2020 Mathematics Subject Classification. Primary 16E30, 18G35, 16S37; Secondary 18E30, 18E35, 16G20, 16E45.

Keywords. Koszul duality, dg Koszul duality, dg orbit categories, triangulated hulls, derived categories, singularity categories, BGG correspondence, finite-dimensional graded algebras, quadratic algebras, Koszul algebras, stable category of Gorenstein-projective modules.

1 Introduction↩︎

Koszul algebras, introduced by Priddy [1] following the seminal work of Koszul on the cohomology of Lie algebras [2], occupy a central position in modern algebra, representation theory, and algebraic geometry. A defining feature of this class of algebras is the existence of a highly structured duality theory, namely Koszul duality, which relates a Koszul algebra to its Koszul dual through deep homological correspondences.

The origins of this theory go back to the work of Bernstein–Gel’fand–Gel’fand, culminating in the celebrated BGG correspondence [3], which identifies the stable category of finite-dimensional graded modules over the exterior algebra with the bounded derived category of coherent sheaves on projective space. This correspondence has had far-reaching consequences; see, for instance, [4]–[6]. From a geometric viewpoint, Koszul duality is closely tied to the structure of derived categories of coherent sheaves. Beilinson [7] showed that the bounded derived category of coherent sheaves on projective space is equivalent to the bounded derived category of modules over a finite-dimensional Koszul algebra, while Kapranov [8] extended this perspective to Grassmannians and other homogeneous spaces.

A major conceptual advance was achieved by Beilinson, Ginzburg, and Soergel [9], who formulated Koszul duality in the framework of derived categories. In particular, for a finite-dimensional Koszul algebra \(\Lambda\) whose Koszul dual \(\Lambda^{!}\) is left Noetherian, they obtained a triangulated equivalence \[\mathsf{D}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{Fg}^{\mathbb{Z}}\bigr) \;\simeq\; \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-gmod}\bigr).\] Subsequently, Martínez–Villa and Saorín [10] weakened these assumptions by replacing Noetherianity with coherence.

Despite these developments, two fundamental problems remain open in general. First, there is, to the best of our knowledge, no non-graded derived Koszul duality describing the bounded derived category \(\mathsf{D}^{b}(\Lambda\mathrm{-mod})\) of a finite-dimensional Koszul algebra in terms of its Koszul dual. Second, even in the graded setting, it is not known in general whether the bounded derived category \(\mathsf{D}^{b}(\Lambda\mathrm{-gmod})\) admits such a description when the Koszul dual \(\Lambda^{!}\) is infinite-dimensional and fails to satisfy finiteness conditions such as Noetherianity or coherence.

The purpose of the present paper is to address both of these problems. For every finite-dimensional Koszul algebra \(\Lambda\), we establish a graded derived Koszul duality without imposing any finiteness conditions on the Koszul dual algebra \(\Lambda^{!}\), and we use this graded theory to construct a genuinely non-graded derived Koszul duality. Thus the present work extends the classical theory in two distinct directions: it removes the usual finiteness assumptions from the graded setting, and it provides a new description of the ungraded bounded derived category in terms of the Koszul dual algebra.

The key idea is to reconstruct the bounded derived category of finite-dimensional modules from graded Koszul duality by means of differential graded orbit constructions and their triangulated hulls. This yields a conceptual framework in which the non-graded bounded derived category arises naturally from the graded theory.

Main results. Let \(\Lambda\) be a finite-dimensional Koszul algebra.

Our first main result establishes a graded derived Koszul duality without any finiteness assumptions on \(\Lambda^{!}\). More precisely, we construct a triangulated equivalence \[\mathsf{D}^b\bigl(\Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}}\bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^b\bigl(\Lambda\mathrm{-gmod}\bigr),\] where \(\Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}}\) denotes the category of locally finite-dimensional graded coperfect \(\Lambda^{!}\)-modules. In particular, this identifies \(\mathsf{D}^{b}(\Lambda\mathrm{-gmod})\) entirely in terms of the Koszul dual algebra.

Moreover, this equivalence admits a singular refinement, yielding a graded singular Koszul duality \[\mathsf{D}^{b}\!\Bigl( \Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}} \big/ \Lambda^{!}\mathrm{-gmod} \Bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}_{\mathrm{sg}}\!\bigl(\Lambda\mathrm{-gmod}\bigr).\] Here \(\Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}}/\Lambda^{!}\mathrm{-gmod}\) denotes the exact quotient category, which may be viewed as a generalization of the dual analogue of the tails category associated with \(\Lambda^{!}\).

When \(\Lambda\) is Iwanaga–Gorenstein, this further induces a graded Bernstein–Gel’fand–Gel’fand type correspondence \[\mathsf{D}^{b}\!\Bigl( \Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}} \big/ \Lambda^{!}\mathrm{-gmod} \Bigr) \;\xrightarrow{\;\sim\;}\; \Lambda\mathrm{-}\underline{\mathrm{Gproj}}^{\mathbb{Z}}.\]

Our second main result provides a non-graded derived Koszul duality. More precisely, we construct a triangulated equivalence \[\mathrm{H}^{0}\!\Bigl( \operatorname{pretr}\Bigl( \mathsf{D}^{b}_{\mathrm{dg}}\!\bigl(\Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}}\bigr) /\langle 1\rangle[-1] \Bigr) \Bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-mod}\bigr).\] Thus the bounded derived category of finite-dimensional \(\Lambda\)-modules is recovered functorially from the graded Koszul dual side.

