Let X be some affine algebraic variety over <mrow class="MJX-TeXAtom-ORD"> <mi mathva

Roland Waters

Roland Waters

Answered question

2022-06-25

Let X be some affine algebraic variety over k (i.e. some closed subset in A k n ). First suppose X to be irreducible. Then the algebra k [ X ] is a domain and we can consider the field of rational functions Q u o t k [ X ] = k ( X ). Could you explain me how to build an analogue of this field in the case when X is not necessarily irreducible? Then k [ X ] must not be a domain and we are to build some kind of localization?
Also, what is the destination of rational functions? Why we cannot be satisfied with only regular maps and regular functions?

Answer & Explanation

Braylon Perez

Braylon Perez

Beginner2022-06-26Added 34 answers

The analogue of the quotient field for a ring with zero divisors is the total ring of fractions: basically, just invert everything that is not a zero divisor. Geometrically, an element of this ring can be viewed as a collection of rational functions, one on each irreducible component of X, such that they coincide on intersections.
Rational functions are important for a wide variety of reasons. Asking this question is like asking why Q is important. Why weren't we happy with Z ? Well, it wasn't big enough for what we wanted to do, so we enlarged it.

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