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The Stacks project

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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