24.8 Sheaves of graded bimodules and tensor-hom adjunction
Please skip this section.
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
multiplication satisfies a(a'x) = (aa')x and (xb)b' = x(bb'),
(ax)b = a(xb),
the identity section 1 of \mathcal{A}^0 acts as the identity by multiplication, and
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.
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