Abstract
We discuss the new notion of a lim Cohen-Macaulay sequence of modules over a local ring, and also a somewhat weaker notion, as well as the theory of content for local cohomology modules. We relate both to the problem of proving the direct summand conjecture and other homological conjectures without using almost ring theory and perfectoid space theory, and we also indicate some other open problems whose solution would yield a new proof of the direct summand conjecture.
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