Timeline for Fiber product arising from reductive group action on varieties

6 events

when toggle format	what		by	license	comment
Nov 14, 2019 at 11:03	comment	added	Minseon Shin		I believe it's called a "contracted product" in the literature.
Nov 14, 2019 at 9:17	comment	added	Angelo		A commonly used notation, which avoids the confusion with fibered products, is $G \times^{P}Y$.
Nov 14, 2019 at 4:40	comment	added	Aaron Landesman		$G \times_P Y$ is smooth because $G \times Y$ is a $P$-torsor over $G \times_P Y$. Then $G$ and $Y$ are both smooth, so $G \times Y$ is smooth. In general if $f: X \to Z$ is a smooth surjection then $X$ being smooth implies $Z$ is smooth. In fact, you only need that $f$ is fppf, though this fact is harder and the homological characterization of regularity. So smoothness of $G \times Y$ implies smoothness of $G \times_P Y$.
Nov 14, 2019 at 3:06	comment	added	Peter McNamara		The notation is unfortunate because of the clash with the fibre product notation. Here one defines $G\times_PY=(G\times Y)/\sim$ where $(gp,y)\sim (g,py)$ for all $p\in P$.
Nov 14, 2019 at 0:15	history	edited	YCor	CC BY-SA 4.0	added tag, specified "reductive" in title
Nov 14, 2019 at 0:05	history	asked	Xuqiang QIN	CC BY-SA 4.0