Example 10.86.2 (0596)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.86: Mittag-Leffler systems
Example 10.86.2 (cite)

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Example 10.86.2. If $(A_ i, \varphi _{ji})$ is a directed inverse system of sets or of modules and the maps $\varphi _{ji}$ are surjective, then clearly the system is Mittag-Leffler. Conversely, suppose $(A_ i, \varphi _{ji})$ is Mittag-Leffler. Let $A'_ i \subset A_ i$ be the stable image of $\varphi _{ji}(A_ j)$ for $j \geq i$. Then $\varphi _{ji}|_{A'_ j}: A'_ j \to A'_ i$ is surjective for $j \geq i$ and $\mathop{\mathrm{lim}}\nolimits A_ i = \mathop{\mathrm{lim}}\nolimits A'_ i$. Hence the limit of the Mittag-Leffler system $(A_ i, \varphi _{ji})$ can also be written as the limit of a directed inverse system over $I$ with surjective maps.

Comments (2)

Comment #5548 by a on October 21, 2020 at 19:44

I was confused but figured out where ML condition was being used in the proof that $\phi_{ji}:A_j' \to A_i'$ is surjective, and thought it would be worth mentioning:

Given $a_i'$ in the stable image $A_i'$ , for all $k>i$ there is a $a_k \in A_k$ mapping to $a_i'$ , ... but these might map to different elements in $A_j$ that only become equal in $A_i$ ! With ML hypothesis, there exists $N$ such that for $k\geq N$ , $A_k$ maps to $A_j'$ ... then $a_N$ maps to an element in the stable image $A_j'$ that maps to $a_i$ .

Comment #5733 by Johan on November 16, 2020 at 13:23

Yes. But I thought just already even using the English phrase "stable image" suggests that the images of the transition maps in a fixed $A_i$ stabilize which only happens for ML systems (and in fact characterizes them).

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