The Stacks project

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. $\square$

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