Remark 75.17.5 (0DK9): Uniqueness of dga

Remark 75.17.5 (Uniqueness of dga). Let $X$ be a quasi-compact and quasi-separated algebraic space over a ring $R$. By the construction of the proof of Theorem 75.17.3 there exists a differential graded algebra $(A, \text{d})$ over $R$ such that $D_\mathit{QCoh}(X)$ is $R$-linearly equivalent to $D(A, \text{d})$ as a triangulated category. One may ask: how unique is $(A, \text{d})$? The answer is (only) slightly better than just saying that $(A, \text{d})$ is well defined up to derived equivalence. Namely, suppose that $(B, \text{d})$ is a second such pair. Then we have

\[ (A, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , K^\bullet ) \]

and

\[ (B, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(L^\bullet , L^\bullet ) \]

for some K-injective complexes $K^\bullet $ and $L^\bullet $ of $\mathcal{O}_ X$-modules corresponding to perfect generators of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Set

\[ \Omega = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , L^\bullet ) \quad \Omega ' = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(L^\bullet , K^\bullet ) \]

Then $\Omega $ is a differential graded $B^{opp} \otimes _ R A$-module and $\Omega '$ is a differential graded $A^{opp} \otimes _ R B$-module. Moreover, the equivalence

\[ D(A, \text{d}) \to D_\mathit{QCoh}(\mathcal{O}_ X) \to D(B, \text{d}) \]

is given by the functor $- \otimes _ A^\mathbf {L} \Omega '$ and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark 22.37.10. If we ever need this remark we will provide a precise statement with a detailed proof here.

The Stacks project

Comments (0)

Post a comment