20.41 Hom complexes
Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{L}^\bullet $ and $\mathcal{M}^\bullet $ be two complexes of $\mathcal{O}_ X$-modules. We construct a complex of $\mathcal{O}_ X$-modules $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet )$. Namely, for each $n$ we set
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n(\mathcal{L}^\bullet , \mathcal{M}^\bullet ) = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{L}^{-q}, \mathcal{M}^ p) \]
It is a good idea to think of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n$ as the sheaf of $\mathcal{O}_ X$-modules of all $\mathcal{O}_ X$-linear maps from $\mathcal{L}^\bullet $ to $\mathcal{M}^\bullet $ (viewed as graded $\mathcal{O}_ X$-modules) which are homogenous of degree $n$. In this terminology, we define the differential by the rule
\[ \text{d}(f) = \text{d}_\mathcal {M} \circ f - (-1)^ n f \circ \text{d}_\mathcal {L} \]
for $f \in \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_{\mathcal{O}_ X}(\mathcal{L}^\bullet , \mathcal{M}^\bullet )$. We omit the verification that $\text{d}^2 = 0$. This construction is a special case of Differential Graded Algebra, Example 22.26.6. It follows immediately from the construction that we have
20.41.0.1
\begin{equation} \label{cohomology-equation-cohomology-hom-complex} H^ n(\Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(\mathcal{L}^\bullet , \mathcal{M}^\bullet [n]) \end{equation}
for all $n \in \mathbf{Z}$ and every open $U \subset X$.
Lemma 20.41.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}_ X$-modules there is an isomorphism
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet )) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ), \mathcal{M}^\bullet ) \]
of complexes of $\mathcal{O}_ X$-modules functorial in $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $.
Proof.
Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.1.
$\square$
Lemma 20.41.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical morphism
\[ \text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{L}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{M}^\bullet ) \]
of complexes of $\mathcal{O}_ X$-modules.
Proof.
Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.3.
$\square$
Lemma 20.41.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical morphism
\[ \text{Tot}\left( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{L}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )) \]
of complexes of $\mathcal{O}_ X$-modules functorial in all three complexes.
Proof.
Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.4.
$\square$
Lemma 20.41.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical morphism
\[ \mathcal{K}^\bullet \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )) \]
of complexes of $\mathcal{O}_ X$-modules functorial in both complexes.
Proof.
Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.5.
$\square$
Lemma 20.41.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical morphism
\[ \text{Tot}(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \otimes _{\mathcal{O}_ X} \mathcal{K}^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{L}^\bullet ), \mathcal{M}^\bullet ) \]
of complexes of $\mathcal{O}_ X$-modules functorial in all three complexes.
Proof.
Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.6.
$\square$
Lemma 20.41.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}_ X$-modules. Let $\mathcal{L}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. 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 $U \subset X$ open.
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 (20.41.0.1). The second equality is true because $\mathcal{I}^\bullet |_ U$ is K-injective by Lemma 20.32.1.
$\square$
Lemma 20.41.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(\mathcal{I}')^\bullet \to \mathcal{I}^\bullet $ be a quasi-isomorphism of K-injective complexes of $\mathcal{O}_ X$-modules. Let $(\mathcal{L}')^\bullet \to \mathcal{L}^\bullet $ be a quasi-isomorphism of complexes of $\mathcal{O}_ X$-modules. Then
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , (\mathcal{I}')^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ((\mathcal{L}')^\bullet , \mathcal{I}^\bullet ) \]
is a quasi-isomorphism.
Proof.
Let $M$ be the object of $D(\mathcal{O}_ X)$ represented by $\mathcal{I}^\bullet $ and $(\mathcal{I}')^\bullet $. Let $L$ be the object of $D(\mathcal{O}_ X)$ represented by $\mathcal{L}^\bullet $ and $(\mathcal{L}')^\bullet $. By Lemma 20.41.6 we see that the sheaves
\[ H^0(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , (\mathcal{I}')^\bullet )) \quad \text{and}\quad H^0(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ((\mathcal{L}')^\bullet , \mathcal{I}^\bullet )) \]
are both equal to the sheaf associated to the presheaf
\[ U \longmapsto \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \]
Thus the map is a quasi-isomorphism.
$\square$
Lemma 20.41.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}_ X$-modules. Let $\mathcal{L}^\bullet $ be a K-flat complex of $\mathcal{O}_ X$-modules. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}_ X$-modules.
Proof.
Namely, if $\mathcal{K}^\bullet $ is an acyclic complex of $\mathcal{O}_ X$-modules, then
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )) & = H^0(\Gamma (X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )))) \\ & = H^0(\Gamma (X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\text{Tot}( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ))) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}( \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ) \\ & = 0 \end{align*}
The first equality by (20.41.0.1). The second equality by Lemma 20.41.1. The third equality by (20.41.0.1). The final equality because $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )$ is acyclic because $\mathcal{L}^\bullet $ is K-flat (Definition 20.26.2) and because $\mathcal{I}^\bullet $ is K-injective.
$\square$
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