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.

Regular functions

Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular function on an algebraic set V contained in {mathbb A}^n is defined to be the restriction of a regular function on {mathbb A}^n, in the sense we defined above.

Related Topics:
Continuous functions - Topological spaces - Smooth functions - Differentiable manifolds

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It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.

Related Topics:
Normal - Topological space - Tietze extension theorem

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Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k. This ring is called the coordinate ring of V.

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Since regular functions on V come from regular functions on {mathbb A}^n, there should be a relationship between their coordinate rings.

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Specifically, to get a function in k we took a function in k, and we said that it was the same as another function if they gave the same values when evaluated on V. This is the same as saying that their difference is zero on V. From this we can see that k is the quotient k/I(V).

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