Mittag-Leffler condition and the vanishing of \(\varprojlim^ 1\)
The author considers inverse limits \(\varprojlim\) of (right) modules over a fixed ring \(k\), with indexing category the natural numbers \(\mathbb{N}\). An inverse system \(\Xi= (X_m)_{m\in\mathbb{N}}\) satisfies the Mittag-Leffler condition if for all \(n\) the decreasing filtration of \(X_n\) formed by the images of \(X_m\) for \(m>n\) is eventually constant. The main results are the following: an inverse system \(\Xi\) satisfies the Mittag-Leffler condition if and only if \(\varprojlim^1 (\Xi\otimes E)=0\) for any left \(k\)-module \(E\) if and only if \(\varprojlim^1 (\Xi \otimes K)=0\) for any ring extension \(K\) of \(k\).
scientific article; zbMATH DE number 863703