Lemma 15.31.2 (0BIQ)—The Stacks project

Lemma 15.31.2. Let $a, a_2, \ldots , a_ r$ be an $H_1$-regular sequence in a ring $R$ (for example a Koszul regular sequence or a regular sequence, see Lemmas 15.30.2 and 15.30.3). With $I = (a, a_2, \ldots , a_ r)$ the blowup algebra $R' = R[\frac{I}{a}]$ is isomorphic to $R'' = R[y_2, \ldots , y_ r]/(a y_ i - a_ i)$.

Proof. By Algebra, Lemma 10.70.6 it suffices to show that $R''$ is $a$-torsion free.

We claim $a, ay_2 - a_2, \ldots , ay_ n - a_ r$ is a $H_1$-regular sequence in $R[y_2, \ldots , y_ r]$. Namely, the map

\[ (a, ay_2 - a_2, \ldots , ay_ n - a_ r) : R[y_2, \ldots , y_ r]^{\oplus r} \longrightarrow R[y_2, \ldots , y_ r] \]

used to define the Koszul complex on $a, ay_2 - a_2, \ldots , ay_ n - a_ r$ is isomorphic to the map

\[ (a, a_2, \ldots , a_ r) : R[y_2, \ldots , y_ r]^{\oplus r} \longrightarrow R[y_2, \ldots , y_ r] \]

used to the define the Koszul complex on $a, a_2, \ldots , a_ r$ via the isomorphism

\[ R[y_2, \ldots , y_ r]^{\oplus r} \longrightarrow R[y_2, \ldots , y_ r]^{\oplus r} \]

sending $(b_1, \ldots , b_ r)$ to $(b_1 - b_2y_2 \ldots - b_ ry_ r, -b_2, \ldots , - b_ r)$. By Lemma 15.28.3 these Koszul complexes are isomorphic. By Lemma 15.30.5 applied to the flat ring map $R \to R[y_2, \ldots , y_ r]$ we conclude our claim is true. By Lemma 15.28.8 we see that the Koszul complex $K$ on $a, ay_2 - a_2, \ldots , ay_ n - a_ r$ is the cone on $a : L \to L$ where $L$ is the Koszul complex on $ay_2 - a_2, \ldots , ay_ n - a_ r$. Since $H_1(K) = 0$ by the claim, we conclude that $a : H_0(L) \to H_0(L)$ is injective, in other words that $R'' = R[y_2, \ldots , y_ r]/(a y_ i - a_ i)$ has no nonzero $a$-torsion elements as desired. $\square$

Comments (2)

Comment #3792 by Kestutis Cesnavicius on November 18, 2018 at 06:51

It seems to me that the Noetherianness assumption is not needed in this lemma: one may reduce to the universal case $R = \mathbb{Z}[T_1, \dotsc, T_r]$ , $a = T_1, a_2 = T_2, \dotsc, a_n = T_n$ by using SGA 6, VII, 1.8 i). A direct proof of the general case is also given by Gabber--Ramero in their "Foundations for almost ring theory": Lemma 7.8.16 (ii) in the v13 of their arXiv manuscript https://arxiv.org/abs/math/0409584.

Comment #3914 by Johan on January 22, 2019 at 18:25

OK, I generalized this to $H_1$ -regular sequences and non-Noetherian rings... See changes here. Thanks for pointing this out.

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