ScalingStacks

Weight structures and weight complex functors can be used to provide a convenient language for the Koszul duality formalism from [BGS96], and slightly more generally [MOS09].

Let $A$ be a $\mathbb{Z}$-graded algebra which is positively graded, that is, $A^{i}=0$ for $i<0$ and assume that $A_{0}$ is semisimple and finite dimensional. Denote by $\langle 1\rangle$ the shift of grading functor on the category $A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}$ of finitely generated graded $A$-modules and let $\mathcal{T}(A)$ be a set of representatives of simple objects concentrated in degree $0.$

The algebra $A$ is called Koszul if $\operatorname{Ext}^{i}(L,L^{\prime}\langle j\rangle)=0\text{ for all }i\neq j$ and $L,L^{\prime}\in\mathcal{T}(A).$ Equivalently, $A$ is Koszul if the family $\widehat{\mathcal{T}}(A)=\bigcup_{i}\mathcal{T}(A)\langle i\rangle[i]$ is tilting in $\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}).$ In this case, the Koszul dual $A^{!}$ of $A$ is the graded algebra

$$A^{!}=\operatorname{End}^{\bullet}_{\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}})}(\mathcal{T}(\mathcal{A}))$$

where the right hand side is defined as in (2.2) using the autoequivalence $\langle 1\rangle[1].$

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].