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.

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