The Stacks project

This is a variant of [Lemma 3.5.4, BS]

Lemma 24.35.3. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be an ideal. Let $K_ n$ be the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ viewed as a differential graded $R$-algebra. Let $(M_ n)$ be an object of $D(\mathbf{N}, (K_ n))$. Then for any $t \geq 1$ we have

\[ R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _ R^\mathbf {L} K_ t) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{K_ n}^\mathbf {L} K_ t) \]

in $D(R)$.

Proof. We fix $t \geq 1$. For $n \geq t$ let us denote ${}_ nK_ t$ the differential graded $R$-algebra $K_ t$ viewed as a left differential graded $K_ n$-module. Observe that

\[ M_ n \otimes _ R^\mathbf {L} K_ t = M_ n \otimes _{K_ n}^\mathbf {L} (K_ n \otimes _ R^\mathbf {L} K_ t) = M_ n \otimes _{K_ n}^\mathbf {L} (K_ n \otimes _ R K_ t) \]

Hence by Lemma 24.35.2 it suffices to show that $({}_ nK_ t)$ and $(K_ n \otimes _ R K_ t)$ are pro-isomorphic in the derived category. The multiplication maps

\[ K_ n \otimes _ R K_ t \longrightarrow {}_ nK_ t \]

are maps of left differential graded $K_ n$-modules. Thus to finish the proof it suffices to show that for all $n \geq 1$ there exists an $N > n$ and a map

\[ {}_ NK_ t \longrightarrow {}_ NK_ n \otimes _ R K_ t \]

in $D(K_ N^{opp}, \text{d})$ whose composition with the multiplication map is the transition map (in either direction). This is done in Divided Power Algebra, Lemma 23.12.4 by an explicit construction. $\square$


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