Permalink
Browse files

Mittag-Leffler formulation improvement

  • Loading branch information...
aisejohan committed Jan 31, 2018
1 parent 3a38c18 commit 6646bc719422f00797f5e0cf842d7ac215f710a7
Showing with 6 additions and 6 deletions.
  1. +6 −6 algebra.tex
@@ -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

0 comments on commit 6646bc7

Please sign in to comment.