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Results tagged with schemes
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user 56878
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.
3
votes
A curve is proper iff the space of global sections is finite-dimensional
As pointed out in the comments, this is false for general bases. Let $k$ be a field, $S = \mathrm{Spec}(k[t])$, let $\overline{X} = \mathbb{P}^1 \times_{\mathrm{Spec} k} S$ with projection $\overline{ …
- 3,058
2
votes
Accepted
Glueing modules over $\{x\}\times \operatorname{Spec} R$
The Beauville-Laszlo theorem holds in much greater generality - see Tag 0BNI on the Stacks Project.
Let $A$ be any ring and let $f\in A$ be a non-zero divisor. Then the category of $f$-torsion free $ …
- 3,058