21.34 Hom complexes
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{L}^\bullet $ and $\mathcal{M}^\bullet $ be two complexes of $\mathcal{O}$-modules. We construct a complex of $\mathcal{O}$-modules $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet )$. Namely, for each $n$ we set
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n(\mathcal{L}^\bullet , \mathcal{M}^\bullet ) = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}^{-q}, \mathcal{M}^ p) \]
It is a good idea to think of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n$ as the sheaf of $\mathcal{O}$-modules of all $\mathcal{O}$-linear maps from $\mathcal{L}^\bullet $ to $\mathcal{M}^\bullet $ (viewed as graded $\mathcal{O}$-modules) which are homogeneous 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 $f \in \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {O}(\mathcal{L}^\bullet , \mathcal{M}^\bullet )$. 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
21.34.0.1
\begin{equation} \label{sites-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 |_ U, \mathcal{M}^\bullet [n]|_ U) \end{equation}
for all $n \in \mathbf{Z}$ and every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Similarly, we have
21.34.0.2
\begin{equation} \label{sites-cohomology-equation-global-cohomology-hom-complex} H^ n(\Gamma (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{L}^\bullet , \mathcal{M}^\bullet [n]) \end{equation}
for the complex of global sections.
Lemma 21.34.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}$-modules there is an isomorphism
\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet )) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ), \mathcal{M}^\bullet ) \]
of complexes of $\mathcal{O}$-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 21.34.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}$-modules there is a canonical morphism
\[ \text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{L}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{M}^\bullet ) \]
of complexes of $\mathcal{O}$-modules.
Proof.
Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.3.
$\square$
Lemma 21.34.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}$-modules there is a canonical morphism
\[ \text{Tot}\left( \mathcal{K}^\bullet \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{L}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )) \]
of complexes of $\mathcal{O}$-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 21.34.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}$-modules there is a canonical morphism
\[ \mathcal{K}^\bullet \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )) \]
of complexes of $\mathcal{O}$-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 21.34.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}$-modules there is a canonical morphism
\[ \text{Tot}(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \otimes _\mathcal {O} \mathcal{K}^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{L}^\bullet ), \mathcal{M}^\bullet ) \]
of complexes of $\mathcal{O}$-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 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)}(\mathcal{L}^\bullet |_ U, \mathcal{I}^\bullet |_ 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).
$\square$
Lemma 21.34.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{I}')^\bullet \to \mathcal{I}^\bullet $ be a quasi-isomorphism of K-injective complexes of $\mathcal{O}$-modules. Let $(\mathcal{L}')^\bullet \to \mathcal{L}^\bullet $ be a quasi-isomorphism of complexes of $\mathcal{O}$-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 $D(\mathcal{O})$ represented by $\mathcal{I}^\bullet $ and $(\mathcal{I}')^\bullet $. Let $L$ be the object of $D(\mathcal{O})$ represented by $\mathcal{L}^\bullet $ and $(\mathcal{L}')^\bullet $. By Lemma 21.34.6 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 21.34.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet $ be a K-flat complex of $\mathcal{O}$-modules. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}$-modules.
Proof.
Namely, if $\mathcal{K}^\bullet $ is an acyclic complex of $\mathcal{O}$-modules, then
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )) & = H^0(\Gamma (\mathcal{C}, \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 (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\text{Tot}( \mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ))) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}( \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ) \\ & = 0 \end{align*}
The first equality by (21.34.0.2). The second equality by Lemma 21.34.1. The third equality by (21.34.0.2). The final equality because $\text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )$ is acyclic because $\mathcal{L}^\bullet $ is K-flat (Definition 21.17.2) and because $\mathcal{I}^\bullet $ is K-injective.
$\square$
Comments (2)
Comment #6369 by Hadi Hedayatzadeh on
Comment #6445 by Johan on