ScalingStacks

Corollary 2.16 implies that the weight complex functor induces an equivalence of categories

(2.4)

$\displaystyle t:\langle\widehat{\mathcal{T}}(A)\rangle_{\cong,\Delta}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D_{perf}}(A^{!}).$

In the case that $A$ is finitely generated as an $A_{0}$-left and right module and $A^{!}$ is left Noetherian, (2.4) specializes to the Koszul duality from [BGS96, Theorem 2.12.5]

(2.5)

$\displaystyle\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D}^{b}(A^{!}\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}).$

The heart of the weight structure defined by $\mathcal{T}(A)$ on the left hand side maps to projective modules in cohomological degree $0$ on the right. Moreover, one can show that the heart of the standard $t$-structure on the right hand side corresponds to the category of linear complexes on the left hand side, see [MOS09].