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 isomorphism1. 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].

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