In Presheaves we have compared functions on a topological space (as an example we considered the complex plane with the Zariski topology) and the functions on open subsets of this space (which in our example would be the complex plane with a finite number of points removed).
In this post we take on this topic again, with an emphasis on the functions which can be expressed in terms of polynomials; in Presheaves we saw that on the entire complex plane we could not admit as a function (we will refer to these functions defined on a space as regular functions on the space) on the complex plane as it was undefined at the point . It can, however, be admitted as a (regular) function on the open subset . We will restrict our topological spaces to the case of varieties (see Basics of Algebraic Geometry).
Note that if we are considering the entire complex plane, the regular functions are only those whose denominators are constants. But on the open subset , we may have polynomials in the denominators as long as their zeroes are not in the open subset, in this case , which is not in . If we take an other open subset, one that is itself a subset of , such as , we can admit even more regular functions on this open subset.
The difference between the properties of a topological space and an open subset of such a space is related to the difference between “local” properties and “global” properties. “Local” means it holds on a smaller part of the space, while “global” means it holds on the entire space. For example, “locally”, the Earth appears flat. Of course, “globally”, we know that the Earth is round. However, ideally we should be able to “patch together” local information to obtain global information. This is what the concept of sheaves (see Sheaves) are for.
We may think about what we will see if we only “look at” a single point, for example, in , we may only look at . We can look at the set of all ratios of polynomials that are always defined at , which means that the polynomial in the denominator is not allowed to have a zero at . However, there are many functions that we can have – for example , , , and so many others aside from those that are already regular on all of . The set of all these functions, which form a ring, is called the local ring at . The local ring at any point of a variety is written . Taking the local ring at is an example of the process of localization.
A single point is not an open subset in our topology, so this does not fit into our definition of a sheaf or a presheaf. Instead, we say that the local ring at a point is the stalk of the sheaf of regular functions at that point. More technically, the stalk of a sheaf (or presheaf) is the set of equivalence classes (see Modular Arithmetic and Quotient Sets) of pairs , under the equivalence relation if there exists an open subset in the intersection for which . The elements of the stalk are called the germs of the sheaf (or presheaf).
An important property of a local ring at a point is that it has only one maximal ideal (see More on Ideals), which is made up of the polynomial functions that vanish at . This maximal ideal we will write as . The quotient (again see Modular Arithmetic and Quotient Sets) is called the residue field.
We recall the Hilbert Nullensatz and the definition of varieties and schemes in Basics of Algebraic Geometry. There we established a correspondence between the points of a variety (resp. scheme) and the maximal ideals (resp. prime ideals) of its “ring of functions”. We can use the ideas discussed here concerning locality, via the concept of presheaves and sheaves, to construct more general varieties and schemes.
One of the great things about algebraic geometry is that it is kind of a “synthesis” of ideas from both abstract algebra and geometry, and ideas can be exchanged between both. For example, we have already mentioned in Basics of Algebraic Geometry that we can start with a ring and look at the set of its maximal (resp. prime) ideals as forming a space. If we look at the set of its prime ideals (usually also referred to as its spectrum, and denoted – again we note that the word spectrum has many meanings in mathematics) then we have a scheme. This ring may not even be a ring of polynomials – we may even consider the ring of integers , and do algebraic “geometry” on the space !
We can also extract the idea of only looking at local information, an idea which has geometric origins, and apply it to abstract algebra. We can then define local rings completely algebraically, without reference to geometric ideas, as a ring with a unique maximal ideal.
A local ring which is also a principal ideal domain (a ring in which every ideal is a principal ideal, again see More on Ideals) and is not a field is called a discrete valuation ring. Discrete valuation rings are localizations of Dedekind domains, which are important in number theory, as we have discussed in Algebraic Numbers; for instance, in Dedekind domains, even though elements may not factor uniquely into irreducibles, ideals will always factor uniquely into prime ideals.
For the ring of integers , an example of a local ring is given by the ring of fractions whose denominator is an integer not divisible by a certain prime number . We denote this local ring by . For , is composed of all fractions whose denominator is an odd number. The unique maximal ideal of this ring is given by the fractions whose numerator is an even number. Since is a Dedekind domain, is also a discrete valuation ring. We refer to the local ring as the localization of at the point (prime ideal) .
We started with the idea of “local” and “global” in geometry, in particular algebraic geometry, and ended up with ideas important to number theory. This is once more an example of how the exchange of ideas between different branches of mathematics leads to much fruitful development of each branch and of mathematics as a whole.
Algebraic Geometry by Robin Hartshorne
Algebraic Number Theory by Jurgen Neukirch