Timeline for What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
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| Mar 17, 2022 at 13:55 | comment | added | Jason Starr | I advise to read Grothendieck's comments on Hilbert's Theorem 90 and descent in FGA. Certainly there do exist $\mathbb{G}_m$-torsors that do not arise via pullback from the Serre twisting sheaf on $\mathbb{P}^n$, since there are non-globally generated invertible sheaves whose dual is also not globally generated. Perhaps a more significant example arises from the method of lifting to the universal torsor to understand rational points (in those cases where the Picard group scheme is a split lattice). | |
| Mar 17, 2022 at 12:19 | comment | added | David E Speyer | I'd say that @JasonStarr's example of $GL_n \to PGL_n$ isn't really another situation: $GL_n$ is an open subset of $\mathbb{A}^{n^2}$ and $PGL_n$ is its preimage in $\mathbb{P}^{n^2-1}$, so the claim follows by argument <b>T</b> for $\mathbb{P}^{n^2-1}$. I'm not sure how to make this precise, but I wonder if there is any $\mathbb{G}_m$ principal bundle which isn't induced from affine space over projective space like this. | |
| Mar 17, 2022 at 12:08 | comment | added | KConrad | David’s comment is an instance of Galois descent, which in turn has a proof using the scaling of coefficients to make a coefficient $1$ that is just like the OP’s proof $\mathbf T$. See equation (2.2) simplified to the case where $a_n = 1$ in kconrad.math.uconn.edu/blurbs/galoistheory/galoisdescent.pdf. The application to showing certain subspaces of $L^n$ have a basis in $K^n$ is Theorem 3.2. | |
| Mar 17, 2022 at 11:05 | comment | added | Jason Starr | Both arguments are versions of the fact that etale (or even fppf) torsors for $\mathbb{G}_m$ are Zariski locally trivial. One advantage of the first argument is that it works for other situations, e.g., every $K$-point of $\textbf{PGL}_n$ lifts to a $K$-point of $\textbf{GL}_n$ (but not necessarily to a $K$-point of $\textbf{SL}_n$ since $\mu_n$-torsors are typically not Zariski locally trivial). | |
| Mar 17, 2022 at 11:05 | comment | added | David E Speyer | I'm not sure what to say about the group cohomology question, but I will point out that the same arguments show: Let $V \subset L^n$ be an $L$-vector space with $\sigma(V) = V$ for all $\sigma \in \text{Gal}(L/K)$, then $V$ has a basis in $K^n$. I use this when I prove Artin's lemma in setting up Galois theory math.stackexchange.com/questions/2386626/… | |
| Mar 17, 2022 at 10:34 | history | asked | Gro-Tsen | CC BY-SA 4.0 |