The Stacks project

53.14 Pushouts

Let $k$ be a field. Consider a solid diagram

\[ \xymatrix{ Z' \ar[d] \ar[r]_{i'} & X' \ar@{..>}[d]^ a \\ Z \ar@{..>}[r]^ i & X } \]

of schemes over $k$ satisfying

  1. $X'$ is separated of finite type over $k$ of dimension $\leq 1$,

  2. $i : Z' \to X'$ is a closed immersion,

  3. $Z'$ and $Z$ are finite over $\mathop{\mathrm{Spec}}(k)$, and

  4. $Z' \to Z$ is surjective.

In this situation every finite set of points of $X'$ are contained in an affine open, see Varieties, Proposition 33.42.7. Thus the assumptions of More on Morphisms, Proposition 37.67.3 are satisfied and we obtain the following

  1. the pushout $X = Z \amalg _{Z'} X'$ exists in the category of schemes,

  2. $i : Z \to X$ is a closed immersion,

  3. $a : X' \to X$ is integral surjective,

  4. $X \to \mathop{\mathrm{Spec}}(k)$ is separated by More on Morphisms, Lemma 37.67.4

  5. $X \to \mathop{\mathrm{Spec}}(k)$ is of finite type by More on Morphisms, Lemmas 37.67.5,

  6. thus $a : X' \to X$ is finite by Morphisms, Lemmas 29.44.4 and 29.15.8,

  7. if $X' \to \mathop{\mathrm{Spec}}(k)$ is proper, then $X \to \mathop{\mathrm{Spec}}(k)$ is proper by Morphisms, Lemma 29.41.9.

The following lemma can be generalized significantly.

Lemma 53.14.1. In the situation above, let $Z = \mathop{\mathrm{Spec}}(k')$ where $k'$ is a field and $Z' = \mathop{\mathrm{Spec}}(k'_1 \times \ldots \times k'_ n)$ with $k'_ i/k'$ finite extensions of fields. Let $x \in X$ be the image of $Z \to X$ and $x'_ i \in X'$ the image of $\mathop{\mathrm{Spec}}(k'_ i) \to X'$. Then we have a fibre product diagram

\[ \xymatrix{ \prod \nolimits _{i = 1, \ldots , n} k'_ i & \prod \nolimits _{i = 1, \ldots , n} \mathcal{O}_{X', x'_ i}^\wedge \ar[l] \\ k' \ar[u] & \mathcal{O}_{X, x}^\wedge \ar[u] \ar[l] } \]

where the horizontal arrows are given by the maps to the residue fields.

Proof. Choose an affine open neighbourhood $\mathop{\mathrm{Spec}}(A)$ of $x$ in $X$. Let $\mathop{\mathrm{Spec}}(A') \subset X'$ be the inverse image. By construction we have a fibre product diagram

\[ \xymatrix{ \prod \nolimits _{i = 1, \ldots , n} k'_ i & A' \ar[l] \\ k' \ar[u] & A \ar[u] \ar[l] } \]

Since everything is finite over $A$ we see that the diagram remains a fibre product diagram after completion with respect to the maximal ideal $\mathfrak m \subset A$ corresponding to $x$ (Algebra, Lemma 10.97.2). Finally, apply Algebra, Lemma 10.97.8 to identify the completion of $A'$. $\square$

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