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