Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always C, the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define {mathbb A}^n_k, called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, {mathbb A}^n_k is, for the moment, just a collection of points.

Related Topics:
Field - Algebraically closed

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Henceforth we will drop the k in {mathbb A}^n_k and instead write {mathbb A}^n.

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Define a function

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:f:{mathbb A}^n o{mathbb A}^1

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to be regular if it can be written as a polynomial, that is, if there is a polynomial p in

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:k

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such that for each point

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:(t1,...,tn)

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of {mathbb A}^n,

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:f(t1,...,tn) = p(t1,...,tn).

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Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will write the regular functions on {mathbb A}^n as k.

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We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k. The vanishing set of S (or vanishing locus) is the set V(S) of all points in mathbb{A}^n where every polynomial in S vanishes. In other words,

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:V(S)={(t1,...,tn) | for all p in S, p(t1,...,tn) = 0}.

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A subset of {mathbb A}^n which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below).

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Given a subset V of {mathbb A}^n which is a variety, can one recover the set of polynomials which generate it? If V is any subset of {mathbb A}^n, define I(V) to be the set of all polynomials whose vanishing set contains V. The I stands for ideal: if two polynomials f and g both vanish on V, then f+g vanishes on V, and if h is any polynomial, then hf vanishes on V, so I(V) is always an ideal of k.

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Two natural questions to ask are: given a subset V of {mathbb A}^n, when is

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:V = V(I(V))?

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Given a set S of polynomials, when is

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:S = I(V(S))?

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The answer to the first question is provided by introducing the Zariski topology, a topology on {mathbb A}^n which directly reflects the algebraic structure of k. Then V = V(I(V)), if and only if V is a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I(V(S)) is the prime radical of the ideal generated by S. In more abstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, and naturally play a basic role in the theory.

Related Topics:
Zariski topology - Hilbert's Nullstellensatz - Prime radical - Galois connection - Closure operator

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For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set V. Hilbert's Basis Theorem implies that ideals in k are always finitely generated.

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An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a prime ideal of the polynomial ring.

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