Definition 24.3.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A *sheaf of graded $\mathcal{O}$-algebras* or a *sheaf of graded algebras* on $(\mathcal{C}, \mathcal{O})$ is given by a family $\mathcal{A}^ n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

called the multiplication maps with the following properties

multiplication is associative, and

there is a global section $1$ of $\mathcal{A}^0$ which is a two-sided identity for multiplication.

We often denote such a structure $\mathcal{A}$. A *homomorphism of graded $\mathcal{O}$-algebras* $f : \mathcal{A} \to \mathcal{B}$ is a family of maps $f^ n : \mathcal{A}^ n \to \mathcal{B}^ n$ of $\mathcal{O}$-modules compatible with the multiplication maps.

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