## 24.3 Sheaves of graded algebras

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.

Given a graded $\mathcal{O}$-algebra $\mathcal{A}$ and an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we use the notation

$\mathcal{A}(U) = \Gamma (U, \mathcal{A}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{A}^ n(U)$

This is a graded $\mathcal{O}(U)$-algebra.

Remark 24.3.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. We have

1. Let $\mathcal{A}$ be a graded $\mathcal{O}_\mathcal {C}$-algebra. The multiplication maps of $\mathcal{A}$ induce multiplication maps $f_*\mathcal{A}^ n \times f_*\mathcal{A}^ m \to f_*\mathcal{A}^{n + m}$ and via $f^\sharp$ we may view these as $\mathcal{O}_\mathcal {D}$-bilinear maps. We will denote $f_*\mathcal{A}$ the graded $\mathcal{O}_\mathcal {D}$-algebra we so obtain.

2. Let $\mathcal{B}$ be a graded $\mathcal{O}_\mathcal {D}$-algebra. The multiplication maps of $\mathcal{B}$ induce multiplication maps $f^*\mathcal{B}^ n \times f^*\mathcal{B}^ m \to f^*\mathcal{B}^{n + m}$ and using $f^\sharp$ we may view these as $\mathcal{O}_\mathcal {C}$-bilinear maps. We will denote $f^*\mathcal{B}$ the graded $\mathcal{O}_\mathcal {C}$-algebra we so obtain.

3. The set of homomorphisms $f^*\mathcal{B} \to \mathcal{A}$ of graded $\mathcal{O}_\mathcal {C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\mathcal{B} \to f_*\mathcal{A}$ of graded $\mathcal{O}_\mathcal {C}$-algebras.

Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules.

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