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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{scheme} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{as_locally_ringed_spaces}{As locally ringed spaces}\dotfill \pageref*{as_locally_ringed_spaces} \linebreak \noindent\hyperlink{as_sheaves_on_}{As sheaves on $CRing^{op}$}\dotfill \pageref*{as_sheaves_on_} \linebreak \noindent\hyperlink{translation_between_the_two_approaches}{Translation between the two approaches}\dotfill \pageref*{translation_between_the_two_approaches} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak \noindent\hyperlink{underlying_topological_space_vs_underlying_locale}{Underlying topological space vs. underlying locale}\dotfill \pageref*{underlying_topological_space_vs_underlying_locale} \linebreak \noindent\hyperlink{simplicial_schemes}{Simplicial schemes}\dotfill \pageref*{simplicial_schemes} \linebreak \noindent\hyperlink{superschemes}{Super-schemes}\dotfill \pageref*{superschemes} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{standard_monographs}{Standard monographs}\dotfill \pageref*{standard_monographs} \linebreak \noindent\hyperlink{other_references}{Other references}\dotfill \pageref*{other_references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{scheme} is a [[space]] that \emph{locally} looks like a particularly simple [[ringed space]]: an [[affine scheme]]. This can be formalised either within the category of [[locally ringed spaces]] or within the category of presheaves of sets on the category of affine schemes $Aff$. The notion of scheme originated in [[algebraic geometry]] where it is, since [[Grothendieck]]`s revolution of that subject, a central notion. However, the idea that A scheme is a ringed space that is locally isomorphic to an affine space. is much more general and need not be restricted to the locality in Zariski topology and to a notion of affine spaces that are [[duality|formal duals]] of rings. If one takes another subcanonical Grothendieck topology $\tau$ on $Aff$ then one talks about \textbf{$\tau$-locally affine spaces}. More generally one can take another ``category of local models'' $Loc$ replacing $Aff$ and suitable topology and consider sheaves on it, as locally affine space in this generalized sense. The category $Loc$ can sometimes be represented by ringed spaces of special type and the gluing can be sometimes made in a genuine (classical, not Grothendieck) topology, within the category of ringed spaces. For instance a smooth [[manifold]] is a ringed space locally isomorphic to a ``smooth affine space'' $\mathbb{R}^n$, with its standard smooth structure. The standard concept of scheme in [[algebraic geometry]] is therefore usefully understood as a special case of [[generalized scheme]]s that naturally appear for instance also in [[differential geometry]], in [[synthetic differential geometry]] and many other topics. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Throughout this article, ``ring'' will mean ``commutative ring with unit''. \hypertarget{as_locally_ringed_spaces}{}\subsubsection*{{As locally ringed spaces}}\label{as_locally_ringed_spaces} A \textbf{scheme} is a [[locally ringed space]] $(X, \mathcal{O}_X)$ such that, for every point $x$ of $X$, there is an open subset $U$ of $X$ with $x \in U$ such that the locally ringed space $(U, \mathcal{O}_{X} | U)$ is isomorphic to an [[affine scheme]], that is to say, a commutative [[prime spectrum|ring spectrum]] $Spec A = (|Spec A|, \mathcal{O}_{Spec A})$. Here $\mathcal{O}_{X} | U$ denotes the restriction of $\mathcal{O}_{X}$ to $U$, that is to say, the sheaf $i^{*}(\mathcal{O}_{X})$, where $i: U \hookrightarrow X$ is the inclusion map, and $i^{*}$ is the corresponding [[inverse image]] functor from the category of sheaves on $X$ to