Lemma 86.3.1. Let $A$ be a Noetherian ring and let $I \subset A$ be a ideal. Let $B$ be an object of (86.2.0.2). Then $\mathop{N\! L}\nolimits ^\wedge _{B/A} = R\mathop{\mathrm{lim}}\nolimits \mathop{N\! L}\nolimits _{B_ n/A_ n}$ in $D(B)$.

**Proof.**
In fact, the presentation $B = A[x_1, \ldots , x_ r]^\wedge / J$ defines presentations

where

By Artin-Rees (Algebra, Lemma 10.50.2) in the Noetherian ring $A[x_1, \ldots , x_ r]^\wedge $ (Lemma 86.2.2) we see that we have canonical surjections

for some $c \geq 0$. It follows that $\mathop{\mathrm{lim}}\nolimits J_ n/J_ n^2 = J/J^2$ as any finite $A[x_1, \ldots , x_ r]^\wedge $-module is $I$-adically complete (Algebra, Lemma 10.96.1). Thus

(termwise limit) and the transition maps in the system are termwise surjective. The two term complex $J_ n/J_ n^2 \longrightarrow \bigoplus B_ n \text{d}x_ i$ represents $\mathop{N\! L}\nolimits _{B_ n/A_ n}$ by Algebra, Section 10.133. It follows that $\mathop{N\! L}\nolimits ^\wedge _{B/A}$ represents $R\mathop{\mathrm{lim}}\nolimits \mathop{N\! L}\nolimits _{B_ n/A_ n}$ in the derived category by More on Algebra, Lemma 15.81.1. $\square$

## 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Your email address will not be published.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.

## Comments (0)