# stacks/stacks-project

Mittag-Leffler formulation improvement

Thanks to Fred Rohrer
https://stacks.math.columbia.edu/tag/0594#comment-2969
 @@ -19704,7 +19704,7 @@ \section{Mittag-Leffler systems} \noindent The purpose of this section is to define Mittag-Leffler systems and why it is a useful property. and why this is a useful notion. \medskip\noindent In the following, $I$ will be a directed set, see @@ -19714,9 +19714,9 @@ \section{Mittag-Leffler systems} Categories, Definition \ref{categories-definition-directed-system}. This is a directed inverse system as we assumed $I$ directed (Categories, Definition \ref{categories-definition-directed-system}). For each $i \in I$, the images $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$ form a decreasing family. Let $A'_i = \bigcap_{j \geq i} \varphi_{ji}(A_j)$. For each $i \in I$, the images $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$ form a decreasing directed family of subsets (or submodules) of $A_i$. Let $A'_i = \bigcap_{j \geq i} \varphi_{ji}(A_j)$. Then $\varphi_{ji}(A'_j) \subset A'_i$ for $j \geq i$, hence by restricting we get a directed inverse system $(A'_i, \varphi_{ji}|_{A'_j})$. From the construction of the limit of an inverse system in the category @@ -19729,8 +19729,8 @@ \section{Mittag-Leffler systems} \label{definition-ML-system} Let $(A_i, \varphi_{ji})$ be a directed inverse system of sets over $I$. Then we say $(A_i, \varphi_{ji})$ is {\it Mittag-Leffler inverse system} if for each $i \in I$, the decreasing family $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$ stabilizes. Explicitly, this means that for each $i \in I$, there each $i \in I$, the family $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$ stabilizes. Explicitly, this means that for each $i \in I$, there exists $j \geq i$ such that for $k \geq j$ we have $\varphi_{ki}(A_k) = \varphi_{ji}( A_j)$. If $(A_i, \varphi_{ji})$ is a directed inverse system of modules over a ring $R$, we say that it is Mittag-Leffler if the underlying