The Stacks project

Lemma 61.29.3. Let $X$ be a weakly contractible affine scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda )$ such that the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ are locally constant. Then there exists a finite disjoint open covering $X = \coprod U_ i$ and for each $i$ a finite collection of finite projective $\Lambda ^\wedge $-modules $M_ a, \ldots , M_ b$ such that $K|_{U_ i}$ is represented by a complex

\[ (\underline{M^ a})^\wedge \to \ldots \to (\underline{M^ b})^\wedge \]

in $D(U_{i, {pro\text{-}\acute{e}tale}}, \Lambda )$ for some maps of sheaves of $\Lambda $-modules $(\underline{M^ i})^\wedge \to (\underline{M^{i + 1}})^\wedge $.

Proof. We freely use the results of Lemma 61.29.2. Choose $a, b$ as in that lemma. We will prove the lemma by induction on $b - a$. Let $\mathcal{F} = H^ b(K)$. Note that $\mathcal{F}$ is a derived complete sheaf of $\Lambda $-modules by Proposition 61.21.1. Moreover $\mathcal{F}/I\mathcal{F}$ is a locally constant sheaf of $\Lambda /I$-modules of finite type. Apply Lemma 61.28.7 to get a surjection $\rho : (\underline{\Lambda }^\wedge )^{\oplus t} \to \mathcal{F}$.

If $a = b$, then $K = \mathcal{F}[-b]$. In this case we see that

\[ \mathcal{F} \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I} = \mathcal{F}/I\mathcal{F} \]

As $X$ is weakly contractible and $\mathcal{F}/I\mathcal{F}$ locally constant, we can find a finite disjoint union decomposition $X = \coprod U_ i$ by affine opens $U_ i$ and $\Lambda /I$-modules $\overline{M}_ i$ such that $\mathcal{F}/I\mathcal{F}$ restricts to $\underline{\overline{M}_ i}$ on $U_ i$. After refining the covering we may assume the map

\[ \rho |_{U_ i} \bmod I : \underline{\Lambda /I}^{\oplus t} \longrightarrow \underline{\overline{M}_ i} \]

is equal to $\underline{\alpha _ i}$ for some surjective module map $\alpha _ i : \Lambda /I^{\oplus t} \to \overline{M}_ i$, see Modules on Sites, Lemma 18.43.3. Note that each $\overline{M}_ i$ is a finite $\Lambda /I$-module. Since $\mathcal{F}/I\mathcal{F}$ has tor amplitude in $[0, 0]$ we conclude that $\overline{M}_ i$ is a flat $\Lambda /I$-module. Hence $\overline{M}_ i$ is finite projective (Algebra, Lemma 10.78.2). Hence we can find a projector $\overline{p}_ i : (\Lambda /I)^{\oplus t} \to (\Lambda /I)^{\oplus t}$ whose image maps isomorphically to $\overline{M}_ i$ under the map $\alpha _ i$. We can lift $\overline{p}_ i$ to a projector $p_ i : (\Lambda ^\wedge )^{\oplus t} \to (\Lambda ^\wedge )^{\oplus t}$1. Then $M_ i = \mathop{\mathrm{Im}}(p_ i)$ is a finite $I$-adically complete $\Lambda ^\wedge $-module with $M_ i/IM_ i = \overline{M}_ i$. Over $U_ i$ consider the maps

\[ \underline{M_ i}^\wedge \to (\underline{\Lambda }^\wedge )^{\oplus t} \to \mathcal{F}|_{U_ i} \]

By construction the composition induces an isomorphism modulo $I$. The source and target are derived complete, hence so are the cokernel $\mathcal{Q}$ and the kernel $\mathcal{K}$. We have $\mathcal{Q}/I\mathcal{Q} = 0$ by construction hence $\mathcal{Q}$ is zero by Lemma 61.28.6. Then

\[ 0 \to \mathcal{K}/I\mathcal{K} \to \underline{\overline{M}_ i} \to \mathcal{F}/I\mathcal{F} \to 0 \]

is exact by the vanishing of $\text{Tor}_1$ see at the start of this paragraph; also use that $\underline{\Lambda }^\wedge /I\overline{\Lambda }^\wedge $ by Modules on Sites, Lemma 18.42.4 to see that $\underline{M_ i}^\wedge /I\underline{M_ i}^\wedge = \underline{\overline{M}_ i}$. Hence $\mathcal{K}/I\mathcal{K} = 0$ by construction and we conclude that $\mathcal{K} = 0$ as before. This proves the result in case $a = b$.

If $b > a$, then we lift the map $\rho $ to a map

\[ \tilde\rho : (\underline{\Lambda }^\wedge )^{\oplus t}[-b] \longrightarrow K \]

in $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$. This is possible as we can think of $K$ as a complex of $\underline{\Lambda }^\wedge $-modules by discussion in the introduction to Section 61.20 and because $X_{pro\text{-}\acute{e}tale}$ is weakly contractible hence there is no obstruction to lifting the elements $\rho (e_ s) \in H^0(X, \mathcal{F})$ to elements of $H^ b(X, K)$. Fitting $\tilde\rho $ into a distinguished triangle

\[ (\underline{\Lambda }^\wedge )^{\oplus t}[-b] \to K \to L \to (\underline{\Lambda }^\wedge )^{\oplus t}[-b + 1] \]

we see that $L$ is an object of $D_{cons}(X, \Lambda )$ such that $L \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has tor amplitude contained in $[a, b - 1]$ (details omitted). By induction we can describe $L$ locally as stated in the lemma, say $L$ is isomorphic to

\[ (\underline{M^ a})^\wedge \to \ldots \to (\underline{M^{b - 1}})^\wedge \]

The map $L \to (\underline{\Lambda }^\wedge )^{\oplus t}[-b + 1]$ corresponds to a map $(\underline{M^{b - 1}})^\wedge \to (\underline{\Lambda }^\wedge )^{\oplus t}$ which allows us to extend the complex by one. The corresponding complex is isomorphic to $K$ in the derived category by the properties of triangulated categories. This finishes the proof. $\square$

[1] Proof: by Algebra, Lemma 10.32.7 we can lift $\overline{p}_ i$ to a compatible system of projectors $p_{i, n} : (\Lambda /I^ n)^{\oplus t} \to (\Lambda /I^ n)^{\oplus t}$ and then we set $p_ i = \mathop{\mathrm{lim}}\nolimits p_{i, n}$ which works because $\Lambda ^\wedge = \mathop{\mathrm{lim}}\nolimits \Lambda /I^ n$.

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