Schemes

Abstract

We have now seen the basic idea of what algebraic geometry aims to investigate, and also some of the commutative algebra needed to prove its basic facts. However, it is clear that the concept of a variety, as we introduced it in Chap. 1, is not satisfactory: It is based on two examples, the affine and projective space, and their subobjects. This would be like defining a topological space as a subset of $\mathbb {R}^n$. For proper foundations, a general concept, based on abstract axioms, is needed.