The Stacks project

Lemma 16.4.2. In Situation 16.4.1 Néron's blowup is functorial in the following sense

  1. if $a \in A$, $a \not\in \mathfrak p$, then Néron's blowup of $A_ a$ is $A'_ a$, and

  2. if $B \to A$ is a surjection of flat finite type $R$-algebras with kernel $I$, then $A'$ is the quotient of $B'/IB'$ by its $\pi $-power torsion.

Proof. Both (1) and (2) are special cases of Algebra, Lemma 10.70.3. In fact, whenever we have $A_1 \to A_2 \to \Lambda $ such that $\mathfrak p_1 A_2 = \mathfrak p_2$, we have that $A_2'$ is the quotient of $A_1' \otimes _{A_1} A_2$ by its $\pi $-power torsion. $\square$

Comments (3)

Comment #3787 by Dario Weißmann on

Typo in the statement: if ...with kernel I then -> if ..., then

In the proof we should also assume that is flat. I don't think it holds quite that general. And maybe mention that one has to use the equational criterion for flatness?

Comment #3788 by Dario Weißmann on

Ok, so I didn't look at 0BIP (properly). All good, please ignore the second part of the above comment

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