Timeline for Over which (graded) rings are all modules decomposable into indecomposables?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 20, 2021 at 12:38 | comment | added | Tilman | Nice construction! | |
| Dec 19, 2021 at 20:11 | comment | added | user164898 | Okay, thanks for your reply. | |
| Dec 19, 2021 at 18:56 | comment | added | Jeremy Rickard | I’m not just taking an infinite product of copies of $A/I$. The structure maps of my module involve $x$: I’m not taking an infinite product of copies of anything Artinian, but of something that, as an $\mathbb{F}_2$ module, is a direct sum of four copies of $\mathbb{F}_2[x]$. | |
| Dec 19, 2021 at 18:38 | comment | added | user164898 | There's something perplexing to me about your argument. I'll write $A$ for the Steenrod algebra. In your graded $A$-module $V$, the Steenrod squares $Sq^n$ act trivially for all $n>3$. So the action of $A$ on $V$ factors through the projection $A \rightarrow A/I$, where $I$ is the ideal generated by all homogeneous elements of degree $>3$. But $A/I$ is certainly Artin. It is also gr-local, i.e., has a unique maximal homogeneous ideal. Isn't it already known that the infinite product of (degree 0, i.e., not suspended) copies of a gr-local Artin ring remains free in the graded module category? | |
| Dec 19, 2021 at 10:40 | history | answered | Jeremy Rickard | CC BY-SA 4.0 |