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Results tagged with schemes
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user 22810
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.
45
votes
1
answer
2k
views
Useful, non-trivial general theorems about morphisms of schemes
I'm trying to compile a list of non-obvious theorems about morphisms of schemes which are useful for general intuition but whose proofs are not easy/technical. … Mnemonic: reasonable schemes have quasi-projective "replacements" and proper schemes have projective "replacements"
Hopefully it's clear now what I'm looking for. …
21
votes
1
answer
2k
views
Are all formal schemes *really* Ind-schemes?
Denote the subcategory of formal schemes by $\FSch\subset \ALRS$.
This raises a problem though. … There's no obvious way to turn a "formal scheme" in this sense into an ind-schemes (which are much more convenient for certain purposes). …
- 7,799
9
votes
1
answer
835
views
Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?
Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
"P …
- 7,799
6
votes
2
answers
939
views
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi...
\operatorname{Spec} A \to S$ where $A$ ranges over all possible rings and whose morphisms are morphisms of schemes above $S$. … Being affine schemes over $S$ they can be considered as $\mathcal{O}_S$-algebras. Their intersection is the pullback which corresponds to their tensor product as $\mathcal{O}_S$-algebras. …
- 7,799
0
votes
What elementary problems can you solve with schemes?
One can prove this by a clever induction on dimension using punctured spectra of local rings and exact sequences in local cohomology both of which are difficult to deal with without schemes. …