Points
A scheme is a locally ringed space, so a fortiori a topological space, but the meanings of point of are threefold:
- a point of the underlying topological space;
- a -valued point of is a morphism from to, for any scheme ;
- 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.
Read more about this topic: Glossary Of Scheme Theory
Famous quotes containing the word points:
“Wi’ joy unfeigned brothers and sisters meet,
An’ each for other’s weelfare kindly spiers:
The social hours, swift-winged, unnoticed fleet;
Each tells the uncos that he sees or hears;
The parents, partial, eye their hopeful years;
Anticipation forward points the view:”
—Robert Burns (1759–1796)
“The dominant metaphor of conceptual relativism, that of differing points of view, seems to betray an underlying paradox. Different points of view make sense, but only if there is a common co-ordinate system on which to plot them; yet the existence of a common system belies the claim of dramatic incomparability.”
—Donald Davidson (b. 1917)
“He is the best sailor who can steer within the fewest points of the wind, and extract a motive power out of the greatest obstacles. Most begin to veer and tack as soon as the wind changes from aft, and as within the tropics it does not blow from all points of the compass, there are some harbors which they can never reach.”
—Henry David Thoreau (1817–1862)