the category of sheaves on $U$. A [[morphism of schemes]] $f : (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ is a morphism of the underlying [[locally ringed space]]s. This means it is a morphism of [[ringed space|ringed spaces]] such that for each point $x \in X$ the induced map of [[local ring]]s \begin{displaymath} (\mathcal{O}_Y)_{f(x)} \to (\mathcal{O}_X)_x \end{displaymath} is \emph{local} (in that it carries the maximal ideal to the maximal ideal). See [[functor of points]]. \hypertarget{as_sheaves_on_}{}\subsubsection*{{As sheaves on $CRing^{op}$}}\label{as_sheaves_on_} \begin{defn} \label{}\hypertarget{}{} (k-ring, k-functor,affine k-scheme) For a ring $k$ the \emph{category of $k$-rings}, denoted by $M_k,$ is defined to be the category of commutative associative $k$-algebras with unit. This is equivalently the [[under category]] $k\downarrow CRing$ of pairs $(R,f:k\to R)$ where $R$ is a commutative ring and $f$ is a ring homomorphism. The \emph{category of $k$-functors}, denoted by $co Psh (M_k)$, is defined to be the category of covariant functors $M_k\to Set$. The forgetful functor $O_k:R\to R$ sending a $k$-ring to its underlying set is called \emph{affine line}. For the full and faithful contravariant functor \begin{displaymath} Sp_k:\begin{cases} M_k&\to& co Psh(M_k) \\ A&\mapsto& M_k(A,-) \end{cases} \end{displaymath} $Sp_k A$ (and every isomorphic functor) is called an \emph{affine $k$-scheme}. $Sp_k$ restricts to an equivalence between the categories of $k$-rings and the category $Aff Sch_k$ of affine $k$-schemes. We think of this category as of $M_k^{op}$. The functor $Sp_k$ commutes with limits and scalar extension (see below). Consequently $Aff Sch_k$ is closed under limits and base change. The affine line $O_k=M_k(k[t],-)$ is an affine $k$-scheme. A \emph{function on a} $k$-functor $X$ is defined to be an object $f\in O(X):=co Psh (M_k)(X,O_k)$. $O(X)$ is a $k$-ring by component-wise addition and -multiplication. \end{defn} \begin{remark} \label{}\hypertarget{}{} The category of $k$-functors has limits. The terminal object is $e:R\mapsto\{\varnothing\}$. Products and pullbacks are computed component-wise. \end{remark} \begin{remark} \label{}\hypertarget{}{} For $\phi:k\to k'$ the `'base change'` functor $(-)\otimes_k k':co Psh(M_k)\to co Psh(M_{k'})$ induced by $(-)\circ \phi:M_k\to M_{k'}$ given by postcompositions with $\phi$ is called \emph{scalar extension}. \end{remark} Now we come to the definition of not necessarily affine k-schemes For a $k$-functor $X\in coPsh(M_k)$ and $E\subseteq O(X)=M_k(X,O_k)$ a set of functions on $X$, Definition in [[k-ring]], we define \begin{displaymath} V(E)(R):=\{x\in X(R) | f\in E, f(x)=0\} \end{displaymath} and \begin{displaymath} D(E)(R):=\{x\in X(R)|f\in E, \text{the } \ f(x) \ \text{ generate the unit ideal of} R\} \end{displaymath} For a transformation $u:Y\to X$ of $k$-functors and $Z\subseteq X$ a subfunctor we define \begin{displaymath} u^{-1}(Z)(U):=\{y\in Y(R)|u(Y)\in Z(R)\} \end{displaymath} A subfunctor $Y\subseteq X$ is called \emph{open subfunctor} resp. \emph{closed subfunctor} if for every transformation $u:T\to X$ we have $u^{-1}(Y)$ is of the form $V(E)$ resp. $D(E)$. \begin{defn} \label{}\hypertarget{}{} A $k$-functor $X$ is called a $k$\emph{-scheme} if the following two conditions hold: \begin{enumerate}% \item ($X$ is a sheaf for the [[Zariski Grothendieck topology]] on $M_k^{op}$) For all $k$-rings and all families $(f_i)_i$ such that $R=\sum_i R f_i$ we have: if for all $x_i\in R[f_i^{-1}]$ such that the images of $x_i$ and $x_j$ coincide in $X(R[f_i^{-1} f_j^{-1}])$ there is a unique $x\in X(R)$ mapping to the $x_i$. \item ($X$ has a cover of Zariski open immersions of affine $k$-schemes) The exists a small family $(U_i)_i$ of open affine