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

$\mathcal{A}^ n \times \mathcal{A}^ m \to \mathcal{A}^{n + m},\quad (a, b) \longmapsto ab$

called the multiplication maps with the following properties

1. multiplication is associative, and

2. 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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