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Results tagged with ag.algebraic-geometry
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user 10605
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
10
votes
Accepted
What is the significance of the Spec of non-affine structure sheaf sections?
Firstly, $\mathcal{O}_X |_U = \mathcal{O}_U$ (restriction of sheaves), so you could just consider $U=X$. Then $\mathrm{Aff}(X) := \mathrm{Spec}\,\mathcal{O}_X(X)$ is the universal affine scheme with a …
- 4,780
2
votes
Expressing properties of graded algebras in terms of the $\mathbb{G}_m$action
Expanding on Dylan Wilson's answer, condition (2) is indeed just a continuation of $\mathbb{G}_m$-action to an $\mathbb{A}^1$-action, since $\mathcal{O}(\mathbb{A}^1) \hookrightarrow \mathcal{O}(\math …
- 4,780
8
votes
Accepted
Is lim R_i = O(colim Spec R_i) true for finite (co)limits?
We have an equivalence of categories $Aff\simeq Ring^{op}$ and a pair of adjoint functors $$\mathcal{O}:Sch\rightleftharpoons Ring^{op} : Spec$$ $$\mathcal{O} \dashv Spec$$
The category of affine sche …
- 4,780
2
votes
Geometric intuition for limits
There's an enlightening example of limit coming from topology. Arguably it was one of the motivating examples for the notion of categorical limit. In general topology it is known as limit over a filte …
- 4,780
0
votes
My first question - on Affine Schemes in Algebraic Geometry
There is a purely category-theoretical way to describe affine schemes. Consider an arbitrary scheme $X$. For any ring $R$ we can consider a set of $R$-points of $X$, that is $X(R)=\mathrm{Hom}_{Schm}\ …
- 4,780