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24.10 Localization and sheaves of graded modules

We advise the reader to skip this section.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote

\[ j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \]

the corresponding localization morphism (Modules on Sites, Section 18.19). Below we will use the following fact: for $\mathcal{O}_ U$-modules $\mathcal{M}_ i$, $i = 1, 2$ and a $\mathcal{O}$-module $\mathcal{A}$ there is a canonical map

\[ j_! : \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}( \mathcal{M}_1 \otimes _{\mathcal{O}_ U} \mathcal{A}|_ U, \mathcal{M}_2) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}( j_!\mathcal{M}_1 \otimes _\mathcal {O} \mathcal{A}, j_!\mathcal{M}_2) \]

Namely, we have $j_!(\mathcal{M}_1 \otimes _{\mathcal{O}_ U} \mathcal{A}|_ U) = j_!\mathcal{M}_1 \otimes _\mathcal {O} \mathcal{A}$ by Modules on Sites, Lemma 18.27.9.

Let $\mathcal{A}$ be a graded $\mathcal{O}$-algebra. We will denote $\mathcal{A}_ U$ the restriction of $\mathcal{A}$ to $\mathcal{C}/U$, in other words, we have $\mathcal{A}_ U = j^*\mathcal{A} = j^{-1}\mathcal{A}$. In Section 24.9 we have constructed adjoint functors

\[ j_* : \textit{Mod}^{gr}(\mathcal{A}_ U) \longrightarrow \textit{Mod}^{gr}(\mathcal{A}) \quad \text{and}\quad j^* : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \textit{Mod}^{gr}(\mathcal{A}_ U) \]

with $j^*$ left adjoint to $j_*$. We claim there is in addition an exact functor

\[ j_! : \textit{Mod}^{gr}(\mathcal{A}_ U) \longrightarrow \textit{Mod}^{gr}(\mathcal{A}) \]

left adjoint to $j^*$. Namely, given a graded $\mathcal{A}_ U$-module $\mathcal{M}$ we define $j_!\mathcal{M}$ to be the graded $\mathcal{A}$-module whose degree $n$ term is $j_!\mathcal{M}^ n$. As multiplication map we use

\[ j_!\mu _{n, m} : j_!\mathcal{M}^ n \times \mathcal{A}^ m \to j_!\mathcal{M}^{n + m} \]

where $\mu _{m, n} : \mathcal{M}^ n \times \mathcal{A}^ m \to \mathcal{M}^{n + m}$ is the given multiplication map. Given a homogeneous map $f : \mathcal{M} \to \mathcal{M}'$ of degree $n$ of graded $\mathcal{A}_ U$-modules, we obtain a homogeneous map $j_!f : j_!\mathcal{M} \to j_!\mathcal{M}'$ of degree $n$. Thus we obtain our functor.

Lemma 24.10.1. In the situation above we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( j_!\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A}_ U)}( \mathcal{M}, j^*\mathcal{N}) \]

Proof. By the discussion in Modules on Sites, Section 18.19 the functors $j_!$ and $j^*$ on $\mathcal{O}$-modules are adjoint. Thus if we only look at the $\mathcal{O}$-module structures we know that

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}))}( j_!\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}_ U))}( \mathcal{M}, j^*\mathcal{N}) \]

(Recall that $\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}))$ denotes the graded category of graded $\mathcal{O}$-modules.) Then one has to check that these identifications map the $\mathcal{A}$-module maps on the left hand side to the $\mathcal{A}_ U$-module maps on the right hand side. To check this, given $\mathcal{O}_ U$-linear maps $f^ n : \mathcal{M}^ n \to j^*\mathcal{N}^{n + d}$ corresponding to $\mathcal{O}$-linear maps $g^ n : j_!\mathcal{M}^ n \to \mathcal{N}^{n + d}$ it suffices to show that

\[ \xymatrix{ \mathcal{M}^ n \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m \ar[r]_{f^ n \otimes 1} \ar[d] & j^*\mathcal{N}^{n + d} \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m \ar[d] \\ \mathcal{M}^{n + m} \ar[r]^{f^{n + m}} & j^*\mathcal{N}^{n + m + d} } \]

commutes if and only if

\[ \xymatrix{ j_!\mathcal{M}^ n \otimes _\mathcal {O} \mathcal{A}^ m \ar[r]_{g^ n \otimes 1} \ar[d] & \mathcal{N}^{n + d} \otimes _\mathcal {O} \mathcal{A}_ U^ m \ar[d] \\ j_!\mathcal{M}^{n + m} \ar[r]^{g^{n + m}} & \mathcal{N}^{n + m + d} } \]

commutes. However, we know that

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{M}^ n \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m, j^*\mathcal{N}^{n + d + m}) & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!(\mathcal{M}^ n \otimes _{\mathcal{O}_ U} \mathcal{A}_ U^ m), \mathcal{N}^{n + d + m}) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{M}^ n \otimes _\mathcal {O} \mathcal{A}^ m, \mathcal{N}^{n + d + m}) \end{align*}

by the already used Modules on Sites, Lemma 18.27.9. We omit the verification that shows that the obstruction to the commutativity of the first diagram in the first group maps to the obstruction to the commutativity of the second diagram in the last group. $\square$

Lemma 24.10.2. In the situation above, let $\mathcal{M}$ be a right graded $\mathcal{A}_ U$-module and let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. Then

\[ j_!\mathcal{M} \otimes _\mathcal {A} \mathcal{N} = j_!(\mathcal{M} \otimes _{\mathcal{A}_ U} \mathcal{N}|_ U) \]

as graded $\mathcal{O}$-modules functorially in $\mathcal{M}$ and $\mathcal{N}$.

Proof. Recall that the degree $n$ component of $j_!\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ is the cokernel of the canonical map

\[ \bigoplus \nolimits _{r + s + t = n} j_!\mathcal{M}^ r \otimes _\mathcal {O} \mathcal{A}^ s \otimes _\mathcal {O} \mathcal{N}^ t \longrightarrow \bigoplus \nolimits _{p + q = n} j_!\mathcal{M}^ p \otimes _\mathcal {O} \mathcal{N}^ q \]

See Section 24.6. By Modules on Sites, Lemma 18.27.9 this is the same thing as the cokernel of

\[ \bigoplus \nolimits _{r + s + t = n} j_!(\mathcal{M}^ r \otimes _{\mathcal{O}_ U} \mathcal{A}^ s|_ U \otimes _{\mathcal{O}_ U} \mathcal{N}^ t|_ U) \longrightarrow \bigoplus \nolimits _{p + q = n} j_!(\mathcal{M}^ p \otimes _{\mathcal{O}_ U} \mathcal{N}^ q|_ U) \]

and we win. An alternative proof would be to redo the Yoneda argument given in the proof of the lemma cited above. $\square$

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