Math 193a: Algebraic Stacks, Fall 2011 - Revision history

Anton at 16:15, 30 November 2011

2011-11-30T16:15:39Z

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Anton: /* Set 8 (in progress) */

2011-11-30T16:14:19Z

Set 8 (in progress)

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	3. Suppose $\X$ is an algebraic stack with quasi-affine diagonal (e.g. any stack with separated and quasi-finite diagonal). Show that for any two ''schemes'' $U$ and $V$ and any morphisms $U\to \X$, $V\to \X$, the fiber product $U\times_\X V$ is a ''scheme''.		3. Suppose $\X$ is an algebraic stack with quasi-affine diagonal (e.g. any stack with separated and quasi-finite diagonal). Show that for any two ''schemes'' $U$ and $V$ and any morphisms $U\to \X$, $V\to \X$, the fiber product $U\times_\X V$ is a ''scheme''.

	== Set 8 (~~in progress~~) ==		== Set 8 (due Dec. 10) ==
	1. ~~Correspondence~~ between $GL_r$-torsors and rank $r$ vector bundles.		1. Show that there is a correspondence between $GL_r$-torsors on a scheme $X$ and rank $r$ vector bundles on $X$. Specifically, given a $GL_r$-torsor $P$ on $X$, show that the sheaf of sections of $P\times^{GL_r}\O_X^r$ is a rank $r$ vector bundle. Given a rank $r$ vector bundle $\E$, show that $Isom(\E,\O^r)$ is a $GL_r$-torsor. Finally, show that these operations are inverse.

	2. $H^1$ ~~and torsors?~~		2. (Stackification). Suppose $\C$ is a site. Suppose $\D\to \C$ is a prestack (i.e. for any covering $X\to Y$ in $\C$, the functor $\D(Y)\to \D(X\to Y)$ is fully faithful). Define $\D^+\to \C$ to be the fibered category whose objects are triples $(Y\in \C, \{X\to Y\}\in Cov(Y), F\in \D(X\to Y))$, and whose morphisms consist of a morphism $f:Y\to Y'$, a common refinement $Z\to Y$ of the covers $X\to Y$ and $X\times_{Y'}Y\to Y$, and a descent-data-respecting isomorphism between the restrictions of $F$ and $f^{-1}F'$ to $Z$. Show that if $\E$ is any stack on $\C$, there is an equivalence $HOM_\C(\D,\E)\to HOM_\C(\D^+,\E)$.

	3~~. Stackification~~. Lemma for guessing stackification.		<!-- 3. Lemma for guessing stackification. -->

	== Other exercises ==		== Other exercises ==

Anton: Reverted edits by 109.230.216.60 (talk) to last revision by Anton

2011-11-22T19:21:47Z

Reverted edits by 109.230.216.60 (talk) to last revision by Anton

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2011-11-22T19:16:42Z

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Anton: Undo revision 786 by 107.20.161.197 (talk)

2011-11-22T18:58:32Z

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2011-11-22T17:30:20Z

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2011-11-22T17:11:39Z

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2011-11-22T13:52:47Z

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2011-11-22T12:50:20Z

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2011-11-22T11:39:24Z

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