Lemma 24.9.1. In the situation above we have
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{B})}( \mathcal{N}, f_*\mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( f^*\mathcal{N}, \mathcal{M}) \]
24.9 Pull and push for sheaves of graded modules
We advise the reader to skip this section.
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. Let $\mathcal{A}$ be a graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a graded $\mathcal{O}_\mathcal {D}$-algebra. Suppose we are given a map
\[ \varphi : f^{-1}\mathcal{B} \to \mathcal{A} \]
of graded $f^{-1}\mathcal{O}_\mathcal {D}$-algebras. By the adjunction of restriction and extension of scalars, this is the same thing as a map $\varphi : f^*\mathcal{B} \to \mathcal{A}$ of graded $\mathcal{O}_\mathcal {C}$-algebras or equivalently $\varphi $ can be viewed as a map
\[ \varphi : \mathcal{B} \to f_*\mathcal{A} \]
of graded $\mathcal{O}_\mathcal {D}$-algebras. See Remark 24.3.2.
Let us define a functor
\[ f_* : \textit{Mod}(\mathcal{A}) \longrightarrow \textit{Mod}(\mathcal{B}) \]
Given a graded $\mathcal{A}$-module $\mathcal{M}$ we define $f_*\mathcal{M}$ to be the graded $\mathcal{B}$-module whose degree $n$ term is $f_*\mathcal{M}^ n$. As multiplication we use
\[ f_*\mathcal{M}^ n \times \mathcal{B}^ m \xrightarrow {(\text{id}, \varphi ^ m)} f_*\mathcal{M}^ n \times f_*\mathcal{A}^ m \xrightarrow {f_*\mu _{n, m}} f_*\mathcal{M}^{n + m} \]
where $\mu _{n, m} : \mathcal{M}^ n \times \mathcal{A}^ m \to \mathcal{M}^{n + m}$ is the multiplication map for $\mathcal{M}$ over $\mathcal{A}$. This uses that $f_*$ commutes with products. The construction is clearly functorial in $\mathcal{M}$ and we obtain our functor.
Let us define a functor
\[ f^* : \textit{Mod}(\mathcal{B}) \longrightarrow \textit{Mod}(\mathcal{A}) \]
We will define this functor as a composite of functors
\[ \textit{Mod}(\mathcal{B}) \xrightarrow {f^{-1}} \textit{Mod}(f^{-1}\mathcal{B}) \xrightarrow { - \otimes _{f^{-1}\mathcal{B}} \mathcal{A}} \textit{Mod}(\mathcal{A}) \]
First, given a graded $\mathcal{B}$-module $\mathcal{N}$ we define $f^{-1}\mathcal{N}$ to be the graded $f^{-1}\mathcal{B}$-module whose degree $n$ term is $f^{-1}\mathcal{N}^ n$. As multiplication we use
\[ f^{-1}\nu _{n, m} : f^{-1}\mathcal{N}^ n \times f^{-1}\mathcal{B}^ m \longrightarrow f^{-1}\mathcal{N}^{n + m} \]
where $\nu _{n, m} : \mathcal{N}^ n \times \mathcal{B}^ m \to \mathcal{N}^{n + m}$ is the multiplication map for $\mathcal{N}$ over $\mathcal{B}$. This uses that $f^{-1}$ commutes with products. The construction is clearly functorial in $\mathcal{N}$ and we obtain our functor $f^{-1}$. Having said this, we can use the tensor product discussion in Section 24.8 to define the functor
\[ - \otimes _{f^{-1}\mathcal{B}} \mathcal{A} : \textit{Mod}(f^{-1}\mathcal{B}) \longrightarrow \textit{Mod}(\mathcal{A}) \]
Finally, we set
\[ f^*\mathcal{N} = f^{-1}\mathcal{N} \otimes _{f^{-1}\mathcal{B}, \varphi } \mathcal{A} \]
as already foretold above.
The functors $f_*$ and $f^*$ are readily enhanced to give functors of graded categories
\[ f_* : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \textit{Mod}^{gr}(\mathcal{B}) \quad \text{and}\quad f^* : \textit{Mod}^{gr}(\mathcal{B}) \longrightarrow \textit{Mod}^{gr}(\mathcal{A}) \]
which do the same thing on underlying objects and are defined by functoriality of the constructions on homogeneous morphisms of degree $n$.
Proof. Omitted. Hints: First prove that $f^{-1}$ and $f_*$ are adjoint as functors between $\textit{Mod}(\mathcal{B})$ and $\textit{Mod}(f^{-1}\mathcal{B})$ using the adjunction between $f^{-1}$ and $f_*$ on sheaves of abelian groups. Next, use the adjunction between base change and restriction given in Section 24.8. $\square$
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