Little space

A small space is a construct from the mathematical sub-areas of algebraic geometry and function theory . A small space consists of a topological space and a set of commutative rings , the elements of which can be understood as functions on the open sets of space.

definition

to the definition opposite

A small space is a topological space together with a sheaf of commutative rings , that is: ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle X}$

• For each open set there is a ring , which is also written as . ${\displaystyle U\subset X}$ ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle \Gamma (U,{\mathcal {O}})}$
• If there are open subsets of , then there is a ring homomorphism such that ${\displaystyle U\supset V}$ ${\displaystyle X}$ ${\displaystyle r_{U,V}:{\mathcal {O}}(U)\rightarrow {\mathcal {O}}(V)}$
  • The following applies to open quantities ${\displaystyle U\supset V\supset W}$ ${\displaystyle r_{U,W}=r_{V,W}\circ r_{U,V}}$
  • For every open set is valid , ${\displaystyle U\subset X}$ ${\displaystyle r_{U,U}=\mathrm {id} _{{\mathcal {O}}(U)}}$
• and fulfills the sheaf conditions: For every open set and every open cover of , that is , and for elements with for all there is exactly one with for all . ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle U\subset X}$ ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle U}$ ${\displaystyle U=\textstyle \bigcup _{i\in I}U_{i}}$ ${\displaystyle s_{i}\in {\mathcal {O}}(U_{i})}$ ${\displaystyle r_{U_{i},U_{i}\cap U_{j}}(s_{i})=r_{U_{j},U_{i}\cap U_{j}}(s_{j})}$ ${\displaystyle i,j\in I}$ ${\displaystyle s\in {\mathcal {O}}(U)}$ ${\displaystyle r_{U,U_{i}}(s)=s_{i}}$ ${\displaystyle i\in I}$

The homomorphisms are called restrictions, since in many applications they are actually restrictions on mapping, as will become clear in the examples below. If the sheaf conditions are not met, then there is only one preassembly of rings. If you are dealing with several small spaces, you can write for a better distinction in order to make the affiliation to the topological space clear. ${\displaystyle r_{U,V}}$ ${\displaystyle {\mathcal {O}}_{X}}$

The above definition can be restricted to a topological basis by explaining the rings and restrictions only for open sets from the topological basis and the above conditions are only required for basic sets. This yields a ringed space in the sense defined above, by any of open sets the ring as a projective limit of with and defined from the given topological basis. ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle r_{U,V}}$ ${\displaystyle U\subset X}$ ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle {\mathcal {O}}(V)}$ ${\displaystyle V\subset U}$ ${\displaystyle V}$

If all the occurring stalks are local , one speaks of a locally reduced space . This case is of great importance in algebraic geometry, as shown in the examples. ${\displaystyle {\mathcal {O}}_{X,x}}$

Examples

• Let it be a topological space and for every open set is the ring of continuous functions as well as the constraint mapping . Then there is a small space; it is called the sheaf of seeds of continuous functions. ${\displaystyle X}$ ${\displaystyle U\subset X}$ ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle U\rightarrow \mathbb {K} }$ ${\displaystyle r_{U,V}}$ ${\displaystyle {\mathcal {O}}(U)\rightarrow {\mathcal {O}}(V),\,f\mapsto f|_{V}}$ ${\displaystyle (X,{\mathcal {O}})}$
• An important example from algebraic geometry is the locally small space defined as follows over the spectrum of a ring . ${\displaystyle \mathrm {Spec} \,R}$ ${\displaystyle R}$
  • The sets form a topological basis of where the non-nilpotent elements pass through; for nilpotent elements is . ${\displaystyle D(f):=\{{\mathfrak {p}}\in \mathrm {Spec} \,R;f\notin {\mathfrak {p}}\}}$ ${\displaystyle \mathrm {Spec} \,R}$ ${\displaystyle f}$ ${\displaystyle D(f)=\emptyset }$
  • ${\displaystyle {\mathcal {O}}_{\mathrm {Spec} \,R}(D(f)):=R_{f}}$be the localization after . ${\displaystyle f}$
  • If so, there is a with a . Then it is well-defined and fulfills the conditions of a small space. ${\displaystyle D(f)\supset D(g)}$ ${\displaystyle s\in R}$ ${\displaystyle g^{n}=sf}$ ${\displaystyle \textstyle n\in \mathbb {N} ^{>0}}$ ${\displaystyle \textstyle r_{D(f),D(g)}({\frac {h}{f^{m}}}):={\frac {hs^{m}}{g^{mn}}}}$

This small space is called an affine scheme. Since the rings are local, there is locally little space. ${\displaystyle {\mathcal {O}}_{\mathrm {Spec} \,R,{\mathfrak {p}}}=R_{\mathfrak {p}}}$