subfunctors of $X$ such that for all fields $K\in M_k$ we have that $X(K)=\bigcup_i U_i(K)$. \end{enumerate} \end{defn} \begin{remark} \label{}\hypertarget{}{} The category of $k$-schemes is closed under finite limits, forming open- and closed subfunctors, and scalar extension. As a subcategory of the category of Zariski sheaves, it is also closed under taking small coproducts. \end{remark} \hypertarget{translation_between_the_two_approaches}{}\subsubsection*{{Translation between the two approaches}}\label{translation_between_the_two_approaches} The \textbf{fundamental theorem on morphisms of schemes} asserts that there is a [[fully faithful functor]] from the category $Sch$ of schemes to $Psh(Aff) \equiv Psh(CRing^{op})$, the category of [[presheaf|presheaves]] on the category of affine schemes, or equivalently on the opposite of the category of commutative rings, given by \begin{displaymath} (X,\mathcal{O}_X)\mapsto Sch((|Spec (-)|,\mathcal{O}_{Spec(-)}),(X,\mathcal{O}_X)) \end{displaymath} This identifies schemes with those presheaves on [[CRing]]${}^{op}$ that \begin{enumerate}% \item are [[sheaf|sheaves]] with respect to the Zariski [[Grothendieck topology]] on $CRing^{op}$; \item have a [[cover]] by Zariski-open immersions of [[affine scheme]]s in the category of presheaves over $Aff$. \end{enumerate} The standard reference for the functor-of-points approach to schemes is \hyperlink{DG}{Demazure-Gabriel}. \begin{remark} \label{}\hypertarget{}{} Different authors take different approaches to the underlying set-theoretic issues. The astute reader will have noticed that we consider the category of \emph{all} functors $CRing \to Set$ -- which is not a locally small category. Nor is the category of sheaves on the site $CRing^{op}$ with its Zariski topology actually a Grothendieck topos. With due regard to such set-theoretic issues, this approach seems to be conceptually the simplest. Or, one could keep one's options open: for some suitable small category of rings left to one's discretion, for example the category of [[finitely presented object|finitely presented]] rings, one can consider schemes locally modelled on that category. Demazure-Gabriel steer a middle course involving [[universes]]: assuming two universes $U$ and $V$ with $\mathbb{N} \in U \in V$, one has a category of ``small rings'' (belonging to $U$) and a category of sets (belonging to $V$) and one considers functors $M \to Set$ from small rings (called ``models'') to (not necessarily small) sets. They remark that the device of using universes is really just a convenience that could mostly be dispensed with: one could work within the standard Bernays-G\"o{}del framework by assuming that the models are inclusive enough to hold various standard commutative algebra constructions (e.g., quotients, localizations, completions) while still remaining a small category. However, since they wish to avail themselves of direct limits in the category of models, they choose to work with universes instead. \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \begin{itemize}% \item [[schemes are sober]] \end{itemize} (\ldots{}) \hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations} In [[algebraic geometry]] this is a basic object of study, since the revolution of [[Grothendieck]]. There are generalizations like [[relative schemes]] (which are just objects in a [[slice category]] $Sch/S$), relative [[noncommutative scheme]]s in [[noncommutative algebraic geometry]] introduced by A. Rosenberg in terms of categories and covers defined using pairs of [[adjoint functors]], the generalized schemes of [[Nikolai Durov]], the [[algebraic stack]]s of [[Deligne-Mumford stack|Deligne-Mumford]] and Artin, the dg-schemes of Kapranov, the [[derived scheme]]s of [[Jacob Lurie]], the higher [[algebraic stack]]s of [[Bertrand Toen|Toën]]--Vezzosi, almost schemes (Ofer Gabber and Lorenzo Ramero), formal schemes (Cartier--Grothendieck), [[locally affine spaces]] in the fpqc, fppf or \'e{}tale topology (Grothendieck), [[algebraic spaces]], etc. See also [[generalized scheme]]. \hypertarget{underlying_topological_space_vs_underlying_locale}{}\subsubsection*{{Underlying topological space vs. underlying locale}}\label{underlying_topological_space_vs_underlying_locale} [[Jacob Lurie]] argues that underlying locale point of view is better than underlying topological space point of view, see [[schemes as locally affine structured (∞,1)-toposes]]. \hypertarget{simplicial_schemes}{}\subsubsection*{{Simplicial schemes}}\label{simplicial_schemes} \begin{itemize}% \item [[simplicial scheme]] \end{itemize} \hypertarget{superschemes}{}\subsubsection*{{Super-schemes}}\label{superschemes} \begin{itemize}% \item [[super-scheme]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Terminology: [[EGA]] says prescheme, for what we call algebraic scheme, and says scheme for what we call [[separated scheme]]. \hypertarget{standard_monographs}{}\subsubsection*{{Standard monographs}}\label{standard_monographs} \begin{itemize}% \item Robin Hartshorne, \emph{Algebraic geometry}, Springer \item Qing Liu, \emph{Algebraic geometry and arithmetic curves}, 592 pp. Oxford Univ. Press 2002 \item D. Eisenbud, J. Harris, \emph{The geometry of schemes}, Springer Grad. Texts in Math. \item [[David Mumford]], \emph{Red book of varieties and schemes} \item Amnon Neeman, \emph{Algebraic and analytic geometry}, London Math. Soc. Lec. Note Series \textbf{345} \item William Fulton, \emph{Intersection theory}, Springer 1984 \item Ulrich G\"o{}rtz, Torsten Wedhorn, \emph{Algebraic geometry I. Schemes with examples and exercises}, Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010. viii+615 pp. \href{http://www.springerlink.com/content/kt5u74/#section=748613&page=1}{Springerlink book} \item M. Demazure, P. Gabriel, \emph{Groupes algebriques}, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 (functor of points approach, mainly) \end{itemize} \begin{itemize}% \item Michel Demazure, lectures on p-divisible groups \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web} \item [[EGA]], [[FGA explained]] \end{itemize} \hypertarget{other_references}{}\subsubsection*{{Other references}}\label{other_references} \begin{itemize}% \item Ravi Vakil's Berkeley \href{http://math.stanford.edu/~vakil/0708-216}{course notes} \item [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]] -- at the beginning one finds a quick introduction in the light of its higher categorical generalizations \item Wikipedia: \href{http://en.wikipedia.org/wiki/Scheme_%28mathematics%29}{scheme (mathematics)}. \end{itemize} MathOverflow: \href{http://mathoverflow.net/questions/9134/arbitrary-products-of-schemes-dont-exist-do-they}{arbitrary-products-of-schemes-dont-exist}, \href{http://mathoverflow.net/questions/32196/model-of-a-scheme-regular-over-the-generic-point}{model-of-a-scheme-regular-over-the-generic-point}, \href{http://mathoverflow.net/questions/26506/categorical-construction-of-the-category-of-schemes}{categorical-construction-of-the-category-of-schemes}, \href{http://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme}{when-is-an-algebraic-space-a-scheme}, \href{http://mathoverflow.net/questions/8918/is-an-algebraic-space-group-always-a-scheme}{is-an-algebraic-space-group-always-a-scheme}, \href{http://mathoverflow.net/questions/89475/connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov}{connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov} category: algebraic geometry [[!redirects schemes]] [[!redirects algebraic scheme]] [[!redirects algebraic schemes]] \end{document}