a sheaf for the Zariski topology,

  2. given a morphism $f : X \to Y$ of $(\mathit{Sch}/S)_ h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $\mathcal{F}(Y)$ is the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times _ Y X)$,

  3. $\mathcal{F}$ turns an almost blow up square as in Example 38.37.10 in the category $(\mathit{Sch}/S)_ h$ into a cartesian diagram of sets, and

  4. $\mathcal{F}$ turns an almost blow up square as in Example 38.37.11 in the category $(\mathit{Sch}/S)_ h$ into a cartesian diagram of sets.

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

\[ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } \]

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

\[ \vcenter { \xymatrix{ \mathcal{F}(E \cap b^{-1}(U_ i)) & \mathcal{F}(b^{-1}(U_ i)) \ar[l] \\ \mathcal{F}(Z \cap U_ i) \ar[u] & \mathcal{F}(U_ i) \ar[u] \ar[l] } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathcal{F}(E \cap b^{-1}(U_{ijk})) & \mathcal{F}(b^{-1}(U_{ijk})) \ar[l] \\ \mathcal{F}(Z \cap U_{ijk}) \ar[u] & \mathcal{F}(U_{ijk}) \ar[u] \ar[l] } } \]

are cartesian, then the same is true for

\[ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } \]

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

\[ \xymatrix{ E \ar[d] \ar[r] & X' \ar[d] \\ E_ k \ar[r] & X'_ k } \]

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

\[ \mathcal{F}(X'_ k) = \mathcal{F}(X') \times _{\mathcal{F}(E)} \mathcal{F}(E_ k) \]

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

\[ \mathcal{F}(X) \longrightarrow \mathcal{F}(X'_ k) \times _{\mathcal{F}(E_ k)} \mathcal{F}(Z) \]

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

\[ \xymatrix{ E_1 \ar[d] \ar[r] & Y \ar[d] \\ Z_1 \ar[r] & X } \]

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

\[ (I) \vcenter { \xymatrix{ Y \times _ X E \ar[r] \ar[d] & Y \times _ X X' \ar[d] \\ Y \times _ X Z \ar[r] & Y } } \quad \text{and}\quad (II) \vcenter { \xymatrix{ E \ar[r] \ar[d] & Z_1 \times _ X X' \ar[d] \\ Z \ar[r] & Z_1 } } \]

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

\[ \mathcal{F}(Y) = \mathcal{F}(Y \times _ X X') \times _{\mathcal{F}(Y \times _ X E)} \mathcal{F}(Y \times _ X Z) \]

and

\[ \mathcal{F}(Z_1) = \mathcal{F}(Z_1 \times _ X X') \times _{\mathcal{F}(E)} \mathcal{F}(Z) \]

By assumption (4) we have

\[ \mathcal{F}(X) = \mathcal{F}(Y) \times _{\mathcal{F}(E_1)} \mathcal{F}(Z_1) \]

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

\[ \xymatrix{ E_1 \times _ X E \ar[r] \ar[d] & E_1 \times _ X X' \ar[d] \\ E_1 \times _ X Z \ar[r] & E_1 } \]

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. $\square$


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