Timeline for Motivation for Karoubi envelope/ idempotent completion
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2020 at 21:10 | answer | added | R. van Dobben de Bruyn | timeline score: 4 | |
| Feb 29, 2020 at 1:41 | history | became hot network question | |||
| Feb 28, 2020 at 19:42 | history | edited | YCor | CC BY-SA 4.0 |
minor formatting of title
|
| Feb 28, 2020 at 19:41 | comment | added | user267839 | @FrançoisBrunault: That was the devil in the detail I was looking for. Indeed $\mathbb{P}^1= [\mathbb{P}^1 \times e] =(\mathbb{P}^1, p) \oplus L$ decomposes by point & Lefschetz motive. This leads to Tate motive as $\mathbb{Z}(1)[2]=L$ are related by the shift. Thank you! | |
| Feb 28, 2020 at 19:33 | vote | accept | user267839 | ||
| Feb 28, 2020 at 19:30 | comment | added | François Brunault | Another example for motives: you need projectors in order to define even very basic objects. For example the Tate motive $\mathbb{Z}(1)$ (or rather its inverse, the Lefschetz motive), you need to decompose the cohomology of the projective line (or the multiplicative group). If you don't have the Tate motive then you can't define motivic cohomology for example. | |
| Feb 28, 2020 at 19:30 | answer | added | Jon Pridham | timeline score: 6 | |
| Feb 28, 2020 at 19:11 | answer | added | Mikhail Bondarko | timeline score: 4 | |
| Feb 28, 2020 at 19:06 | comment | added | user267839 | @SamHopkins: that's a really nice one! +1 | |
| Feb 28, 2020 at 19:02 | comment | added | user267839 | so if we recall that the motivation of study motives was to develop a "universal " cohomology theory that it seems that passing to idempotent closure allows some nice features. More simply question is if we don't pass to idempotent closure in the construction of pure motives, will we obtain still a theory that is just not so powerful or does the construction fails completly. I was reading Manin's paper on construction of motives but I nowhere found a step which would "totally" fail if we not make pass to the completion. So the question is if it is of "vital" or "enhencing" nature? | |
| Feb 28, 2020 at 18:56 | comment | added | Sam Hopkins | I think a very illuminating example is the construction of Deligne's category $\mathrm{Rep}(S_t)$ of "representations of the symmetric group $S_t$" where $t$ is a complex parameter. The idea is that for integer $n$, every representation of $S_n$ is a direct summand of a tensor power of the defining representation. So to mimic this for an arbitrary parameter $t$, first you create via diagrammatic rules a category whose objects correspond to tensor powers of the defining representation; then you take the Karoubian envelope to get all representations. | |
| Feb 28, 2020 at 18:49 | comment | added | user267839 | @Sam Hopkins: that's true. The point is when we think about the couple of constructions I mentioned (e.g. the motivic case) where it become neccessary to deal with a category beeing closed under taking direct summands? What might fail if not not do it? | |
| Feb 28, 2020 at 18:47 | comment | added | François Brunault | Idempotents appear very naturally when studying the cohomology of algebraic varieties. For example if some finite group acts on your variety then you can consider the piece of cohomology cut out by characters of this group (corresponding to idempotents in the group algebra). | |
| Feb 28, 2020 at 18:42 | comment | added | Sam Hopkins | The point of the idempotent completion is so that direct summands of objects of your category are now objects as well. | |
| Feb 28, 2020 at 18:28 | history | edited | LSpice | CC BY-SA 4.0 |
Language & TeX editing
|
| Feb 28, 2020 at 17:38 | history | asked | user267839 | CC BY-SA 4.0 |