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Results tagged with schemes
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user 5513
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.
11
votes
What are the epimorphisms in the category of schemes?
$\DeclareMathOperator\Spec{Spec}$Actually, your suggested categorical characterization of spectra of fields does work.
Edit: (I had written something incorrect here)
By Martin's comment below, we just …
- 12.9k
6
votes
Accepted
The restriction of a global section which is not a zero divisor is still a non-zero divisor?
Here's a counterexample.
Let $P=\mathbb{P}^1$, $X=\mathbb{A}^1$, and attach $X$ to $P$ along a single point $\{x\}$. Then there is a global section $f$ which is nonzero on $X$ except at $x$, and is i …
- 4,790
17
votes
Accepted
Is every regular (excellent) scheme separated?
- Separated, excellent, regular: Spec$(k)$.
- Separated, excellent, not regular: Spec$(k[\epsilon]/\epsilon^2)$.
- Separated, not excellent, regular: See http://en.wikipedia.org/wiki/Excellent_ring
…
- 4,790