Definition 24.8.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. A graded $(\mathcal{A}, \mathcal{B})$-bimodule is given by a family $\mathcal{M}^ n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

$\mathcal{M}^ n \times \mathcal{B}^ m \to \mathcal{M}^{n + m},\quad (x, b) \longmapsto xb$

and

$\mathcal{A}^ n \times \mathcal{M}^ m \to \mathcal{M}^{n + m},\quad (a, x) \longmapsto ax$

called the multiplication maps with the following properties

1. multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$,

2. $(ax)b = a(xb)$,

3. the identity section $1$ of $\mathcal{A}^0$ acts as the identity by multiplication, and

4. the identity section $1$ of $\mathcal{B}^0$ acts as the identity by multiplication.

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

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