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.

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

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

This is a graded $(\mathcal{A}(U), \mathcal{B}(U))$-bimodule.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right graded $\mathcal{A}$-module and let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. In this case the graded tensor product defined in Section 24.6

$\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$

is a right graded $\mathcal{B}$-module with obvious multiplication maps. This construction defines a functor and a functor of graded categories

$\otimes _\mathcal {A} \mathcal{N} : \textit{Mod}(\mathcal{A}) \longrightarrow \textit{Mod}(\mathcal{B}) \quad \text{and}\quad \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \textit{Mod}^{gr}(\mathcal{B})$

by sending homomorphisms of degree $n$ from $\mathcal{M} \to \mathcal{M}'$ to the induced map of degree $n$ from $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ to $\mathcal{M}' \otimes _\mathcal {A} \mathcal{N}$.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right graded $\mathcal{B}$-module. In this case the graded internal hom defined in Section 24.7

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L})$

is a right graded $\mathcal{A}$-module with multiplication maps1

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {B}(\mathcal{N}, \mathcal{L}) \times \mathcal{A}^ m \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{n + m}_\mathcal {B}(\mathcal{N}, \mathcal{L})$

sending a section $f = (f_{p,q})$ of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {B}(\mathcal{N}, \mathcal{L})$ over $U$ and a section $a$ of $\mathcal{A}^ m$ over $U$ to the section $f a$ if $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{n + m}_\mathcal {B}(\mathcal{N}, \mathcal{L})$ over $U$ defined as the family of maps

$\mathcal{N}^{-q - m}|_ U \xrightarrow {a \cdot -} \mathcal{N}^{-q}|_ U \xrightarrow {f_{p, q}} \mathcal{M}^ p|_ U$

We omit the verification that this is well defined. This construction defines a functor and a functor of graded categories

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, -) : \textit{Mod}(\mathcal{B}) \longrightarrow \textit{Mod}(\mathcal{A}) \quad \text{and}\quad \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, -) : \textit{Mod}^{gr}(\mathcal{B}) \longrightarrow \textit{Mod}^{gr}(\mathcal{A})$

by sending homomorphisms of degree $n$ from $\mathcal{L} \to \mathcal{L}'$ to the induced map of degree $n$ from $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L})$ to $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L}')$.

Lemma 24.8.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right graded $\mathcal{A}$-module. Let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right graded $\mathcal{B}$-module. With conventions as above we have

$\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{B})}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{L}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( \mathcal{M}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L}))$

and

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {A}^{gr}( \mathcal{M}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L}))$

functorially in $\mathcal{M}$, $\mathcal{N}$, $\mathcal{L}$.

Proof. Omitted. Hint: This follows by interpreting both sides as $\mathcal{A}$-bilinear graded maps $\psi : \mathcal{M} \times \mathcal{N} \to \mathcal{L}$ which are $\mathcal{B}$-linear on the right. $\square$

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. As a special case of the above, suppose we are given a homomorphism $\varphi : \mathcal{A} \to \mathcal{B}$ of graded $\mathcal{O}$-algebras. Then we obtain a functor and a functor of graded categories

$\otimes _{\mathcal{A}, \varphi } \mathcal{B} : \textit{Mod}(\mathcal{A}) \longrightarrow \textit{Mod}(\mathcal{B}) \quad \text{and}\quad \otimes _{\mathcal{A}, \varphi } \mathcal{B} : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \textit{Mod}^{gr}(\mathcal{B})$

On the other hand, we have the restriction functors

$res_\varphi : \textit{Mod}(\mathcal{B}) \longrightarrow \textit{Mod}(\mathcal{A}) \quad \text{and}\quad res_\varphi : \textit{Mod}^{gr}(\mathcal{B}) \longrightarrow \textit{Mod}^{gr}(\mathcal{A})$

We can use the lemma above to show these functors are adjoint to each other (as usual with restriction and base change). Namely, let us write ${}_\mathcal {A}\mathcal{B}_\mathcal {B}$ for $\mathcal{B}$ viewed as a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Then for any right graded $\mathcal{B}$-module $\mathcal{L}$ we have

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}({}_\mathcal {A}\mathcal{B}_\mathcal {B}, \mathcal{L}) = res_\varphi (\mathcal{L})$

as right graded $\mathcal{A}$-modules. Thus Lemma 24.8.2 tells us that we have a functorial isomorphism

$\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{B})}( \mathcal{M} \otimes _{\mathcal{A}, \varphi } \mathcal{B}, \mathcal{L}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( \mathcal{M}, res_\varphi (\mathcal{L}))$

We usually drop the dependence on $\varphi$ in this formula if it is clear from context. In the same manner we obtain the equality

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{gr}_\mathcal {B}( \mathcal{M} \otimes _\mathcal {A} \mathcal{B}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {A}^{gr}(\mathcal{M}, \mathcal{L})$

of graded $\mathcal{O}$-modules.

 Our conventions are here that this does not involve any signs.

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