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Results tagged with schemes
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user 141600
The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.
2
votes
Accepted
Proving the representability of a functor that is covered by open subfunctors
I found another way to prove this Theorem, which then also shows that this map is injective. Consider two Zariski sheaves $F,G:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ and for each sheaf a cover b …
- 215
3
votes
1
answer
737
views
Proving the representability of a functor that is covered by open subfunctors
A cover by open subfunctors means, that for every scheme $T$ and for every morphism $h_T\rightarrow F$, the pullback $F_i\times_F h_T$ is representable, say by $Y_i$ and the morphism of schemes $Y_i\rightarrow …
- 215
4
votes
0
answers
96
views
Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian
Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set:
$$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{ …
- 215