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 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 $f \in \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {O}(\mathcal{L}^\bullet , \mathcal{M}^\bullet )$. 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
$$\label{sites-cohomology-equation-cohomology-hom-complex} H^ n(\Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(\mathcal{L}^\bullet |_ U, \mathcal{M}^\bullet [n]|_ U)$$

for all $n \in \mathbf{Z}$ and every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Similarly, we have

21.34.0.2
$$\label{sites-cohomology-equation-global-cohomology-hom-complex} H^ n(\Gamma (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{L}^\bullet , \mathcal{M}^\bullet [n])$$

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)}(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). $\square$

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. $\square$

1. On the right hand side of the equation (21.33.0.1), shouldn't one restrict the complexes to $U$?
2. What are $L$ and $M$ in Lemma 21.33.6? Are these the objects represented by $\mathcal{L}^{\bullet}$ and $\mathcal{I}^{\bullet}$?

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