A scheme is a space that 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 formal duals of rings. If one takes another subcanonical Grothendieck topology $\tau$ on $Aff$ then one talks about $\tau$-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 schemes that naturally appear for instance also in differential geometry, in synthetic differential geometry and many other topics.
Throughout this article, “ring” will mean “commutative ring with unit”.
A scheme is a locally ringed space $(X, \mathcal{O}_X)$ with an open cover (as locally ringed spaces), by affine schemes: the spectra $Spec A = (|Spec A|, \mathcal{O}_{Spec A})$ of unital commutative rings.
A morphism $f : X \to Y$ of schemes is a morphism of the underlying locally ringed spaces. This means it is a morphism of ringed spaces such that for each point $x \in X$ the induced map of local rings
is local (in that it carries the maximal ideal to the maximal ideal). See functor of points.
(k-ring, k-functor,affine k-scheme)
For a ring $k$ the 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 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 affine line.
For the full and faithful contravariant functor
$Sp_k A$ (and every isomorphic functor) is called an 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 function on a $k$-scheme $X$ 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.
The category of $k$-functors has limits.
The terminal object is $e:R\mapsto\{\varnothing\}$. Products and pullbacks are computed component-wise.
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 scalar extension.
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
and
For a transformation $u:Y\to X$ of $k$-functors and $Z\subseteq X$ a subfunctor we define
A subfunctor $Y\subseteq X$ is called open subfunctor resp. closed subfunctor if for every transformation $u:T\to X$ we have $u^{-1}(Y)$ is of the form $V(E)$ resp. $D(E)$.
A $k$-functor $X$ is called a $k$-scheme if the following two conditions hold:
($X$ is a sheaf for the Zariski Grothendieck topology? on $M_k^{op}$) For all $k$-rings and all families $(f_i)_i$ such that $R=\coprod_i R f_i$ 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 $x_i$.
($X$ 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 $X(K)=\coprod_i U_i(K)$.
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.
The fundamental theorem on morphisms of schemes asserts that there is a fully faithful functor from the category $Sch$ of schemes to the category $Aff:=Psh(C Ring^{op})$ of presheaves on the opposite category of commutative rings given by
This identifies schemes with those presheaves on CRing${}^{op}$ that
The standard reference for the functor-of-points approach to schemes is Demazure-Gabriel.
Different authors take different approaches to the underlying set-theoretic issues. The astute reader will have noticed that we consider the category of 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 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-Gödel 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.
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 schemes 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 stacks of Deligne-Mumford and Artin, the dg-schemes of Kapranov, the derived schemes of Jacob Lurie, the higher algebraic stacks of Toën–Vezzosi, almost schemes (Ofer Gabber and Lorenzo Ramero), formal schemes (Cartier–Grothendieck), locally affine spaces in the fpqc, fppf or étale topology (Grothendieck), algebraic spaces, etc.
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.
Terminology: EGA says prescheme, for what we call algebraic scheme, and says scheme for what we call separated scheme.
Robin Hartshorne, Algebraic geometry, Springer
Qing Liu, Algebraic geometry and arithmetic curves, 592 pp. Oxford Univ. Press 2002
D. Eisenbud, J. Harris, The geometry of schemes, Springer Grad. Texts in Math.
David Mumford, Red book of varieties and schemes
Amnon Neeman, Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345
William Fulton, Intersection theory, Springer 1984
Ulrich Görtz, Torsten Wedhorn, Algebraic geometry I. Schemes with examples and exercises, Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010. viii+615 pp. Springerlink book
M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 (functor of points approach, mainly)
Michel Demazure, lectures on p-divisible groups web
Ravi Vakil’s Berkeley course notes
Paul Goerss, Topological Algebraic Geometry - A Workshop – at the beginning one fins a quick introduction in the light of its higher categorical generalizations
Wikipedia: scheme (mathematics).
MathOverflow: arbitrary-products-of-schemes-dont-exist, model-of-a-scheme-regular-over-the-generic-point, categorical-construction-of-the-category-of-schemes, when-is-an-algebraic-space-a-scheme, is-an-algebraic-space-group-always-a-scheme, connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov