Timeline for Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?
Current License: CC BY-SA 3.0
9 events
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| Feb 16, 2014 at 11:18 | comment | added | David E Speyer | @QiaochuYuan An earlier version of this post contained a sentence like: "This assumes that you are teaching from the perspective that a scheme is a locally ringed space whose underlying set is the set of prime ideals; the functor-of-points perspective makes this issue much less confusing." Then I thought "wait a minute, no one teaches a first course from the functor-of-points perspective!" and deleted it. But, of course, you are right that this is one of the things functor of points is good for. | |
| Feb 16, 2014 at 4:40 | comment | added | Qiaochu Yuan | @David: I think this is only confusing if you've already internalized the idea that the underlying set of an affine scheme is its set of prime ideals. From a functor-of-points perspective you're still just gluing points together, you're just using a richer notion of point than prime ideals. | |
| Feb 16, 2014 at 4:36 | comment | added | Qiaochu Yuan | ...of a pair of maps $f, g : X \to Y$ is the quotient of $Y$ by the equivalence relation generated by the relation $f(x) \sim g(x)$ for all $x \in X$. | |
| Feb 16, 2014 at 4:36 | comment | added | Qiaochu Yuan | @Drew: roughly speaking, the equalizer of a pair of maps $f, g : X \to Y$ is $\{ x \in X : f(x) = g(x) \}$ (at least this is how it is calculated in many familiar concrete categories). Here the pair of maps are the two homomorphisms $\mathbb{Z}[\alpha] \to \mathbb{F}_2$ (where $\alpha = \frac{1 + \sqrt{-7}}{2}$) whose kernels are the two prime ideals over $(2)$, namely $(2, \alpha)$ and $(2, 1 - \alpha)$, and these become the single ideal $(2, 2\alpha)$ in $\mathbb{Z}[2\alpha]$. (This equalizer becomes a coequalizer in the opposite category of affine schemes. Roughly speaking, the coequalizer | |
| Feb 16, 2014 at 4:05 | comment | added | Drew Armstrong | I'm unfamiliar with the "equalizer" idea. Is there a more pedestrian way to say that $\mathrm{Spec}\,\mathbb{Z}[\sqrt{-7}]$ is obtained from $\mathrm{Spec}\,\mathbb{Z}[(1+\sqrt{-7})/2]$ by identifying two points? Which two ideals are collapsing to which ideal? Thanks. | |
| Feb 16, 2014 at 3:59 | vote | accept | Drew Armstrong | ||
| Feb 16, 2014 at 1:00 | history | edited | David E Speyer | CC BY-SA 3.0 |
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| Feb 16, 2014 at 0:53 | history | edited | David E Speyer | CC BY-SA 3.0 |
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| Feb 16, 2014 at 0:48 | history | answered | David E Speyer | CC BY-SA 3.0 |