Timeline for push-forward of linear algebraic group schemes
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 6, 2017 at 21:13 | answer | added | anon19 | timeline score: 3 | |
| Nov 6, 2017 at 15:19 | vote | accept | Danny | ||
| Nov 6, 2017 at 13:38 | vote | accept | Danny | ||
| Nov 6, 2017 at 15:13 | |||||
| Nov 5, 2017 at 22:15 | comment | added | Jason Starr | @R.vanDobbendeBruyn. Representability follows, for instance, from Lemma 2.3.3 of Max Lieblich's article, "Remarks on the stack of coherent algebras": arxiv.org/pdf/math/0603034.pdf I believe that it can also be deduced from Cor. 7.7.8 of EGA $\textrm{III}_2,$ but I cannot quite make that work . . . | |
| Nov 5, 2017 at 21:57 | comment | added | R. van Dobben de Bruyn | To me it's not even clear whether $\pi_* G$ is representable under your assumptions. Is this standard? | |
| Nov 5, 2017 at 21:41 | answer | added | Jason Starr | timeline score: 9 | |
| Nov 5, 2017 at 21:30 | comment | added | nfdc23 | What is the motivation for the question? Already with $S$ a geometric point and $X$ a smooth connected proper curve there will typically be non-split $X$-tori $T$ (arising from units of non-trivial connected etale covers) that admit no non-trivial $X$-point, so $\pi_{\ast}(T) = 1$; that trivial pushforward is affine (even a torus!) but is it something you would consider to be useful? | |
| Nov 5, 2017 at 21:19 | comment | added | Jason Starr | @nfdc23. You are completely correct. I was thinking of $\pi_*(B \mathbb{G}_m)$, the pushforward of the classifying stack, not $\pi_* \mathbb{G}_m,$ the pushforward of the group scheme. Sorry about that! | |
| Nov 5, 2017 at 21:16 | comment | added | Danny | right -- to clarify, I didn't mean $R^1\pi_*$, but just $\pi_*$. | |
| Nov 5, 2017 at 21:13 | comment | added | nfdc23 | @JasonStarr: did you mean to refer to ${\rm{R}}^1\pi_{\ast}(\mathbf{G}_m)$? For the class of $\pi$ you mention (or more generally smooth proper with geometrically connected fibers of any dimension), $\pi_{\ast}(\mathbf{G}_{m,X})$ is identified with $\mathbf{G}_{m,S}$, as you know. Likewise, $\pi_{\ast}({\rm{GL}}_{n,X}) = {\rm{GL}}_{n,S}$ for such $\pi$ for any $n > 0$. | |
| Nov 5, 2017 at 21:00 | comment | added | Jason Starr | No, it is not affine. Already for $X\to S$ a smooth, projective morphism whose geometric fibers are connected curves of genus $g>0$, for $G$ the multiplicative group, the pushforward $\pi_*G$ is an extension of an etale group scheme (basically $\mathbb{Z}$) by the relative Jacobian of $X/S,$ and this is an Abelian scheme over $S.$ | |
| Nov 5, 2017 at 20:50 | history | asked | Danny | CC BY-SA 3.0 |