Timeline for Why the stable module category?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 4, 2023 at 15:57 | answer | added | Cihan | timeline score: 8 | |
| May 2, 2023 at 21:43 | comment | added | John Palmieri | Modules are simpler than chain complexes, and it is sometimes useful to represent an element of Ext as an actual module homomorphism. | |
| May 2, 2023 at 20:54 | comment | added | Tim Campion | @JohnPalmieri In the derived category, we also have $Hom(\Omega^m M, N) = Ext^m_R(M,N)$ (where $\Omega^m$ means the inverse shift operator in that category) -- why is this a selling point for the stable module category as opposed to the derived category? | |
| May 1, 2023 at 16:08 | comment | added | John Palmieri | Because of the role projective modules play in homological algebra, "killing them" provides a way of studying homological algebra. The inverse of the shift operator in the triangulated structure for the stable module category is the syzygy operator $\Omega$, and if $m > 0$, then the set of morphisms from $\Omega^m M$ to $N$ is $\operatorname{Ext}^m_R(M,N)$. (Neil Strickland already said this, but it's worth highlighting.) | |
| May 1, 2023 at 15:16 | comment | added | Tim Campion | @DenisT I guess one of the main things I'm wondering is why we quotient by this particular categorical ideal. | |
| May 1, 2023 at 15:12 | history | edited | Tim Campion | CC BY-SA 4.0 |
edited body
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| S May 1, 2023 at 13:15 | history | suggested | CommunityBot | CC BY-SA 4.0 |
Fix math spacing with \mathit; tweak wording slightly for idiomaticity; add comma
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| May 1, 2023 at 12:45 | review | Suggested edits | |||
| S May 1, 2023 at 13:15 | |||||
| May 1, 2023 at 10:14 | answer | added | Neil Strickland | timeline score: 13 | |
| May 1, 2023 at 6:37 | history | became hot network question | |||
| May 1, 2023 at 2:57 | comment | added | Denis T | I'd add some (maybe too informal) analogy with Koszul dualty: instead of looking at "formal" situation (taking derived endomorphisms of trivial module), you're trying to look at "compact" one; in absence of a particular distinguished object with interesting cohomological behaviour, you consider whole R-mod as a multi-object "algebra" with Ext as multiplication, and then take a quotient by zero divisor categorical ideal. | |
| Apr 30, 2023 at 22:49 | answer | added | Mare | timeline score: 17 | |
| Apr 30, 2023 at 22:36 | history | asked | Tim Campion | CC BY-SA 4.0 |