The Stacks project

Lemma 52.4.3. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi _ j : I^ c \to A$, $j = 1, \ldots , r$ be as in Lemma 52.4.1. Let $A \to B$ be a ring map with $B$ Noetherian and let $N$ be a finite $B$-module. Then, after possibly increasing $c$ and adjusting $\varphi _ j$ accordingly, there is a unique unique graded $B$-module map

\[ \Phi _ N : \bigoplus \nolimits _{n \geq 0} I^{nc}N \to N[T_1, \ldots , T_ r] \]

with $\Phi _ N(g x) = \Phi (g) x$ for $g \in I^{nc}$ and $x \in N$ where $\Phi $ is as in Lemma 52.4.2. The composition of $\Phi _ N$ with the map $N[T_1, \ldots , T_ r] \to \bigoplus _{n \geq 0} I^ nN$, $T_ j \mapsto f_ j$ is the inclusion map $\bigoplus _{n \geq 0} I^{nc}N \to \bigoplus _{n \geq 0} I^ nN$.

Proof. The uniqueness is clear from the formula and the uniqueness of $\Phi $ in Lemma 52.4.2. Consider the Noetherian $A$-algebra $B' = B \oplus N$ where $N$ is an ideal of square zero. To show the existence of $\Phi _ N$ it is enough (via Lemma 52.4.1) to show that $\varphi _ j$ extends to a map $\varphi '_ j : I^ cB' \to B'$ after possibly increasing $c$ to some $c'$ (and replacing $\varphi _ j$ by the composition of the inclusion $I^{c'} \to I^ c$ with $\varphi _ j$). Recall that $\varphi _ j$ corresponds to a section

\[ h_ j \in \Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \mathcal{O}_{\mathop{\mathrm{Spec}}(A)}) \]

see Cohomology of Schemes, Lemma 30.10.5. (This is in fact how we chose our $\varphi _ j$ in the proof of Lemma 52.4.1.) Let us use the same lemma to represent the pullback

\[ h'_ j \in \Gamma (\mathop{\mathrm{Spec}}(B') \setminus V(IB'), \mathcal{O}_{\mathop{\mathrm{Spec}}(B')}) \]

of $h_ j$ by a $B'$-linear map $\varphi '_ j : I^{c'}B' \to B'$ for some $c' \geq c$. The agreement with $\varphi _ j$ will hold for $c'$ sufficiently large by a further application of the lemma: namely we can test agreement on a finite list of generators of $I^{c'}$. Small detail omitted. $\square$

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