The Stacks project

Lemma 87.17.7. Let $A \to B \to C$ be arrows in $\textit{WAdm}^{Noeth}$ which are adic and topologically of finite type. If $B \to C$ is rig-smooth, then the kernel of the map

\[ H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C) \to H^{-1}(\mathop{N\! L}\nolimits _{C/A}^\wedge ) \]

(see Lemma 87.3.5) is annihilated by an ideal of definition.

Proof. Let $\overline{\mathfrak q} \subset C$ be a prime ideal which does not contain an ideal of definition. Since the modules in question are finite it suffices to show that

\[ H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C)_{\overline{\mathfrak q}} \to H^{-1}(\mathop{N\! L}\nolimits _{C/A}^\wedge )_{\overline{\mathfrak q}} \]

is injective. As in the proof of Lemma 87.3.5 choose presentations $B = A\{ x_1, \ldots , x_ r\} /J$, $C = B\{ y_1, \ldots , y_ s\} /J'$, and $C = A\{ x_1, \ldots , x_ r, y_1, \ldots , y_ s\} /K$. Looking at the diagram in the proof of Lemma 87.3.5 we see that it suffices to show that $J/J^2 \otimes _ B C \to K/K^2$ is injective after localization at the prime ideal $\mathfrak q \subset A\{ x_1, \ldots , x_ r, y_1, \ldots , y_ s\} $ corresponding to $\overline{\mathfrak q}$. Please compare with More on Algebra, Lemma 15.33.6 and its proof. This is the same as asking $J/KJ \to K/K^2$ to be injective after localization at $\mathfrak q$. Equivalently, we have to show that $J_\mathfrak q \cap K^2_\mathfrak q = (KJ)_\mathfrak q$. By Lemma 87.17.6 we know that $(K/J)_\mathfrak q = J'_\mathfrak q$ is generated by a regular sequence. Hence the desired intersection property follows from More on Algebra, Lemma 15.32.5 (and the fact that an ideal generated by a regular sequence is $H_1$-regular, see More on Algebra, Section 15.32). $\square$


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