Timeline for Closure of flat monoid acts under Rees extensions
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| Jun 26 at 14:39 | comment | added | Sean Cox | The 3rd sentence of my previous comment is stated in Lemma 2.1 of "Stability and Flatness in acts over monoids", but does not match the statement of its citation (Corollary 4.9 of Renshaw "Flatness and amalgamation in monoids"); the latter assumes B is flat. However I believe it is in fact true, if one goes through the proof of Theorem 4.8 of the latter reference. There, for the "only if" direction, the injectivity of $1 \otimes \lambda$ appears to not be used. | |
| Jun 18 at 15:12 | comment | added | Sean Cox | Good point. I found some relevant work of Renshaw. In "Stability and flatness in acts over monoids" he proves that if $A \subset B$ (with $A$ nonempty) and $B/A$ is flat, then the monoid is right-reversible, and moreover the inclusion $A \to B$ is "stable", a weakening of pure. So my question is equivalent to: are flat acts closed under stable extensions? He proves flat acts are closed under pure extensions in Chapter IV Theorem 1.5 of his thesis (research-repository.st-andrews.ac.uk/handle/10023/11071). But I don't see how to weaken "pure" to "stable" in that argument. | |
| Jun 18 at 13:52 | comment | added | Benjamin Steinberg | I think it will be very rare for $B/A$ to be flat. Notice this has the one point $S$-set as a retract and so that should be flat as well. But then $S$ must be right reversible, that is, any two principal left ideals of $S$ must intersect. So this quite strong | |
| Jun 16 at 23:53 | history | asked | Sean Cox | CC BY-SA 4.0 |