### Localizations of a ring at specific elements correspond to sections of a sheaf

One fundamental observation in algebraic geometry is that there are a number of powerful correspondences between ring localizations and algebraic sheaves. Although elementary, this post will outline one such correspondence, which is that the localizations of a ring to a specific element can be exactly thought of as the sections of the sheaf of regular functions over the distinguished open set of . (Consider this post to be a self-directed refresher! It has been a long time since I learned this material.)

Suppose that is a finitely generated algebra over an algebraically closed field . One might naturally wonder if there is any geometric interpretation of the localization of this ring at a single one of its elements. Say we take (where ) and consider the local ring , i.e. quotients of the form for and nonnegative integers . If we consider and as polynomials to be evaluated, then these quotients only make sense if is always nonzero. We can in fact show that the local ring corresponds exactly to the sections of the sheaf over the distinguished open set .

Consider the natural homomorphism which maps a formal fraction to the regular function which is given by the actual quotient of polynomials . This mapping is straightforwardly injective: suppose that on ; then on and on the complement of by assumption, so everywhere. Hence the preimage of such a function is exactly 0.

Surjectivity is less trivial. Obviously, every function of the form is regular, but is every function in globally representable in the form ? Yes. Suppose that is a regular function in . I will sketch out the argument in broad strokes.

First, by the definition of a regular function, is representable as a quotient of polynomials, , in the neighborhood of every point . Distinguished open sets are in a sense the “smallest open sets” of the Zariski topology (i.e. they form a basis of the topology), so if we shrink these neighborhoods enough, they become distinguished open sets in their own right, say for every such point . These open neighborhoods together cover and, conversely, we may say that the variety is the intersection of every or, equivalently, the single variety .

We have essentially managed to glue together all our little pieces of information about the regular function into a single piece of information about , but how can we go further? Translating into the algebraic setting, we know of course that , but here we can use the Nullstellensatz to yield a more tractable result; in particular, it tells us that , or that (as an element of ) is a member of the radical of the ideal which is generated by finitely many elements .

Now we know that for finitely many . Let us define , recalling from before that each comes from the local representation of as near . Now taking any arbitrary point with a local representation of , we can check algebraically that and hence . Therefore is a valid global representation of over .

February 10th, 2023 | Posted in Math