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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09C5. Beware of the difference between the letter 'O' and the digit '0'.