Basics of Algebraic Geometry

Update: A more technical discussion of varieties and schemes can be found in Varieties and Schemes Revisited.

Consider polynomial functions (of positive degree) on the complex plane. By the fundamental theorem of algebra, any polynomial function can be factored, up to a constant, into linear factors over the complex numbers:

f(z)=C(z-a_{1})(z-a_{2})...(z-a_{n})

The a_{i} are of course referred to as the roots of the polynomial. They are also called the zeroes of the polynomial, because we always have f(a_{i})=0 for any of the a_{i}. These linear factors are in one-to-one correspondence with the points of the complex plane \mathbb{C}; the factor {z-a_{i}} of course corresponds to the point  a_{i} in \mathbb{C}.

Making an analogy with the factorization of the integers, the linear factors can be thought of as the “primes” of polynomial functions with complex coefficients (the constant factors play the role of the “units” –  see The Fundamental Theorem of Arithmetic and Unique Factorization).

Therefore, one can see some kind of relation between the “primes” of a ring (see Rings, Fields, and Ideals) of polynomial functions on a space and the “points” of this space.

A linear factor z-a_{i} “generates” an ideal in the ring of polynomial functions with coefficients in \mathbb{C}, namely the set of “multiples” of z-a_{i}, written (z-a_{i}) alternatively, one may also describe this ideal as the set of polynomial functions that “vanish” at the point a_{i} in the complex plane \mathbb{C}. This ideal (z-a_{i}) is actually a maximal ideal (see More on Ideals) in this particular ring.

By a famous theorem called Hilbert’s Nullstellensatz (it is named after the German mathematician David Hilbert, while “nullstellensatz” is German for “theorem of the zeroes”), the maximal ideals generated by the linear factors in the ring of polynomial functions with complex coefficients are in one-to-one correspondence with the points of the complex plane \mathbb{C}.

We leave the complex plane for a while and recall some concepts from high school mathematics. We learned that there are certain “shapes” that can be described by polynomial equations. For example, in the xy plane, the equation y=x describes a line, y=x^2 describes a parabola, and the equation x^2+y^2=1 describes a circle. What this means is that the points comprising these “shapes” have (real number) coordinates x and y which satisfy their respective polynomial equations. We rewrite these equations as

y-x=0 (line)

y-x^2=0 (parabola)

x^2+y^2-1=0 (circle)

We can then say that the “line”, the “parabola”, and the “circle” are the sets of points whose coordinates x and y make their respective equations

y-x

y-x^2

x^2+y^2-1

equal to zero. Hence we also refer to these sets of points as the zero sets of their respective equations.

We can also have polynomial functions “on” these “shapes”, or rather, zero sets. These are just like ordinary polynomial functions, but we recall that these functions must also obey the equations that describe these sets. For instance, we can have the polynomial function y^2-5x^3+1 on the line which is the zero set of the equation y=x, but since all points on this line must have coordinates that satisfy the equation y=x, we can also write the polynomial function as x^2-5x^3+1 or y^2-5y^3+1. Note that a polynomial function such as 5x^3-5y^3 on this line is also the same as the polynomial function 0.

Using what we know from Modular Arithmetic and Quotient Sets, we can also rewrite the set of polynomial functions (with real coefficients) on the line described by the equation y=x as the quotient set (or quotient ring)

\mathbb{R}[x]/(y-x).

We now go back to the complex plane. The complex plane itself can also be considered as some kind of “shape”, and one that is described by an equation, namely one that is trivial, such as 1=1 or z=z. We can describe more complicated “shapes” using polynomial equations involving more than one complex variable. On these “shapes” there are also polynomial functions (with complex coefficients), and as in the case above, we can express the set of these polynomial functions as a quotient ring. For instance, if our “shape” is described by a single polynomial f(w,z) in two complex variables w and z, the polynomial functions in two complex variables on this “shape” form the quotient ring

\mathbb{C}[w,z]/(f(w,z))

Even in this more general case, Hilbert’s Nullstellensatz still provides us with a one-to-one correspondence between the points of this “shape” (the elements of the zero set of f(w,z)) and the maximal ideals of the quotient ring \mathbb{C}[w,z]/(f(w,z)).

A “shape” which is the zero set of a polynomial (or polynomials, for higher dimensions) is also referred to technically as an algebraic variety, or simply a variety. We can put a topology on a variety, called the Zariski topology (see Basics of Topology and Continuous Functions and Presheaves) by declaring the zero sets of polynomial functions on the variety to be the closed sets. For example, when the variety is the complex plane, the closed sets are the finite sets of points, which are the zero sets of polynomial functions on the complex plane.

Because of the one-to-one correspondence between the maximal ideals of a ring of polynomial functions on the variety and the points of the variety, we can take the point of view that the points of the variety are given by the maximal ideals of the variety. In other words, we need not draw or visualize anything anymore; the only thing we need to do is study the ideals of the ring we are given. This also means that the only thing we need to study the variety is the ring. One can also perhaps say that we are doing geometry, i.e. studying shapes, not by looking at the shapes themselves, but by looking at the functions on the shapes.

The idea of a variety can be further generalized by thinking of a “shape” whose points are given not only by the maximal ideals of a ring, but by its prime ideals. This leads into the concept of a scheme, which has “generic points” in addition to its “ordinary points” (those which are given by the maximal ideals).

The study of varieties and schemes, as well as the polynomial functions on them, are part of the branch of mathematics called algebraic geometry. The name comes from the use of concepts from abstract algebra, such as rings, fields, and ideals, to study geometry, but it should also be reminiscent of the algebra that is more familiar from high school mathematics, for example in the motivating example of polynomials and coordinates. One may also recognize algebraic geometry as a high-powered version of Cartesian geometry, once again from high school mathematics, also known as analytic geometry; however, in higher-level mathematics the word “analytic” has another meaning (related to calculus, in particular complex calculus), therefore we are further justified in using the name “algebraic geometry”.

References:

Algebraic Geometry on Wikipedia

Algebraic Variety on Wikipedia

Scheme on Wikipedia

Algebra by Michael Artin

Algebraic Geometry by Robin Hartshorne

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