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1. • CommentRowNumber1.
   • CommentAuthorKevin Lin
   • CommentTimeJul 20th 2010
   • PermaLink
   Author: Kevin Lin
   Format: HtmlAdded stub for [[GAGA]].

   Added stub for GAGA.
2. • CommentRowNumber2.
   • CommentAuthorJohn Baez
   • CommentTimeJul 25th 2010
   • PermaLink
   Author: John Baez
   Format: MarkdownItexGreat!

   Great!

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI'd like to boost the _[[GAGA]]_-related stuff on the $n$Lab a bit. So far I have at [[GAGA]] itself: * polished the Idea-section just a tad * started a subsection "Theorems", currently it only points to [[comparison theorem (étale cohomology)]] (which has been existing for a while) and [[Chow's theorem]] (which is a stub I just created) * added a pointer to [[Jean-Pierre Demailly]], _Analytic methods in algebraic geometry_, lecture notes 2009 ([pdf](http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth.pdf)) At _[[analytic space]]_, in the Idea-section, I tried to bring the point out more clearly by expanding slight. Now it reads as such: > Analytic spaces are [[spaces]] that are locally modeled on [[Isbell duality|formal duals]] of sub-[[associative algebra|algebras]] of [[power series]] algebras on elements with certain [[convergence]] properties with respect to given [[seminorms]]. This is in contrast to [[algebraic spaces]] ([[algebraic varieties]], [[schemes]]) where no [[convergence]] properties are considered > In [[complex number|complex]] [[analytic geometry]] analytic spaces -- [[complex analytic space]] -- are a vast generalization of complex [[analytic manifolds]] and are usually treated in the formalism of [[locally ringed spaces]]. In this case the _[[GAGA]]-principle_ closely relates [[complex analytic geometry]] with [[algebraic geometry]] over the [[complex numbers]]. I've been also cross-linking a bit further. More needs to be done here.

   I’d like to boost the GAGA-related stuff on the $n$Lab a bit. So far I have

   at GAGA itself:

   • polished the Idea-section just a tad

   • started a subsection “Theorems”, currently it only points to comparison theorem (étale cohomology) (which has been existing for a while) and Chow’s theorem (which is a stub I just created)

   • added a pointer to Jean-Pierre Demailly, Analytic methods in algebraic geometry, lecture notes 2009 (pdf)

   At analytic space, in the Idea-section, I tried to bring the point out more clearly by expanding slight. Now it reads as such:

   Analytic spaces are spaces that are locally modeled on formal duals of sub-algebras of power series algebras on elements with certain convergence properties with respect to given seminorms. This is in contrast to algebraic spaces (algebraic varieties, schemes) where no convergence properties are considered

   In complex analytic geometry analytic spaces – complex analytic space – are a vast generalization of complex analytic manifolds and are usually treated in the formalism of locally ringed spaces. In this case the GAGA-principle closely relates complex analytic geometry with algebraic geometry over the complex numbers.

   I’ve been also cross-linking a bit further. More needs to be done here.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded under "Theorems" also a pointer to _[[analytification]]_ and added the references discussing GAGA in higher geometry (stacks) * {#Lurie04} [[Jacob Lurie]], _[[Tannaka duality for geometric stacks]]_, ([arXiv:math.AG/0412266](http://arxiv.org/abs/math/0412266)) * {#Hall11} [[Jack Hall]], _Generalizing the GAGA Principle_ ([arXiv:1101.5123](http://arxiv.org/abs/1101.5123)) * {#GeraschenkoZureickBrown12} [[Anton Geraschenko]], David Zureick-Brown, _Formal GAGA for good moduli spaces_ ([arXiv:1208.2882](http://arxiv.org/abs/1208.2882))

   added under “Theorems” also a pointer to analytification and added the references discussing GAGA in higher geometry (stacks)

   • {#Lurie04} Jacob Lurie, Tannaka duality for geometric stacks, (arXiv:math.AG/0412266)

   • {#Hall11} Jack Hall, Generalizing the GAGA Principle (arXiv:1101.5123)

   • {#GeraschenkoZureickBrown12} Anton Geraschenko, David Zureick-Brown, Formal GAGA for good moduli spaces (arXiv:1208.2882)

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeMay 22nd 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> This is in contrast to algebraic spaces (algebraic varieties, schemes) where no convergence properties are considered This is quite a nonsense. In algebraic geometry one has regular, thus polynomial function, hence they __do__ converge. But one can consider formal spaces, like formal schemes, formal stacks...then one has formal power series without convergence and for them a similar statement makes sense. The present one should be removed.

   This is in contrast to algebraic spaces (algebraic varieties, schemes) where no convergence properties are considered

   This is quite a nonsense. In algebraic geometry one has regular, thus polynomial function, hence they do converge. But one can consider formal spaces, like formal schemes, formal stacks…then one has formal power series without convergence and for them a similar statement makes sense. The present one should be removed.