I asked a question about the commutativity of inverse limits before. Since I am interested in using inverse system to describe certain subgroups of a finite group, I always assume the groups and posets are finite.
Having reading some material on inverse systems, a new question about exactness arise in my mind. Generally, the exactness of surjective inverse systems over a finite poset is not preserved by their limits. I am curious about the specific conditions, except being "directed," under which the limits of surjective inverse systems can preserve exactness. Note that a directed finite poset has unique maximal elements, which makes the case trivial. In other words, being "directed" is too strong.
For the condition, I have an idea on it. Let $I$ be a poset satisfying for any $x,y\in I$, $$\exists z\in I, z\ge x,z\ge y\iff x\ge y \mbox{ or }y\ge x.$$ I believe that limits of surjective systems over $I$ preserve exactness. In summary, my question is as follows:
- Is this true that such an $I$ can ensure the preservation of exactness? If so, does this $I$ or this condition have a name?
- Are there any other conditions under which the limits of surjective system preserve the exactness?