Section 27.7 (062P): Cones—The Stacks project

27.7 Cones

In algebraic geometry cones correspond to graded algebras. By our conventions a graded ring or algebra

$A$ comes with a grading

$A = \bigoplus _{d \geq 0} A_ d$ by the nonnegative integers, see Algebra, Section 10.56.

Definition 27.7.1. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$ -algebra. Assume that $\mathcal{O}_ S \to \mathcal{A}_0$ is an isomorphism ¹. The cone associated to $\mathcal{A}$ or the affine cone associated to $\mathcal{A}$ is

$C(\mathcal{A}) = \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}).$

The cone associated to a graded sheaf of

$\mathcal{O}_ S$ -algebras comes with a bit of extra structure. Namely, we obtain a grading

$\pi _*\mathcal{O}_{C(\mathcal{A})} = \bigoplus \nolimits _{n \geq 0} \mathcal{A}_ n$

Thus we can define an abstract cone as follows.

Definition 27.7.2. Let $S$ be a scheme. A cone $\pi : C \to S$ over $S$ is an affine morphism of schemes such that $\pi _*\mathcal{O}_ C$ is endowed with the structure of a graded $\mathcal{O}_ S$ -algebra $\pi _*\mathcal{O}_ C = \bigoplus \nolimits _{n \geq 0} \mathcal{A}_ n$ such that $\mathcal{A}_0 = \mathcal{O}_ S$ . A morphism of cones from $\pi : C \to S$ to $\pi ' : C' \to S$ is a morphism $f : C \to C'$ such that the induced map

$f^* : \pi '_*\mathcal{O}_{C'} \longrightarrow \pi _*\mathcal{O}_ C$

is compatible with the given gradings.

Any vector bundle is an example of a cone. In fact the category of vector bundles over

$S$ is a full subcategory of the category of cones over

$S$ .

[1] Often one imposes the assumption that

$\mathcal{A}$ is generated by

$\mathcal{A}_1$ over

$\mathcal{O}_ S$ . We do not assume this in order to be consistent with [II, (8.3.1), EGA].

The Stacks project

27.7 Cones

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