Scheme (mathematics)

The classical algebraic geometry deals with subsets of the affine or projective space that arise as zero sets of finitely many polynomials ( algebraic varieties ). Thus, the geometric objects are the solution sets of algebraic equations. The conceptual scheme motivated by the fact, not only to consider solutions in a fixed algebraically closed field, but solutions in arbitrary rings, simultaneously. As an example we consider the equation. She has no solutions or, in contrast, or two; here are the solutions of course the images of solutions. These data together give a functor (rings ) → ( sets), the amount of a ring

Of solutions or points assigns. This functor is representable, ie, there is a ring, so that

Applies ( denotes the set of ring homomorphisms, in our example, it turns out that the point functors are to classical algebraic varieties if and displayed ( on the category of rings and k - algebras ) if the varieties are affine If now. the conceptual scheme is to be a far-reaching as possible generalization of the concept of variety, it is an affine scheme is nothing but a ring ( at least from a categorical point of view), and the general term " scheme " should be framed so as to all the varieties represented in the category schemes are.

Because it is not readily possible to generalize the concept of the ring adapted to the conceptual scheme based on the range, instead of a ring. The construction of the spectrum is a ( contravariant ) faithful embedding of the category of rings to the category of ringed spaces, ie topological spaces together with a sheaf of rings, and the essential part of the definition of a schema consists only in the "right " to choose the sub-category.

Definition

A schema is a local ringed space locally isomorphic to the spectrum of a ring. If a schema globally isomorphic to the spectrum of a ring, it is said affine.

For more details: The spectrum of a ring is the set of all prime ideals in, in characters

The closed sets are by definition of the sets of the form

For an ideal. The so- defined topology of the space is called for historical reasons Zariski topology. The structure sheaf of maps the ring of rational functions on, by definition, any Zariski open set. A ringed space is a pair consisting of a topological space and a sheaf of rings by definition. A locally ringed space is a ringed space, where are the seeds of local rings, ie have a unique maximal ideal. In particular, the spectrum of a ring with its structure sheaf is a locally ringed space. An affine scheme is a locally ringed space, by definition, is isomorphic to the spectrum of a ring. A scheme is a locally ringed space which can be cover by open sets, such that for all the restriction is an affine scheme.

Term variations

In the original version of Alexander Grothendieck called the objects defined above Präschemata and sat for the naming scheme yet separateness advance. However, in the second edition of the first chapter of Éléments de géométrie algébrique he changed the terminology to the generally used today.

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