Scheme (algebraic geometry)

Classical algebraic geometry deals with subsets of affine or projective space that arise as zero sets of finitely many polynomials ( algebraic varieties ). The geometric objects are therefore solution sets of algebraic systems of equations. The term scheme is motivated by considering not only solutions in a fixed algebraically closed field , but solutions in arbitrary rings , and at the same time. As an example, consider the equation . It has above or no solutions, in or against each two; the solutions in are of course the pictures of the solutions in . These data together result in a functor (rings) → (sets), which gives a ring the set ${\displaystyle x^{2}-2=0}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle R}$

${\displaystyle F(R)=\{r\in R\mid r^{2}=2\}}$

which assigns solutions or points . This functor can be represented ; i.e., there is a ring so that ${\displaystyle S}$

${\displaystyle F(R)=\operatorname {Hom} (S,R)}$

applies. denotes the set of ring homomorphisms ; In our example it turns out that the point functors for classical algebraic varieties can be represented (over the category of rings or k -algebras) if the varieties are affine. If the term scheme is to be the broadest possible generalization of the term variety, then an affine scheme is nothing more than a ring (at least from a categorical point of view), and the general term “scheme” should be understood in such a way that all varieties can be represented in the Category of schemes are. ${\displaystyle \operatorname {Hom} (S,R)}$ ${\displaystyle S\to R}$ ${\displaystyle S=\mathbb {Z} [T]/(T^{2}-2).}$

Since it is not easily possible to generalize the concept of the ring appropriately, the term scheme is based instead on the spectrum of a ring . The construction of the spectrum is a (contravariant) true embedding of the category of the rings in the category of the smallest spaces, i.e. the topological spaces together with a sheaf of rings, and the essential part of the definition of a scheme consists only in finding the "correct" Select sub-category.

definition

A scheme is a locally small space that is locally isomorphic to the spectrum of a ring . If a scheme is globally isomorphic to the spectrum of a ring, it is said to be affine .

More detailed:
The spectrum of a ring is the set of all prime ideals in , in signs ${\displaystyle R}$ ${\displaystyle X:=\operatorname {Spec} (R)}$ ${\displaystyle R}$

${\displaystyle \mathrm {Spec} (R)=\{{\mathfrak {p}}\mid {\mathfrak {p}}\subseteq R{\text{ mit }}{\mathfrak {p}}{\text{ Primideal}}\}}$.

The closed sets of are by definition the sets of the form ${\displaystyle X}$

${\displaystyle V(I)=\{{\mathfrak {p}}\mid I\subseteq {\mathfrak {p}}\subseteq R{\text{ mit }}{\mathfrak {p}}{\text{ Primideal}}\}}$

for an ideal . The topology of the room defined in this way is also called the Zariski topology for historical reasons . By definition, the structural structure of assigns the ring of rational functions to every Zariski-open set . A small space is by definition a pair of a topological space and a sheaf of rings . A local small space is a small space for which the stalks of are local rings , i.e. H. have a clear maximum ideal. In particular, the spectrum of a ring with its structural grain is a locally small area. An affine scheme is by definition a locally small space that is isomorphic to the spectrum of a ring. A schema is a locally small space that can be covered by open sets , so that the restriction is an affine schema for all . ${\displaystyle I\subseteq R}$ ${\displaystyle X}$
${\displaystyle O_{X}}$ ${\displaystyle \mathrm {Spec} (R)}$ ${\displaystyle U\subset X}$ ${\displaystyle O_{X}(U)}$ ${\displaystyle U}$
${\displaystyle (X,O_{X})}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle (X,O_{X})}$ ${\displaystyle O_{X}}$

${\displaystyle U_{i}}$ ${\displaystyle i}$ ${\displaystyle (U_{i},O_{X}|_{U_{i}})}$

Properties of schemes

Schemes can have numerous special properties, some of which are discussed below.

Related schemes. A schema is called connected if the underlying topological space is connected.

Quasi-compact schemes. A schema is called quasi-compact if the underlying topological space is quasi-compact.

Irreducible schemes. A schema is called irreducible if the underlying topological space is irreducible, that is, it is not empty and not the union of two different closed subsets.

