Prelude to a Scheme
Schemes are fundamental objects in modern algebraic geometry, introduced in the 1960s by the renowned French mathematician Alexander Grothendieck. Schemes are built up from affine schemes, which generalise affine varieties, the central object of classical algebraic geometry.
Affine varieties are solution sets of systems of polynomial equations in n variables over some base field. The simplest polynomial we can consider is the zero polynomial, say f(x,y) = 0 over the complex numbers. The solution set of this polynomial is the whole affine plane C2.
In addition to being coordinates of the affine plane, we can also view x and y as polynomials. These polynomials form a ring, an abstract number system which generalises the additive and multiplicative properties of the integers (whole numbers). Since this ring is formed by the coordinates of the affine plane, we refer to this polynomial ring as the coordinate ring of the affine variety C2. We thus have a pair: a set of points and a corresponding ring of functions acting on this set of points.
What happens when our polynomial is slightly less simple, say f(x,y) = x2 + y2 – 1? Over the real numbers, the solution set is a circle centred at the origin with radius 1, naturally embedded in the affine plane. What is the coordinate ring of this set? Since we only care about polynomials in algebraic geometry, we would do well to at least start with the coordinate ring on the whole plane; i.e., polynomials in two variables x and y. We can then simply restrict the inputs of each polynomial to the points on the variety. One may notice that if we started with the polynomial x2 + y2 – 1 and restricted inputs to points on the circle, then the restricted polynomial is identical to the zero polynomial. In this case our coordinate ring consists of polynomials in two variables, with the extra condition that x2 + y2 – 1 is deemed as equivalent to the zero polynomial.
A sensible question to consider is whether every ring is the coordinate ring of some affine variety. It turns out that the answer is no, and that coordinate rings must satisfy quite restrictive properties. The integers are an example of a ring which is not the coordinate ring of any affine variety.
In the modern setting of algebraic geometry, we instead start with a ring and attempt to reconstruct the relationship between an affine variety (a space of points) and its coordinate ring (a ring of functions). The space which we end up constructing from the ring is the affine scheme. Affine schemes can then be glued to form schemes. This reverse approach of starting with a ring turns out to be very powerful and unifying. One example of this is the ability to work with the ring of integers, which is the main object of study in number theory. Through the theory of schemes, techniques from algebraic geometry can then be applied to solve problems in number theory.
Alexander Lai De Oliveira was a recipient of a 2018/19 AMSI Vacation Research Scholarship.