The Stacks project

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

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

where

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

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

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

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

\[ \mathop{N\! L}\nolimits ^\wedge _{B/A} = \mathop{\mathrm{lim}}\nolimits (J_ n/J_ n^2 \longrightarrow \bigoplus B_ n \text{d}x_ i) \]

(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. $\square$


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