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). $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GH2. Beware of the difference between the letter 'O' and the digit '0'.