The Stacks project

Lemma 16.9.2. Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation 16.9.1. Let $r \geq 1$ and $\pi _1, \ldots , \pi _ r \in R$ map to elements of $\mathfrak q$. Assume

  1. for $i = 1, \ldots , r$ we have

    \[ \text{Ann}_{R/(\pi _1^8, \ldots , \pi _{i - 1}^8)R}(\pi _ i) = \text{Ann}_{R/(\pi _1^8, \ldots , \pi _{i - 1}^8)R}(\pi _ i^2) \]


    \[ \text{Ann}_{\Lambda /(\pi _1^8, \ldots , \pi _{i - 1}^8)\Lambda }(\pi _ i) = \text{Ann}_{\Lambda /(\pi _1^8, \ldots , \pi _{i - 1}^8)\Lambda }(\pi _ i^2) \]
  2. for $i = 1, \ldots , r$ the element $\pi _ i$ maps to a strictly standard element in $A$ over $R$.

Then, if

\[ R/(\pi _1^8, \ldots , \pi _ r^8)R \to A/(\pi _1^8, \ldots , \pi _ r^8)A \to \Lambda /(\pi _1^8, \ldots , \pi _ r^8)\Lambda \supset \mathfrak q/(\pi _1^8, \ldots , \pi _ r^8)\Lambda \]

can be resolved, so can $R \to A \to \Lambda \supset \mathfrak q$.

Proof. We are going to prove this by induction on $r$.

The case $r = 1$. Here the assumption is that there exists a factorization $A/\pi _1^8 \to \bar C \to \Lambda /\pi _1^8$ which resolves the situation modulo $\pi _1^8$. Conditions (1) and (2) are the assumptions needed to apply Lemma 16.7.3. Thus we can “lift” the resolution $\bar C$ to a resolution of $R \to A \to \Lambda \supset \mathfrak q$.

The case $r > 1$. In this case we apply the induction hypothesis for $r - 1$ to the situation $R/\pi _1^8 \to A/\pi _1^8 \to \Lambda /\pi _1^8 \supset \mathfrak q/\pi _1^8\Lambda $. Note that property (2) is preserved by Lemma 16.2.7. $\square$

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Your email address will not be published.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 07F8. Beware of the difference between the letter 'O' and the digit '0'.