"Every scheme as a sheaf" references? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:48:32Z http://mathoverflow.net/feeds/question/10314 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references "Every scheme as a sheaf" references? Ilya Nikokoshev 2009-12-31T19:39:08Z 2010-01-12T17:32:34Z <p>I have sometimes hard time reading papers that are written in the language of schemes being replaced by the functors they represent (I have especially homotopy scheme theory in mind). </p> <p>I think the topic is connected to topoi and Grothendieck topologies, but for now I'm looking for something simple, just the working overview of the language of representable functors, what is a scheme, etc.</p> http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10316#10316 Answer by Charles Siegel for "Every scheme as a sheaf" references? Charles Siegel 2009-12-31T19:55:43Z 2009-12-31T19:55:43Z <p>You can start with <a href="http://homepage.sns.it/vistoli/descent.pdf" rel="nofollow">these</a> notes by Vistoli, which talk about that stuff in the direction of doing stacks and descent theory. The other articles in FGA explained might be useful, as they do a lot of moduli space construction (ie, prove that a given functor is representable). Another place to look would be in Eisenbud and Harris's "Geometry of Schemes" where they characterize schemes among sheaves on CommRing, towards the end of the book.</p> http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10317#10317 Answer by Anton Geraschenko for "Every scheme as a sheaf" references? Anton Geraschenko 2009-12-31T20:12:11Z 2009-12-31T20:12:11Z <p>Chapter I of Demazure and Gabriel's <a href="http://books.google.com/books?id=RDKRyP00aoMC&amp;printsec=frontcover&amp;f=false" rel="nofollow">Introduction to Algebraic Geometry and Algebraic Groups</a> develops the basic notions of algebraic geometry from the functor of points approach.</p> http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10330#10330 Answer by Shizhuo Zhang for "Every scheme as a sheaf" references? Shizhuo Zhang 2009-12-31T22:53:36Z 2010-01-01T20:58:39Z <p>It is a very nice question. Functor view point algebraic geometry was proposed by Gabriel and later developed by Grothendieck. </p> <p>Actually, Kontsevich and Rosenberg developed noncommutaive algebraic geometry completely based on this point of view explicitly. They take the presheaf $\text{Alg}^{op} \rightarrow \text{Set}$ as a noncommutative space and developed flat descent theory, the theory of noncommutative smooth space, theory of noncommutative stack. As an interesting example, they defined noncommutative grassmannian, group scheme, general flag vairety as a presheaves and using descent theory of quasi coherent sheaves to glue affine presheaves together according to the so called "smooth topology".</p> <p>I attended a lecture course last semester, he proved a theorem which "shows" that "One can do algebraic geometry only using presheaves rather than sheaves, if one need sheaves, just take sheafification and all the properties will hold".</p> <p>There are the following references:</p> <ul> <li><a href="http://webdoc.sub.gwdg.de/ebook/serien/e/mpi%5Fmathematik/2004/35.ps" rel="nofollow">Noncommutative spaces</a> (from page 15)</li> <li><a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=2333" rel="nofollow">Noncommutative stacks</a></li> <li><a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=3621" rel="nofollow">Noncommutative Grassmannian and related constructions</a> (from page 12 is interesting)</li> </ul> <p>All these papers are available in Max Plank preprint series (search Kontsevich or Rosenberg in "author" and leave other blank empty).</p> <p>If you take a look at the first paper, just disregard the notion of "Q-category" which is a technique tool to generalize grothendieck topologies because in noncommutative case, flat morphism does not respect to base change in general. </p> http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10428#10428 Answer by Anweshi for "Every scheme as a sheaf" references? Anweshi 2010-01-02T00:37:04Z 2010-01-02T00:37:04Z <ol> <li><p>Read the excellent proof of Yoneda's lemma in wikipedia.</p></li> <li><p>Read the short section on "functor of points" in Mumford's "Lectures on curves on an algebraic surface. For reviewing the basics of schemes etc., Mumford's Red book will be useful.</p></li> </ol> <p>I think this is the simplest and most compact way of getting started. Then one can move on with the sources suggested by other people in other answers to this question.</p> <p>I hope my answer is not "too introductory".</p> http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10646#10646 Answer by Kevin Lin for "Every scheme as a sheaf" references? Kevin Lin 2010-01-04T00:50:36Z 2010-01-04T00:50:36Z <p>I think Eisenbud-Harris is probably the best place to get started. Alternatively, I think <a href="http://www.math.ucdavis.edu/~osserman/classes/256A/notes/yoneda.pdf" rel="nofollow">these</a> <a href="http://www.math.ucdavis.edu/~osserman/classes/256A/notes/fiber-prods.pdf" rel="nofollow">notes</a> by Brian Osserman are nice.</p> http://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/11563#11563 Answer by mdeland for "Every scheme as a sheaf" references? mdeland 2010-01-12T17:32:34Z 2010-01-12T17:32:34Z <p>You may also find the notes "Introduction to Functorial Algebraic Geometry" useful. They are based on a course given by Groethendieck - and can be found <a href="http://www.math.jussieu.fr/~leila/grothendieckcircle/FuncAlg.pdf" rel="nofollow">here</a>. Unfortunately they're a little rough around the edges sometimes and slightly dizzying to read because of the scanning.</p>