Glossary of Scheme Theory - Points

Points

A scheme is a locally ringed space, so a fortiori a topological space, but the meanings of point of are threefold:

1. a point of the underlying topological space;
2. a -valued point of is a morphism from to, for any scheme ;
3. a geometric point, where is defined over (is equipped with a morphism to), where is a field, is a morphism from to where is an algebraic closure of .

Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points of the underlying space include analogues of the generic points (in the sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The -valued points are thought of, via Yoneda's lemma, as a way of identifying with the representable functor it sets up. Historically there was a process by which projective geometry added more points (e.g. complex points, line at infinity) to simplify the geometry by refining the basic objects. The -valued points were a massive further step.

As part of the predominating Grothendieck approach, there are three corresponding notions of fiber of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a geometric fiber of a morphism is thought of as

.

This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.