Functorial point of view for formal schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:30:36Z http://mathoverflow.net/feeds/question/33070 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33070/functorial-point-of-view-for-formal-schemes Functorial point of view for formal schemes Ricky 2010-07-23T09:28:17Z 2010-07-23T10:57:25Z <p>Giving a scheme is the same as giving the corresponding functor from the category of rings to the category of set, and there are characterization of what functors arise in this way. This is explained in the book by Demazure and Gabriel. This "functorial point of view" is sometimes very useful, so I was wondering whether there is something similar for formal schemes rather that algebraic schemes. I searched for this but I wasn't able to find anything, it seems that Demazure and Gabriel don't speak about formal schemes at all.</p> <p>Ricky</p> http://mathoverflow.net/questions/33070/functorial-point-of-view-for-formal-schemes/33076#33076 Answer by Timo Schürg for Functorial point of view for formal schemes Timo Schürg 2010-07-23T10:15:30Z 2010-07-23T10:15:30Z <p>You can try having a look at this paper:</p> <p><a href="http://arxiv.org/abs/math.AT/0011121" rel="nofollow">http://arxiv.org/abs/math.AT/0011121</a></p> <p>It's the most functorial-minded paper on algebraic geometry I'ver ever seen. It's written by an algebraic topologist. He cares mostly about affine and formal schemes. </p> <p>The definition you're looking for is in section 4 of the paper. The functorial point of view for a formal scheme is a small filtered colimit of schemes, the colimit taken in the functor category. </p> http://mathoverflow.net/questions/33070/functorial-point-of-view-for-formal-schemes/33079#33079 Answer by Zoran Škoda for Functorial point of view for formal schemes Zoran Škoda 2010-07-23T10:57:25Z 2010-07-23T10:57:25Z <p>There are many nonequivalent generalities in which one can define a formal scheme, for example the definitions in Hartshorne and in EGA are not quite the same. (I use in this answer some parts of my own editing in nlab's <a href="http://ncatlab.org/nlab/show/formal+scheme" rel="nofollow">entry</a> where more references can be found). In my understanding, whatever the definition is, the category of formal schemes is a realization of certain subcategory of Ind-schemes. Typically one requires at least that the [[ind-object]] in the subcategory may be represented by a diagram whose connecting morphisms are closed immersions of schemes. A pretty modern treatment is in </p> <ul> <li>A. Beilinson, V. Drinfel'd, <em>Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system</em>, preliminary version (<a href="http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf" rel="nofollow">pdf</a>)</li> </ul> <p>Some subcategories of Ind-objects in many algebraic categories can be described by putting the topology on algebraic objects. Thus the complete local rings, or more general the pseudocompact case, in the Grothendieck's approach to local schemes. One can use a topological version of Yoneda on rings to get a nice theory of formal schemes, over an arbitrary ring:</p> <ul> <li>B. Pareigis, R. A. Morris, <em>Formal groups and Hopf algebras over discrete rings</em>, Trans. Amer. Math. Soc. <strong>197</strong> (1974), 113--129 (<a href="http://dx.doi.org/10.2307/1996930" rel="nofollow">doi</a>).</li> </ul> <p><a href="http://ncatlab.org/nlab/show/Nikolai+Durov" rel="nofollow">Nikolai Durov</a> suggests to use directly the Gabriel-Demazure approach but not over Aff but over the opposite to the category of pairs (commutative ring, nilpotent ideal). Formal schemes should be an appropriate subcategory of that category of presheaves. That larger category (but without singling out there the smaller subcategory which would correspond more precisely to Grothendieck's formal schemes) is sketched in ch. 7-9 of </p> <ul> <li>N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, <em>A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra</em>, Journal of Algebra 309, n. 1, 318--359 (2007) (<a href="http://dx.doi.org/10.1016/j.jalgebra.2006.08.025" rel="nofollow">doi:jalgebra</a>) (<a href="http://front.math.ucdavis.edu/math.RT/0604096" rel="nofollow">math.RT/0604096</a>).</li> </ul>