The Stacks project

Lemma 61.11.6. Let $A \to B$ be a finite and finitely presented ring map. If $A$ is w-contractible, so is $B$.

Proof. We will use the criterion of Lemma 61.11.2. Set $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$ and denote $f : Y \to X$ the induced morphism. As $f : Y \to X$ is a finite morphism, we see that the set of closed points $Y_0$ of $Y$ is the inverse image of the set of closed points $X_0$ of $X$. Let $y \in Y$ with image $x \in X$. Then $x$ specializes to a unique closed point $x_0 \in X$. Say $f^{-1}(\{ x_0\} ) = \{ y_1, \ldots , y_ n\} $ with $y_ i$ closed in $Y$. Since $R = \mathcal{O}_{X, x_0}$ is strictly henselian and since $f$ is finite, we see that $Y \times _{f, X} \mathop{\mathrm{Spec}}(R)$ is equal to $\coprod _{i = 1, \ldots , n} \mathop{\mathrm{Spec}}(R_ i)$ where each $R_ i$ is a local ring finite over $R$ whose maximal ideal corresponds to $y_ i$, see Algebra, Lemma 10.153.3 part (10). Then $y$ is a point of exactly one of these $\mathop{\mathrm{Spec}}(R_ i)$ and we see that $y$ specializes to exactly one of the $y_ i$. In other words, every point of $Y$ specializes to a unique point of $Y_0$. Thus $Y$ is w-local. For every $y \in Y_0$ with image $x \in X_0$ we see that $\mathcal{O}_{Y, y}$ is strictly henselian by Algebra, Lemma 10.153.4 applied to $\mathcal{O}_{X, x} \to B \otimes _ A \mathcal{O}_{X, x}$. It remains to show that $Y_0$ is extremally disconnected. To do this we look at $X_0 \times _ X Y \to X_0$ where $X_0 \subset X$ is the reduced induced scheme structure. Note that the underlying topological space of $X_0 \times _ X Y$ agrees with $Y_0$. Now the desired result follows from Lemma 61.11.5. $\square$

Comments (2)

Comment #5952 by Owen on

'Moreover, every point of specializes to a unique point of as (a) this is true for and (b) the map is separated.' It should be , not , but I still don't follow. It's possible for a specialization in to lift to more than one specialization along a separated map .

Comment #6136 by on

Thanks for catching this! I have fixed the proof in this commit. Thanks again!

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