Ringed space
A ringed space is a construct of the mathematical branches of algebraic geometry and the theory of functions. A ringed space consists of a topological space and a set of commutative rings whose elements can be understood as functions on the open sets of the space.
Definition
A ringed space is a topological space together with a sheaf of commutative rings, ie:
- For every open set is a ring given to you as well as writes.
- If there are open subsets of, so there is a ring homomorphism such that For open sets,
- For every open set,
The homomorphisms are called constraints, as it is in many applications is actually restrictions of pictures, as will become clear in the examples below. Are the sheaf conditions are not met, then there is only a presheaf of rings. If you're dealing with multiple ringed spaces, so you can for better differentiation write to make membership in the topological space significantly.
You can restrict above definition to a topological basis by the rings and restrictions stated only for open sets of the topological basis and the above conditions are required only for base quantities. This gives it a ringed space in the sense defined above, by the ring as a projective limit of the defined for arbitrary open sets with and from the given topological basis.
Are all occurring locally straws, one speaks of a local ringed space. This is the case in the algebraic geometry, as shown in the examples.
Examples
- It is a topological space and for each open set is the ring of continuous functions and the restriction mapping. Then is a ringed space, it is called the sheaf of germs of continuous functions.
- An important example from algebraic geometry is defined as follows locally ringed space over the spectrum of a ring. The amounts based on a topological form, wherein the non nilpotent elements passes; is nilpotent elements.
- Is according to the location.
- If so there is a with a. Then is well defined, and meets the conditions of a ringed space.
- Ringed spaces play an important role in the theory of functions of several variables. Once a site, one defined as the ring of holomorphic functions. In the below textbook, the authors of a ringed space require in addition that is Hausdorff and is included in the sheaf of germs of continuous functions. There the concept of ringed space is thus narrower, as in the theory of Riemann surfaces.
Limitations
Is a ringed space and open so you get a ringed space when looking for every open set ( a topological basis) of states that, because is indeed an open set of. This is called the restriction of to.
Morphisms between ringed spaces
A morphism between ringed spaces and is a pair consisting of a continuous map and a family, each of which is a ring homomorphism and for open sets in the diagram
Is commutative, where the restrictions are indicated in both sheaves with. They say it in short, that the ring homomorphisms are compatible with the constraints.
In the category of locally ringed spaces is demanded in addition that the ring homomorphisms are local, ie, the maximal ideal of in the maximal ideal of map.
With these morphisms, we obtain the category ringed spaces. One can therefore speak of isomorphic ringed spaces. This is very important for some conceptions. So we define a schema as a ringed space, in which each point of the topological space has an open environment so that the restriction is to an affine scheme to that environment isomorphic.
Similarly, we define an analytic space as a ringed space, in which each point has a neighborhood such that the restriction is it isomorphic to a ringed space of holomorphic functions on a complex manifold in.
Module sheaves
Is a ringed space, a module is a sheaf of abelian groups over so that every abelian group carries the structure of a module and the restrictions of the sheaf Modulmorphismen are, that is for all open sets, Ringlemente and module elements. These objects, which are also called module sheaves are studied in algebraic geometry and function theory, the coherent sheaves play an important role.