# Tag Info

Accepted

### Building algebraic geometry without prime ideals

Actually, you have rediscovered a nice motivation of using prime ideals as points. Indeed, your collection of points are triples $(R, k_x, \mathrm{ev}_x)$ where , $\mathrm{ev}_x \colon R \to k_x$ is a ...
• 8,406
Accepted

Accepted

### Are higher etale homotopy groups topological groups in a natural way?

TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as ...
• 16k

### The underlying space of a scheme remembers its affineness?

Here is a counterexample. Fix a field $k$, and let $Y$ be built from two copies of the affine nodal curve $y^2=x^3+x^2$, glued together on the complement of the singular point. In other words $Y$ is a ...
• 8,871
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### An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras

No, it's not true in general (EGA 2, (1.2.3)). The following example is taken from EGA 2, (1.3.3). Over a field $K$, let $S$ be the affine plane with a doubled origin. Then $S$ is the union of two ...
• 12.6k
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• 34k

### Building algebraic geometry without prime ideals

This nice approach to points on schemes in fact becomes crucial once one leaves the world of schemes and travels to the galaxy of stacks. For an algebraic stack $X$, one defines a point of $X$ to be a ...
• 19.3k

### On Q-Cartier Divisors

Maybe I can say something useful here. The main confusion seems to be how to find the sheaves/ideals/modules associated to multiples of divisors. As Martin Bright points out, symbolic power of a ...
• 19.7k

### Why would one "attempt" to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

In the spirit of You Could Have Invented Spectral Sequences by T.Chow, I claim you could have invented $\operatorname{Ext}^{1}(\mathbb Q(0),M)$ as group of "rational points" of a motive. Here is how. ...
• 9,680
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### A book on elliptic curves using scheme theory?

Not exactly a book, but there are course notes on abelian varieties from a course that Brian Conrad taught a few years ago. It is definitely from the perspective of scheme theory/functor of points.
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### The Serre duality theorem intuition

First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions. For pointers to more recent ...
• 8,406
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### Smallness of the category of schemes of finite type

You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows: If $S$ is affine and we only consider affine finite ...
• 5,334
### Can $\mathcal O_X$ be recognized abstract-nonsensically?
I think you are a little confused about what your characterization of $R$ does, and this causes problems as you generalize to sheaves. There is no characterization of $R$ as an element of the ...