Properties of morphisms - Revision history

Anton: /* Bjorn Poonen's table */

2012-02-18T02:11:07Z

Bjorn Poonen's table

← Older revision		Revision as of 18:11, 17 February 2012
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	The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.		The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.

			{{todo\|add column for local on the ''source''}}

	{\|class="wikitable"		{\|class="wikitable"

132.239.145.162: /* Effective descent classes */

2011-10-26T22:43:01Z

Effective descent classes

← Older revision		Revision as of 14:43, 26 October 2011
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	addthm(['separated','locally quasi-finite'],'representable')		addthm(['separated','locally quasi-finite'],'representable')
	</pre></code>		</pre></code>

	x

132.239.145.162: /* Effective descent classes */

2011-10-26T22:42:05Z

Effective descent classes

← Older revision		Revision as of 14:42, 26 October 2011
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	addthm(['separated','locally quasi-finite'],'representable')		addthm(['separated','locally quasi-finite'],'representable')
	</pre></code>		</pre></code>

			x

Anton: /* Effective descent classes */

2011-10-17T01:51:08Z

Effective descent classes

← Older revision		Revision as of 17:51, 16 October 2011
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	addthm(['quasi-affine'],'representable')		addthm(['quasi-affine'],'representable')
	addthm(['immersion'],'representable')		addthm(['immersion'],'representable')
			addthm(['separated','locally quasi-finite'],'representable')
	</pre></code>		</pre></code>

Anton: /* Bjorn Poonen's table */

2011-10-16T18:39:03Z

Bjorn Poonen's table

Show changes

Anton: /* Bjorn Poonen's table */

2011-10-16T18:26:59Z

Bjorn Poonen's table

← Older revision		Revision as of 10:26, 16 October 2011
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	\|-		\|-
	\|fpqc		\|fpqc
	\|Vis05, 2.34		\|[http://arxiv.org/abs/math/0412512 Vis05], 2.34
	\|Vis05, 2.35(i)		\|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(i)
	\|Vis05, 2.35(v)		\|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(v)
	\|		\|
	\|		\|

Anton: /* Bjorn Poonen's table */

2011-10-16T18:26:00Z

Bjorn Poonen's table

← Older revision		Revision as of 10:26, 16 October 2011
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	=Bjorn Poonen's table=		=Bjorn Poonen's table=

	The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page.		The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.

	{\|class="wikitable"		{\|class="wikitable"
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	\|Vis05, 2.35(v)		\|Vis05, 2.35(v)
	\|		\|
			\|
			\|-
			\|good moduli space
			\|[http://arxiv.org/abs/0804.2242 Alp08] 4.1
			\|YES
			\|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(i)
			\|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(ii)
	\|		\|
	\|-		\|-

Anton: /* Theorems */

2011-10-08T21:33:17Z

Theorems

← Older revision		Revision as of 13:33, 8 October 2011
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	addthm(['good moduli space'],'f_*(coh)=coh')		addthm(['good moduli space'],'f_*(coh)=coh')
	addthm(['good moduli space'],'universally closed')		addthm(['good moduli space'],'universally closed')
	addthm(['fppf'],'open')		addthm(['fppf'],'open') # EGAIV 2.4.6
	addthm(['smooth','regular base'],'regular source')		addthm(['smooth','regular base'],'regular source')
	define('smooth',['fppf','geometrically regular fibers']) # flat + geometrically regular fibers doesn't imply formally smooth, right?		define('smooth',['fppf','geometrically regular fibers']) # flat + geometrically regular fibers doesn't imply formally smooth, right?
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	addthm(['proper','birational','quasi-finite','normal base','representable'],'isomorphism') #Zariski's Main Theorem ... deduce from other one?		addthm(['proper','birational','quasi-finite','normal base','representable'],'isomorphism') #Zariski's Main Theorem ... deduce from other one?
	addthm(['proper','quasi-finite'],'finite') # somewhere in EGA IV.18.12		addthm(['proper','quasi-finite'],'finite') # somewhere in EGA IV.18.12
			addthm(['affine','universally closed'],'integral') # EGAIV 18.12.8
			addthm(['quasi-finite','separated'],'quasi-affine') # EGAIV 18.12.12
	</pre></code>		</pre></code>

			==Desirables==
			* It's be nice to handle compositions of morphisms. For example, ZMT (EGAIV 18.12.13) says that a quasi-compact quasi-separated quasi-finite separated morphism is an open immersion followed by a finite morphism. (note: must have representable) Perhaps the right thing to do is to simply have a syntax where a composition is represented by a tuple:
			addthm(['representable','quasi-separated','quasi-compact','quasi-finite','separated'],('open immersion','finite'))
			then we could also say things like
			define('quasi-affine',('open immersion','affine'))
			addthm(('','isomorphism'),'')
			addthm(('isomorphism',''),'')
			Note that we would want to redefine immersions as compositions of open then closed, and make open/closed immersions atomic.

			The problem is that this could lead to combinatorial explosion, with properties like <code>(['quasi-compact','immersion'],[('smooth','separated'),'finite type'])</code> to mean "a quasi-compact open immersion, followed by a finite type morphisms which is a composition of a smooth morphism and a separated morphism".

			* Can prefixes like "locally" and "universally" be handled well? For example, we should have <code>addthm('','locally ')</code>. This seems like it would be hard, since it would require trying to handle base change in general, which would be tricky. However, if there were a way to do it, it would make sense to encode descent theorems.

	==Effective descent classes==		==Effective descent classes==

Anton: /* Definitions and non-theorems */

2011-10-08T21:06:11Z

Definitions and non-theorems

← Older revision		Revision as of 13:06, 8 October 2011
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	It would be nice to set things up so that the following trivial theorems didn't have to be loaded.		It would be nice to set things up so that the following trivial theorems didn't have to be loaded.
	<code><pre>		<code><pre>
	addthm(['locally finite type','noetherian base'],'noetherian source')		addthm(['locally finite type','locally noetherian base'],'locally noetherian source')
	addthm(['locally finite type','noetherian base'],'locally of finite presentation')		addthm(['locally finite type','locally noetherian base'],'locally of finite presentation')
	addthm(['affine'],'separated')		addthm(['affine'],'separated')
	addthm(['closed immersion'],'finite type')		addthm(['closed immersion'],'finite type')

Anton: /* Definitions and non-theorems */

2011-10-08T17:25:46Z

Definitions and non-theorems

← Older revision		Revision as of 09:25, 8 October 2011
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	define('quasi-separated',['separated diagonal'])		define('quasi-separated',['separated diagonal'])
	define('separated',['proper diagonal'])		define('separated',['proper diagonal'])
	# define('separated',['quasi-compact diagonal']) ~~# right definition?~~		define('quasi-separated',['quasi-compact diagonal'])
	define('smooth',['formally smooth','locally of finite presentation'])		define('smooth',['formally smooth','locally of finite presentation'])
	define('unramified',['formally unramified','locally of finite presentation'])		define('unramified',['formally unramified','locally of finite presentation'])