Lemma 21.34.6 (0A94)—The Stacks project

Lemma 21.34.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $L$ and $M$ be objects of $D(\mathcal{O})$. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules representing $M$. Let $\mathcal{L}^\bullet $ be a complex of $\mathcal{O}$-modules representing $L$. Then

\[ H^0(\Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \]

for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Similarly, $H^0(\Gamma (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(L, M)$.

Proof. We have

\begin{align*} H^0(\Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet ))) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(L|_ U, M|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \end{align*}

The first equality is (21.34.0.1). The second equality is true because $\mathcal{I}^\bullet |_ U$ is K-injective by Lemma 21.20.1. The proof of the last equation is similar except that it uses (21.34.0.2). $\square$

Comments (1)

Comment #8674 by ZL on September 24, 2023 at 12:21

There is a typo " $H^ 0(\Gamma (U, \mathop{\mathcal{H}\mathit{om}} \nolimits ^\bullet (\mathcal{L}^ \bullet , \mathcal{I} ^ \bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O} _ U)}(L| _ U, M| _ U)$ " should be " $H^ 0(\Gamma (U, \mathop{\mathcal{H}\mathit{om}} \nolimits ^\bullet (\mathcal{L}^ \bullet , \mathcal{I} ^ \bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O} _ U)}(L| _ U, \mathcal{I} ^ \bullet| _ U)$ "

There are also:

2 comment(s) on Section 21.34: Hom complexes

Comments (1)

Post a comment