• Small spaces also play an important role in the function theory of several variables . If a domain is defined as the ring of holomorphic functions . In the textbook given below, the authors also require a small space that is Hausdorff-like and that the sheaf contains the seeds of continuous functions. There the concept of the smallest space is therefore more narrowly defined, as is the theory of Riemann surfaces . ${\displaystyle X\subset \mathbb {C} ^{n}}$ ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle U\rightarrow \mathbb {C} }$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}}$

restrictions

If a small space is open, then a small space is obtained if one stipulates that for every open set (a topological basis) of , because there is also an open set of . One calls the restriction of to . ${\displaystyle (X,{\mathcal {O}}_{X})}$ ${\displaystyle Y\subset X}$ ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ ${\displaystyle U}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {O}}_{Y}(U):={\mathcal {O}}_{X}(U)}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ ${\displaystyle (X,{\mathcal {O}}_{X})}$ ${\displaystyle Y}$

Morphisms between small spaces

to define the morphism of small spaces

A morphism between small spaces and is a pair consisting of a continuous map and a family , where each is a ring homomorphism and for open sets in the diagram ${\displaystyle (X,{\mathcal {O}}_{X})}$ ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ ${\displaystyle (f,\varphi )}$ ${\displaystyle f:X\rightarrow Y}$ ${\displaystyle \varphi =(\varphi _{V})_{V}}$ ${\displaystyle \varphi _{V}:{\mathcal {O}}_{Y}(V)\rightarrow {\mathcal {O}}_{X}(f^{-1}(V))}$ ${\displaystyle V\supset W}$ ${\displaystyle Y}$

${\displaystyle {\begin{array}{ccc}{\mathcal {O}}_{Y}(V)&{\stackrel {\varphi _{V}}{\longrightarrow }}&{\mathcal {O}}_{X}(f^{-1}(V))\\\downarrow r_{V,W}&&\downarrow r_{f^{-1}(V),f^{-1}(W)}\\{\mathcal {O}}_{Y}(W)&{\stackrel {\varphi _{W}}{\longrightarrow }}&{\mathcal {O}}_{X}(f^{-1}(W))\end{array}}}$

is commutative, with the restrictions in both sheaves denoted by. In short, one says that the ring homomorphisms are compatible with the restrictions. ${\displaystyle r}$ ${\displaystyle \varphi _{V}}$

In the category of locally small spaces, it is also required that the ring homomorphisms are local, i.e. map the maximum ideal of into the maximum ideal of . ${\displaystyle \varphi _{x}\colon {\mathcal {O}}_{Y,f(x)}\to {\mathcal {O}}_{X,x}}$ ${\displaystyle {\mathcal {O}}_{Y,f(x)}}$ ${\displaystyle {\mathcal {O}}_{X,x}}$

With these morphisms we get the category of small spaces. One can therefore speak of isomorphic small spaces. This is very important for some concepts. A scheme is defined as a small space in which every point of the topological space has an open environment, so that the restriction to this environment is isomorphic to an affine scheme. ${\displaystyle (X,{\mathcal {O}}_{X})}$

Similarly, one defines an analytic space as a small space in which every point has a neighborhood, so that the restriction to it is isomorphic to a small space of holomorphic functions on a complex manifold im . ${\displaystyle \mathbb {C} ^{n}}$

Module sheaves

Is a ringed space, a is module a sheaf of abelian groups over so that each abelian group the structure of a - module transmits and the restrictions of the sheaf are Modulmorphismen, that is for all open sets , Ringlemente and module members . These objects, which are also called module sheaves , are investigated in algebraic geometry and function theory, with the coherent sheaves playing an important role. ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle \rho }$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle \rho _{U,V}(ax)=r_{U,V}(a)\rho _{U,V}(x)}$ ${\displaystyle U\supset V}$ ${\displaystyle a\in {\mathcal {O}}(U)}$ ${\displaystyle x\in {\mathcal {F}}(U)}$

Individual evidence

1. ↑ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Chapter 4: "Presheaves and Sheaves", paragraph "Ringed Spaces"
2. ↑ Ina Kersten: Lineare Algebraische Gruppen , Verlag: Niedersächsische Staats- und Universitätsbibliothek (2007), ISBN 3-9403-4405-2 , Chapter 2.12: "Geringte Räme"
3. ^ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Chapter 5: "Affine Schemes"
4. ^ R. Gunning, H. Rossi: Analytic functions of several complex variables . Prentice-Hall 1965, Chapter V, Section A, Definition 1
5. ↑ Klaus Lamotke: Riemannsche surfaces , Springer-Verlag (2009), ISBN 3-6420-1710-X , Chapter 4.4.2: "Garben"
6. ^ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Chapter 4: "Affine Schemes", paragraph "Ringed Spaces"
7. ^ R. Gunning, H. Rossi: Analytic functions of several complex variables . Prentice-Hall 1965, Chapter V, Paragraph A, Definition 6
8. ^ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Chapter 7: "Operations on Sheaves, Quasi-coherent and Coherent Sheaves"