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.

The modern viewpoint

The modern approach to algebraic geometry redefines the basic objects. Varieties are subsumed in Alexander Grothendieck's concept of a scheme. Schemes start with the observation that if finitely generated reduced k-algebras are geometrical objects, then perhaps arbitrary commutative rings should also be geometrical objects. As such, schemes become both a more general algebro-geometric object, and a convenient language to describe those objects. This language of schemes has proved to be a valuable way of dealing with geometric concepts and has become a cornerstone of modern algebraic geometry.

Related Topics:
Alexander Grothendieck - Scheme

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