The Stacks project

Lemma 88.3.3. Let $A$ be a Noetherian ring and let $I \subset A$ be a ideal. Let $B$ be an object of (88.2.0.2). Then

  1. the pro-objects $\{ \mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B B/I^ nB\} $ and $\{ \mathop{N\! L}\nolimits _{B_ n/A_ n}\} $ of $D(B)$ are strictly isomorphic (see proof for elucidation),

  2. $\mathop{N\! L}\nolimits _{B/A}^\wedge = R\mathop{\mathrm{lim}}\nolimits \mathop{N\! L}\nolimits _{B_ n/A_ n}$ in $D(B)$.

Here $B_ n$ and $A_ n$ are as in Section 88.2.

Proof. The statement means the following: for every $n$ we have a well defined complex $\mathop{N\! L}\nolimits _{B_ n/A_ n}$ of $B_ n$-modules and we have transition maps $\mathop{N\! L}\nolimits _{B_{n + 1}/A_{n + 1}} \to \mathop{N\! L}\nolimits _{B_ n/A_ n}$. See Algebra, Section 10.134. Thus we can consider

\[ \ldots \to \mathop{N\! L}\nolimits _{B_3/A_3} \to \mathop{N\! L}\nolimits _{B_2/A_2} \to \mathop{N\! L}\nolimits _{B_1/A_1} \]

as an inverse system of complexes of $B$-modules and a fortiori as an inverse system in $D(B)$. Furthermore $R\mathop{\mathrm{lim}}\nolimits \mathop{N\! L}\nolimits _{B_ n/A_ n}$ is a homotopy limit of this inverse system, see Derived Categories, Section 13.34.

Choose a presentation $B = A[x_1, \ldots , x_ r]^\wedge / J$. This defines presentations

\[ B_ n = B/I^ nB = A_ n[x_1, \ldots , x_ r]/J_ n \]

where

\[ J_ n = JA_ n[x_1, \ldots , x_ r] = J/(J \cap I^ nA[x_1, \ldots , x_ r]^\wedge ) \]

The two term complex $J_ n/J_ n^2 \longrightarrow \bigoplus B_ n \text{d}x_ i$ represents $\mathop{N\! L}\nolimits _{B_ n/A_ n}$, see Algebra, Section 10.134. By Artin-Rees (Algebra, Lemma 10.51.2) in the Noetherian ring $A[x_1, \ldots , x_ r]^\wedge $ (Lemma 88.2.2) we find a $c \geq 0$ such that we have canonical surjections

\[ J/I^ nJ \to J_ n \to J/I^{n - c}J \to J_{n - c},\quad n \geq c \]

for all $n \geq c$. A moment's thought shows that these maps are compatible with differentials and we obtain maps of complexes

\[ \mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B B/I^ nB \to \mathop{N\! L}\nolimits _{B_ n/A_ n} \to \mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B B/I^{n - c}B \to \mathop{N\! L}\nolimits _{B_{n - c}/A_{n - c}} \]

compatible with the transition maps of the inverse systems $\{ \mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B B/I^ nB\} $ and $\{ \mathop{N\! L}\nolimits _{B_ n/A_ n}\} $. This proves part (1) of the lemma.

By part (1) and since pro-isomorphic systems have the same $R\mathop{\mathrm{lim}}\nolimits $ in order to prove (2) it suffices to show that $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is equal to $R\mathop{\mathrm{lim}}\nolimits \mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B B/I^ nB$. However, $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is a two term complex $M^\bullet $ of finite $B$-modules which are $I$-adically complete for example by Algebra, Lemma 10.97.1. Hence $M^\bullet = \mathop{\mathrm{lim}}\nolimits M^\bullet /I^ nM^\bullet = R\mathop{\mathrm{lim}}\nolimits M^\bullet /I^ n M^\bullet $, see More on Algebra, Lemma 15.87.1 and Remark 15.87.6. $\square$


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 0AJS. Beware of the difference between the letter 'O' and the digit '0'.