Timeline for Are Azumaya algebras of trivial Brauer class isomorphic to $\mathcal{E}nd(\mathcal{H})$?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2019 at 11:38 | vote | accept | CommunityBot | ||
| Oct 27, 2019 at 11:18 | answer | added | Uriya First | timeline score: 5 | |
| Oct 27, 2019 at 11:16 | comment | added | user39380 | @MartinBright Thanks for the nice suggestion! Let me check it.. | |
| Oct 27, 2019 at 9:17 | comment | added | Martin Bright | The usual way to recover $k^n$ from $M_n(k)$ is to take a minimal left ideal. Does that work? | |
| Oct 27, 2019 at 6:58 | history | edited | user39380 | CC BY-SA 4.0 |
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| Oct 27, 2019 at 6:55 | comment | added | user39380 | @Eoin Thanks! Furthermore, would there be a way to express this $\mathcal{H}$ in terms of $\mathcal{F},\mathcal{G}$? (The result seems to claim we can always pick some line bundle $\mathcal{L}$, so that we can write $\mathcal{G}\otimes\mathcal{L}$ as $\mathcal{F}\otimes\mathcal{H}$? I am a bit confused how to show a module has a specific "tensor summand") | |
| Oct 27, 2019 at 5:33 | comment | added | Eoin | I think so. I'm looking at Milne's Etale cohomology book, Chapter IV, Section 2. For X a scheme you have an exact sequence $1\rightarrow G_{m,X}\rightarrow GL_{n,X}\rightarrow PGL_{n,X}\rightarrow 1$ (of etale sheaves on X). The group of azumaya algebras includes into $H^2_{et}(X,G_m)$ by Theorem 2.5 of loc. cit., so if one is trivial then it comes from an element of $H^1_{et}(X,GL_n)$. I think this correspondence sends a locally free sheaf $\mathcal{F}$ of rank $n$ to $\mathcal{E}nd(\mathcal{F})$. | |
| Oct 27, 2019 at 3:22 | history | asked | user39380 | CC BY-SA 4.0 |