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The Stacks project

Lemma 88.17.7. 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 88.3.5) 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 88.3.5 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 88.3.5 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 88.17.6 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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