The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

20.35 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 22.19.6. It follows immediately from the construction that we have

20.35.0.1
\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 20.35.1. 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 15.67.1. $\square$

Lemma 20.35.2. 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 15.67.2. $\square$

Lemma 20.35.3. 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

\[ \text{Tot}(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \otimes _{\mathcal{O}_ X} \mathcal{K}^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{L}^\bullet ), \mathcal{M}^\bullet ) \]

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 15.67.3. $\square$

Lemma 20.35.4. 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 15.67.5. $\square$

Lemma 20.35.5. 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 15.67.6. $\square$

Lemma 20.35.6. 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 (20.35.0.1). The second equality is true because $\mathcal{I}^\bullet |_ U$ is K-injective by Lemma 20.30.1. $\square$

Lemma 20.35.7. 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 20.35.6 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 20.35.8. 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 (20.35.0.1). The second equality by Lemma 20.35.1. The third equality by (20.35.0.1). 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 20.27.2) and because $\mathcal{I}^\bullet $ is K-injective. $\square$


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