4
votes
0answers
84 views

Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
4
votes
2answers
128 views

To what extent is a scheme morphism determined by its topological map?

I am just beginning to learn scheme theory. This question is aimed at getting a feel for something so apologies in advance for the lack of precision. I am struck by the following difference from the ...
9
votes
2answers
392 views

Varieties as schemes

Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne. Firstly, my main question. I understood that Grothendiecks introduction of schemes ...
3
votes
1answer
96 views

Varieties given by non-algebraic equations

In algebraic geometry one (mostly) studies varieties given by polynomial equations. Such equations define algebraic varieties and there are many "dictionaries" available. For example, the category ...
6
votes
2answers
408 views

on the generic points of a scheme

This question may be a little bit metaphysical:are there any important properties about the generic points on a scheme?Or rather,why do we introduce the concept of generic point?I am not very clear ...