Introduction to Algebraic Geometry 3

This article is part of a series of posts introducing the ideas in modern Algebraic Geometry to a general audience, The preceding article is here.

Photo by Evgeny Tkachenko on Unsplash

In this article and the next, I am going to try an introduce one of the central ideas of modern Algebraic Geometry (AG), namely that of a scheme.

First in this article we are going to define what is called a coordinate ring and we will talk about schemes in the next.

In the spirit of this series of articles trying to keep on board as wide an audience as possible, I will start with a word game and show how the idea ports over to the study of polynomial systems.

So, suppose we tried to understand the adjective ‘round’.

Now, one way is to see what this adjective applies to. We can have ‘round circle’, ’round wheel’, ‘round sun’, and a whole host of other round things.

In other words, we can (start to) define a function APPLIES_TO which maps ‘round’ to the set of words that it applies to. It is a very large set for sure, possibly infinite even, but it is possible nonetheless.

APPLIES_TO (round) = { circle, wheel, sun, … more round things … }

Similarly we can do the same thing for ‘yellow’ :

`APPLIES_TO (yellow) = { lemon, sun, canary, … more yellow things … }`

Now to understand the adjectives ‘round’, and ‘yellow’, we can study the set of nouns they map to through the APPLIES_TO function.

For example we can observe that ‘sun’ is there in both sets, but that ‘canary’ is one set and not the other.

Moreover, if the adjectives are related, then it is reasonable to expect that the sets they map to are related as well.

Consider the adjective ‘yellow-ish’:

APPLIES_TO (yellow-ish) = { pumpkin, lime, lemon, sun, canary, … more yellow-ish things … }

Now since yellow things are certainly yellow-ish, let us informally say that ‘yellow > yellow-ish’ and observe that the corresponding sets are included in one another :

`APPLIES_TO (yellow) = { lemon, sun, canary, … more yellow things … } ⊂ APPLIES_TO (yellow_ish) = { pumpkin, lime, lemon, sun, canary, … more yellow-ish things … }`

Sure, whatever Alain. What has that got to do with polynomials eh ?

Well suppose we wanted to study this polynomial equation

x² + y² = 1  -- with (x,y) in the real plane

Solutions for this include (0,1), (1,0), (0,-1), (1/√2, 1/√2) and many more besides.

Ok I know, I know, it is the unit circle. I am trying to keep things simple here remember.

Now the big idea is to study what happens when you evaluate polynomials for these solutions.

The set of all 2-variable polynomials with real coefficients is denoted R[x,y]. One example of such a polynomial is for example 7xy+y³.

The set of all polynomials when evaluated on the set of solutions for x^2+y^2=1 is called the coordinate ringof x^2+y^2=1

So for example, 7xy+y³ can certainly be evaluated for any point in the solution for x^2+y^2=1 . 7xy+y³ can be evaluated for (0,1) — one of the solutions for x^2+y^2=1 — and that evaluation is just 7x0x1+1³=1. We can do something similar for all the points on the unit circle. So 7xy+y³ is in the coordinate ring for x^2+y^2=1

Hey Alain, get serious ! Any polynomials in (x,y) can be evaluated for the points on the unit circle so your coordinate ring thingy is just R[x,y]. Do you have any point at all to make here or what ?

Wait !

Let us evaluate the constant polynomial 23on the unit circle. No big mystery here, every point on the unit circle evaluates to the same value, namely 23, so 23 is in the coordinate ring for the unit circle.

Now let us evaluate the polynomial 2x²+2y²+21 on the same unit circle. Well for (0,1) we get 2x0²+2x1²+21=23 , and for (1,0) we 2x1²+2x0²+21=23, and for (1/√2, 1/√2) we get 2x(1/√2)²+2x(1/√2)²+21=23, in fact we get 23 for every point on the unit circle.

So the polynomials 23 and 2x²+2y²+21 are ‘the same’ when evaluated on the unit circle. They are not ‘the same’ in general of course.

This means that the coordinate ring for x^2+y^2=1 is NOT R[x,y] — it is strictly smaller.

So notice what we did here: we attached a set of polynomials to the thing that we want to study (in this case the unit circle).

Now (fortunately) it turns out that the study of this set will tell us a lot about the unit circle — and that is one of the ideas of Algebraic Geometry : we study the set of functions defined on the polynomial system of interest, NOT the polynomial system directly.

Ok that’s it for today , see you (hopefully) in the next post where we go on to define a scheme !

(This article closely followed an exposition in the Princeton Companion to Mathematics section IV.5. The book is a great read for anyone interested in modern mathematics.)