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I've been working through Vakil's MATH 216 notes and have ran into a wall when he discusses the affine cone of a projective scheme in section 8.2.12. Namely, if S is a finitely generated graded ring, the affine cone of ProjS is defined to be SpecS. The first exercise in this section asks to show that if ProjS is a scheme over a field k (or in general if it's a finitely generated graded ring) we get a natural morphism from $SpecS\setminus V(S_+) \rightarrow ProjS$.

I'm not sure what this morphism should look like. Classically, if we wanted to go from $A^{n+1}_k$ to $P_k^n$ we would send a point $(a_0, ..., a_n)$ to $[a_0:...:a_n]$. But in general what should we be doing?

For example, if $S=k[x,y]$ so that $ProjS = P_k^1$, then in $SpecS$ we have the point $(y-x^2)$. Should I try to homogenize this ideal? I'm not sure how to do that without ending up in $P^2$ (i.e. $(zy-x^2)$). What point of ProjS should this point be sent to?

Regards,

Garnet

Garnet
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  • Algebraically it's sort of clear what to do: if $f$ is a positive homogeneous element then there's an inclusion $(S_f)_0 \to S_f$ inducing a map of affine schemes, and these are the building blocks of $\operatorname{Proj}$ and $\operatorname{Spec}$ – Hoot May 25 '15 at 17:42

1 Answers1

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Let $f$ be homogenous of degree $\geq1$. We can cover $\text{Proj}\space S$ by $\text{Spec} \space S_{(f)}$ where $S_{(f)}$ are homogenous degree 0 elements of $S_f$[Liu 2.3.36]. Then by functoriality, the ring maps $S_{(f)} \to S_f$ induces a map of affine schemes $ \text{Spec} \space S_f \to \text{Spec} \space S_{(f)}$. Now since $S$ is finitely generated over $S_{0}$, we know that $S \cong S_0[x_1,\dots,x_n]/I$ with the natural grading. Thus $V(S_{+}) = V((x_0,\dots,x_n)+I)$ in $S$. Thus the $S_{f}$ over all $f$ homogeneous of degree $\geq1$ cover $\text{Spec} \space S \backslash V(S_{+})$. The natural maps $ \text{Spec} \space S_f \to \text{Spec} \space S_{(f)}$ glue to give a map $\text{Spec} \space S \backslash V(S_{+}) \to \text{Proj} \space S$.

As a consequence the point $(y-x^2)$ is mapped (pulled back) to the point $(0)$ of $\text{Proj} \space {S}$.

Complexification
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