Scheme (mathematics)

The theory of schemes was pioneered by Alexander Grothendieck. The foundations of scheme theory were initially organized in Grothendieck's multi-volume work Éléments de Géométrie Algébrique with the assistance of Jean Dieudonné.

Roughly speaking, a scheme is a topological space which is locally affine; that is, a scheme has the local structure of the so-called affine schemes, i.e. of spectra of rings endowed with Zariski topologies.

The Category of Schemes

A scheme $(X,{\mathcal {O}}_{X})$ consists of a topological space $X$ together with a sheaf ${\mathcal {O}}_{X}$ of rings (called the structural sheaf on $X$) such that every point of $X$ has an open neighborhood $U$ such that the locally ringed space $(U,{\mathcal {O}}_{X}\vert _{U})$ is isomorphic to an affine scheme.

Projective Schemes constitute an important class of schemes, especially for the study of curves.

The category of schemes is defined by taking morphisms of schemes to be morphisms of locally ringed spaces. Many kinds of morphisms of schemes are characterized affine-locally, in the sense that

Gluing Properties

The notion of "gluing" is one of the central ideas in the theory of schemes. Let $S$ be a scheme, and $(X_{i})_{i\in I}$ a family of $S$-schemes. If we're given families $(X_{ij})_{j\in I}$ and $S$-isomorphisms $f_{ij}:X_{ij}\to X_{ji}$ such that: $f_{ii}=id_{X_{i}}$, $f_{ij}(X_{ij}\cap X_{ik})=X_{ji}\cap X_{jk}$, and $f_{ik}=f_{jk}\circ f_{ij}$ on $X_{ij}\cap X_{ik}$ for all $i,j,k\in I$, then there is an $S$-scheme $X$ together with $S$-immersions $g_{i}:X_{i}\to X$ such that $g_{i}=g_{j}\circ f_{i}j$ on $X_{ij}$ and so that $X=\bigcup _{i\in I}g_{i}(X_{i})$. This scheme $X$ is called the gluing over $S$ of the $X_{i}$ along the $X_{ij}$.

The $S$-scheme $X$ is universal for the property above: i.e., for any $S$-scheme $Z$ and family of morphisms $u_{i}:X_{i}\to Z$ such that $u_{i}=u_{j}\circ f_{ij}$ on $X_{ij}$, then there is a unique morphism $u:X\to Z$ such that $u_{i}=u\circ g_{i}$. Moreover, if $Z$ is a scheme, then giving a morphism $u:Z\to X$ is equivalent to giving an open covering $(Z_{i})_{i\in I}$ of $Z$ and morphisms $u_{i}:Z_{i}\to X_{i}$ such that $u_{j}=f_{ij}\circ u_{i}$ on $Z_{i}\cap Z_{j}$.