A Note on Surjective Inverse Systems

Given an upward directed set $I$ we consider surjective $I$-inverse systems ${X_\al,f_{\al\be}:X_\be\lra X_\al| \al\leq\be\in I}$, namely those inverse systems that have all $f_{\al\be}$ surjective. A number of properties of $I$-inverse systems have been investigated; such are the Mittag-Leffler condition, investigated by Grothendieck and flabby and semi-flabby $I$-inverse systems studied by Jensen. We note that flabby implies semi-flabby implies surjective implies Mittag-Leffler. Some of the results about surjective inverse systems have been known for some time. The aim of this note is to give a series of equivalent statements and implications involving surjective inverse systems and the systems satisfying the Mittag-Leffler condition, together with improvements of established results, as well as their relationships with the already known, but scattered facts. The most prominent results relate cardinalities of the index sets with right exactness of the inverse limit functor and the non-vanishing of the inverse limit – connections related to cohomological dimensions.

💡 Research Summary

The paper investigates inverse systems indexed by an upward directed set (I) under the minimal hypothesis that every bonding map (f_{\alpha\beta}:X_{\beta}\to X_{\alpha}) is surjective. Such systems are called surjective (I)-inverse systems. The author begins by recalling three classical notions: the Mittag‑Leffler (ML) condition introduced by Grothendieck, and Jensen’s concepts of flabby and semi‑flabby inverse systems. By carefully analysing the definitions, the paper establishes a strict hierarchy of implications:

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