Noetherian schemes. A scheme is called locally Noetherian if it has an open affine cover such that the (affine) rings are all Noetherian. If it is also quasi-compact, it is called noetherian . ${\displaystyle X}$ ${\displaystyle \textstyle \bigcup _{i}U_{i}}$ ${\displaystyle \Gamma (U_{i},{\mathcal {O}}_{X})}$ ${\displaystyle X}$

Reduced schemes. A scheme is called reduced if the local rings are reduced for all . ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {O}}_{X,x}}$

Whole schemes. A scheme is called whole if it is reduced and irreducible. One can show that this is equivalent to saying that for every open subset the ring is zero-divisor-free. Furthermore, in an entire scheme, all stalks are zero-divisor-free, but the reverse generally does not have to apply. ${\displaystyle \emptyset \not =U\subset X}$ ${\displaystyle \Gamma (U,{\mathcal {O}}_{X})}$

Normal schemes. Be a scheme. Then it is normal at one point if the stalk is completely closed over its quotient body. A scheme is called normal , if it is normal in every point, compare also normal variety . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {O}}_{X,x}}$

Regular schemes. Be a Noetherian scheme. A point is then called regular if the stalk is regular. The scheme is called regular if every point in is regular. ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {O}}_{X,x}}$ ${\displaystyle X}$ ${\displaystyle X}$

Schema morphisms

Schemas form a category . A schema morphism is a morphism of locally small spaces between schemas.

More precisely: Be and locally small spaces. A morphism between them, a pair consisting of a continuous mapping and a Ringgarbenhomomorphismus possesses the following property: for each point of the of is induced homomorphism between local rings locally , d. H. leads the maximum ideal of into the maximum ideal of over. ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ ${\displaystyle (X,{\mathcal {O}}_{X})}$ ${\displaystyle (f,f^{\#})}$ ${\displaystyle f\colon Y\to X}$ ${\displaystyle f^{\#}\colon {\mathcal {O}}_{X}\to f_{*}{\mathcal {O}}_{Y}}$ ${\displaystyle y\in Y}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {O}}_{X,f(y)}\to {\mathcal {O}}_{Y,y}}$ ${\displaystyle {\mathcal {O}}_{X,f(y)}}$ ${\displaystyle {\mathcal {O}}_{Y,y}}$

Note: If a sheaf is generally open , the so-called direct image is referred to below . It is given by the data collection and defines a sheaf . ${\displaystyle G}$ ${\displaystyle Y}$ ${\displaystyle f_{*}(G)}$ ${\displaystyle f}$ ${\displaystyle U\mapsto G(f^{-1}(U))}$ ${\displaystyle X}$

Separate schemes

As can be shown, a topological space is separated in the topological sense (i.e. Hausdorffsch ) if and only if the diagonal is closed in (with regard to the product topology). The concept of the separation of schemata is based on this fact. ${\displaystyle \Delta (X)\subseteq X\times X}$ ${\displaystyle X\times X}$

A schema morphism is called separated if the associated diagonal morphism is a closed immersion. A schema is called separated if the canonical schema morphism is separated. ${\displaystyle f\colon Y\to X}$ ${\displaystyle f}$ ${\displaystyle \Delta _{Y/X}\colon Y\to Y\times _{X}Y}$ ${\displaystyle X\to \mathrm {Spec} (\mathbb {Z} )}$

Variations of terms

In the original version, Alexander Grothendieck called the objects defined above pre-schemas and assumed that the term schema was still separate. In the second edition of the first chapter of the Éléments de géométrie algébrique , however, he changed the terminology to that commonly used today.

A generalization of the concept of schemes was proposed by Shinichi Mochizuki in his work on the abc conjecture in 2012 .

literature

• Tom Gannon: What is a scheme? , Notices AMS, 2017, No. 11, pdf
• Robin Hartshorne : Algebraic Geometry. Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9 , Chapter II Schemes .
• Ulrich Görtz , Torsten Wedhorn : Algebraic Geometry I. Vieweg-Teubner Verlag, Springer Fachmedien Wiesbaden GmbH 2010, ISBN 978-3-8348-0676-5 .

Web links

• David Harari: Schemes (PDF; 330 kB)
• What is ... a scheme? , Notices of the AMS