Lemma 61.29.5. 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 $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$ is isomorphic in $D(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$ to a complex of constant sheaves of $\Lambda /I^ n$-modules. Then

$H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n)$

has the Mittag-Leffler condition.

Proof. Say $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$ is isomorphic to $\underline{E_ n}$ for some object $E_ n$ of $D(\Lambda /I^ n)$. Since $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has finite tor dimension and has finite type cohomology sheaves we see that $E_1$ is perfect (see More on Algebra, Lemma 15.74.2). The transition maps

$K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^{n + 1}} \to K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$

locally come from (possibly many distinct) maps of complexes $E_{n + 1} \to E_ n$ in $D(\Lambda /I^{n + 1})$ see Cohomology on Sites, Lemma 21.53.3. For each $n$ choose one such map and observe that it induces an isomorphism $E_{n + 1} \otimes _{\Lambda /I^{n + 1}}^\mathbf {L} \Lambda /I^ n \to E_ n$ in $D(\Lambda /I^ n)$. By More on Algebra, Lemma 15.97.4 we can find a finite complex $M^\bullet$ of finite projective $\Lambda ^\wedge$-modules and isomorphisms $M^\bullet /I^ nM^\bullet \to E_ n$ in $D(\Lambda /I^ n)$ compatible with the transition maps.

Now observe that for each finite collection of indices $n > m > k$ the triple of maps

$H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n) \to H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ m) \to H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ k)$

is isomorphic to

$H^0(X, \underline{M^\bullet /I^ nM^\bullet }) \to H^0(X, \underline{M^\bullet /I^ mM^\bullet }) \to H^0(X, \underline{M^\bullet /I^ kM^\bullet })$

Namely, choose any isomorphism

$\underline{M^\bullet /I^ nM^\bullet } \to K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n$

induces similar isomorphisms module $I^ m$ and $I^ k$ and we see that the assertion is true. Thus to prove the lemma it suffices to show that the system $H^0(X, \underline{M^\bullet /I^ nM^\bullet })$ has Mittag-Leffler. Since taking sections over $X$ is exact, it suffices to prove that the system of $\Lambda$-modules

$H^0(M^\bullet /I^ nM^\bullet )$

has Mittag-Leffler. Set $A = \Lambda ^\wedge$ and consider the spectral sequence

$\text{Tor}_{-p}^ A(H^ q(M^\bullet ), A/I^ nA) \Rightarrow H^{p + q}(M^\bullet /I^ nM^\bullet )$

By More on Algebra, Lemma 15.27.3 the pro-systems $\{ \text{Tor}_{-p}^ A(H^ q(M^\bullet ), A/I^ nA)\}$ are zero for $p > 0$. Thus the pro-system $\{ H^0(M^\bullet /I^ nM^\bullet )\}$ is equal to the pro-system $\{ H^0(M^\bullet )/I^ nH^0(M^\bullet )\}$ and the lemma is proved. $\square$

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