Lemma 24.25.5. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let (\mathcal{A}, \text{d}) be a sheaf of differential graded algebras on (\mathcal{C}, \mathcal{O}). There exists a set T and for each t \in T an injective map \mathcal{M}_ t \to \mathcal{M}'_ t of acyclic differential graded \mathcal{A}-modules such that for an object \mathcal{I} of \textit{Mod}(\mathcal{A}, \text{d}) the following are equivalent
\mathcal{I} is graded injective, and
for every solid diagram
\xymatrix{ \mathcal{M}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ t \ar@{..>}[ru] }
a dotted arrow exists in \textit{Mod}(\mathcal{A}, \text{d}) making the diagram commute.
Proof.
Let T and \mathcal{N}_ t \to \mathcal{N}'_ t be as in Lemma 24.25.1. Denote F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A}) the forgetful functor. Let G be the left adjoint functor to F as in Lemma 24.24.1. Set
\mathcal{M}_ t = G(\mathcal{N}_ t) \to G(\mathcal{N}'_ t) = \mathcal{M}'_ t
This is an injective map of acyclic differential graded \mathcal{A}-modules by Lemma 24.24.2. Since G is the left adjoint to F we see that there exists a dotted arrow in the diagram
\xymatrix{ \mathcal{M}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ t \ar@{..>}[ru] }
if and only if there exists a dotted arrow in the diagram
\xymatrix{ \mathcal{N}_ t \ar[r] \ar[d] & F(\mathcal{I}) \\ \mathcal{N}'_ t \ar@{..>}[ru] }
Hence the result follows from the choice of our collection of arrows \mathcal{N}_ t \to \mathcal{N}_ t'.
\square
Comments (0)