Timeline for Proving the graded structure of the tensor algebra from only the universal property
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 19, 2023 at 14:23 | answer | added | Theo Johnson-Freyd | timeline score: 3 | |
| Mar 18, 2023 at 0:36 | comment | added | LSpice | @TheoJohnson-Freyd, right. I like that argument better than my not-quite-there approach. (I tried to do something like that, but, despite the close connections between algebraic groups and my research, couldn't manage to see them here.) Would you be willing to post it as an answer? | |
| Mar 17, 2023 at 22:33 | comment | added | Theo Johnson-Freyd | @LSpice Let's see. Probably I don't remember exactly what I meant more than a year ago. But conveniently I was very vague in my hint, so probably I can salvage it. Let's say our base ring is $R$. For every $R$-ring $R \to S$, consider the base-change $T V \otimes S := (T_R V) \otimes S = T_S(V \otimes S)$. [This equivalence is by universal property.] Then $T V \otimes S$ carries an action by $S^\times$. In other words, $TV$ carries an action by the algebraic group $\mathbb{G}_m$, which acts via the rescaling action on $V$. Now prove that actions by $\mathbb{G}_m$ are the same as gradings. | |
| Mar 16, 2023 at 4:29 | comment | added | LSpice | @TheoJohnson-Freyd, that surely works for infinite fields, but, if I take your meaning correctly, what about, say, a tensor algebra over $\mathbb F_p$, in which it seems that we could not use homogeneity to distinguish elements in degrees $1$ and $p$? | |
| S Mar 16, 2023 at 3:07 | vote | accept | sudgy | ||
| Mar 16, 2023 at 3:07 | answer | added | sudgy | timeline score: 1 | |
| Oct 4, 2021 at 3:54 | vote | accept | sudgy | ||
| S Mar 16, 2023 at 3:07 | |||||
| Oct 2, 2021 at 21:21 | comment | added | Theo Johnson-Freyd | I disagree with @DanieleTampieric because when MO was originally created (before M.SE, indeed before we were integrated into the SE network) it was understood that “research level” meant PhD-student-and-above, and I for one learned category theory as a PhD student. As for the OP’s question, the trick is to use the rescaling action on V. | |
| Oct 2, 2021 at 7:55 | comment | added | Daniele Tampieri | First of all, hi and welcome to the MathOverflow. Your question is nicely written and well motivated, but is bit out of topic here since this Q&A site is devoted to mathematics research level questions. Why don't you try our sister site Math.SE? It is devoted to general (even very difficult) questions in mathematics. | |
| Oct 1, 2021 at 20:41 | answer | added | LSpice | timeline score: 6 | |
| Oct 1, 2021 at 20:31 | review | Close votes | |||
| Oct 3, 2021 at 0:59 | |||||
| S Oct 1, 2021 at 19:55 | review | First questions | |||
| Oct 1, 2021 at 20:14 | |||||
| S Oct 1, 2021 at 19:55 | history | asked | sudgy | CC BY-SA 4.0 |