Timeline for push-forward of linear algebraic group schemes
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
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| Nov 7, 2017 at 19:48 | comment | added | Jason Starr | The previous comment was wrong. Again I was thinking of the pushforward of the classifying stack $BG$, not the pushforward of $G.$ For the pushforward of $G,$ the obstruction space is $H^1(X_k,\sigma^*(\Omega_{W/k})^\vee).$ If $W$ is a smooth, linear group scheme of $X$, then there exists a locally free sheaf $\mathfrak{w}$ on $X$ such that $\Omega_{W/k}$ is the pullback of $\mathfrak{w}^\vee.$ Thus, the obstruction space is $H^1(X_k,\mathfrak{w}^\vee).$ This is typically nonzero. So you were correct: it seems difficult to deduce smoothness (unless $X/S$ is a family of genus $0$ curves). | |
| Nov 7, 2017 at 11:27 | comment | added | Jason Starr | If $W$ is smooth over $S$, and if $\pi$ is proper, finitely presented, and flat of relative dimension $\leq 1,$ then it should be possible to prove smoothness of $\pi_*W\to S$ by infinitesimal deformation theory. For a geometric point $\text{Spec}\ k\to S$ and for a section $\sigma:X_k\to W_k$, the obstruction space is $H^2(X_k,\sigma^*(\Omega_{W/K})^\vee).$ This is zero if the relative dimension is $\leq 1.$ | |
| Nov 6, 2017 at 21:13 | history | answered | anon19 | CC BY-SA 3.0 |