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Tag: 0367

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This tag has label descent-lemma-finite-type-local-source-fppf-algebra and it points to

• Lemma 10.2 on page 32 of Section 10: Descent of finiteness properties of morphisms, in Chapter 31: Descent
• Lemma 31.10.2 on page 1913 of the book version

The corresponding content:

code

Lemma 31.10.2. Let $R \to A \to B$ be ring maps. Assume $R \to B$ is of finite type and $A \to B$ faithfully flat and of finite presentation. Then $R \to A$ is of finite type.

Proof. By Algebra, Lemma 9.154.2 there exists a commtutative diagram $$ \xymatrix{ R \ar[r] \ar@{=}[d] & A_0 \ar[d] \ar[r] & B_0 \ar[d] \\ R \ar[r] & A \ar[r] & B } $$ with $R \to A_0$ of finite presentation, $A_0 \to B_0$ faithfully flat of finite presentation and $B = A \otimes_{A_0} B_0$. Since $R \to B$ is of finite type by assumption, we may add some elements to $A_0$ and assume that the map $B_0 \to B$ is surjective! In this case, since $A_0 \to B_0$ is faithfully flat, we see that as $$ (A_0 \to A) \otimes_{A_0} B_0 \cong (B_0 \to B) $$ is surjective, also $A_0 \to A$ is surjective. Hence we win. $\square$

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\begin{lemma}
\label{lemma-finite-type-local-source-fppf-algebra}
Let $R \to A \to B$ be ring maps.
Assume $R \to B$ is of finite type and
$A \to B$ faithfully flat and of finite presentation.
Then $R \to A$ is of finite type.
\end{lemma}

\begin{proof}
By
Algebra, Lemma \ref{algebra-lemma-descend-faithfully-flat-finite-presentation}
there exists a commtutative diagram
$$
\xymatrix{
R \ar[r] \ar@{=}[d] &
A_0 \ar[d] \ar[r] &
B_0 \ar[d] \\
R \ar[r] & A \ar[r] & B
}
$$
with $R \to A_0$ of finite presentation,
$A_0 \to B_0$ faithfully flat of finite presentation
and $B = A \otimes_{A_0} B_0$. Since $R \to B$ is of finite
type by assumption, we may add some elements to $A_0$ and assume
that the map $B_0 \to B$ is surjective!
In this case, since $A_0 \to B_0$ is faithfully flat, we see
that as
$$
(A_0 \to A) \otimes_{A_0} B_0 \cong (B_0 \to B)
$$
is surjective, also $A_0 \to A$ is surjective. Hence we win.
\end{proof}
    

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