We also establish a singular analogue: \[\mathrm{H}^{0}\!\left( \operatorname{pretr}\!\left( \mathsf{D}^{b}_{\mathrm{dg}} \bigl( \Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}} / \Lambda^{!}\mathrm{-gmod} \bigr) / \langle 1\rangle[-1] \right) \right) \;\xrightarrow{\;\sim\;}\; \mathsf{D}_{\mathrm{sg}}\!\bigl(\Lambda\mathrm{-mod}\bigr).\]

When \(\Lambda\) is Iwanaga–Gorenstein, this yields a non-graded form of stable Koszul duality and recovers, in particular, a non-graded analogue of the Bernstein–Gel’fand–Gel’fand correspondence.

These dualities have several applications. First, they provide new descriptions of the bounded derived categories \(\mathsf{D}^{b}(\mathcal{O}_{\lambda})\) for all integral blocks of the Bernstein–Gel’fand–Gel’fand category \(\mathcal{O}\), including singular blocks, thereby extending the classical picture to the non-graded setting.

Second, we obtain analogous derived Koszul dualities for certain categories of perverse sheaves arising in geometric representation theory.

Finally, the preceding results suggest a broader picture. Motivated by the dg orbit descriptions established in this paper, we formulate conjectural descriptions of bounded derived categories and singularity categories of finite-dimensional graded algebras in terms of dg orbit categories and their triangulated hulls, and we provide supporting evidence for this perspective.

Organization of the paper. Section 2 fixes notation and reviews the necessary background. Section 3 develops graded derived Koszul duality and its singular and dg refinements. Section 4 treats the non-graded setting and establishes the main results. Section 5 presents applications to category \(\mathcal{O}\) and perverse sheaves. Section 6 studies explicit classes of algebras, and Section 7 formulates conjectural extensions and supporting evidence.

Acknowledgements↩︎

This work was completed in 2021; its preparation for publication was delayed due to personal circumstances and other commitments.

The author dedicates this work to the memory of I. M. Gel’fand, whose profound contributions have had a lasting influence on modern mathematics. The author first encountered Gel’fand’s work during graduate studies in Morocco through functional analysis, in particular Banach algebras, and later through homological algebra via the monograph Methods of Homological Algebra by S. I. Gel’fand and Y. I. Manin.

2 Preliminaries and Background↩︎

In this section we recall some basic material and fix the notation used throughout the paper.

Let \(k\) be a field. We consider \(\mathbb{Z}_{\ge 0}\)-graded \(k\)-algebras of the form \[\Lambda = kQ/I,\] where \(Q\) is a finite quiver and \(I\subset kQ\) is a homogeneous ideal. The grading on \(\Lambda\) is induced by the path-length grading on the path algebra \(kQ\).

The ideal \(I\) is not assumed to be admissible; in particular, the algebra \(\Lambda\) may be either finite-dimensional or locally finite-dimensional. This level of generality is required for our purposes. Although in most applications \(\Lambda\) will be finite-dimensional and Koszul, its Koszul dual \(\Lambda^{!}\) need not be finite-dimensional. Nevertheless, \(\Lambda^{!}\) is locally finite-dimensional in the sense that each homogeneous component \(\Lambda^{!}_{n}\) is finite-dimensional for \(n\ge 0\). Equivalently, for all vertices \(i,j\in Q_{0}\) the graded component \(e_{j}\Lambda_{n}e_{i}\) is finite-dimensional.

We view such algebras as small \(k\)-linear graded categories whose objects are the vertices of \(Q\). For vertices \(i,j\in Q_{0}\), the morphism space \[\Lambda(i,j)=e_{j}\Lambda e_{i}\] is the \(k\)-vector space spanned by paths from \(i\) to \(j\) modulo the relations in \(I\), endowed with the grading induced from \(kQ\).

2.1 Modules and their categories↩︎

A quiver \(Q\) is a quadruple \[Q=(Q_{0},Q_{1},s,t),\] where \(Q_{0}\) is a finite set of vertices, \(Q_{1}\) is a finite set of arrows, and \(s,t\colon Q_{1}\to Q_{0}\) assign to each arrow \(\alpha\in Q_{1}\) its source \(s(\alpha)\) and target \(t(\alpha)\), respectively.

We say that \(Q\) is well directed if there exists a total order on \(Q_0\) such that every arrow connects consecutive vertices and all arrows are oriented in the same direction with respect to this order. In this case, \(Q_0\) can be identified with a finite subset of \(\mathbb{Z}\). In particular, once a direction (either increasing or decreasing) is fixed, no arrow between two consecutive vertices is oriented oppositely. It follows that a well directed quiver contains no oriented cycles. In the infinite setting, the same condition is often referred to as gradability; see [11], [12].

The path algebra \(kQ\) of a quiver \(Q\) over \(k\) is the \(k\)-algebra whose underlying vector space has a basis consisting of all paths in \(Q\), including the trivial paths \(e_x\) at vertices \(x\in Q_0\). Multiplication is given by concatenation of composable paths and is defined to be zero otherwise.

An ideal \(I\subset kQ\) is said to be admissible if there exists an integer \(m\ge 2\) such that \[(kQ)^{m}\subseteq I\subseteq (kQ)^{2},\] where \((kQ)^{n}\) denotes the ideal generated by all paths of length at least \(n\). The quotient algebra \[\Lambda = kQ/I\] is called the bound path algebra associated with the bound quiver \((Q,I)\). If \(I\) is homogeneous with respect to the path-length grading, then \(\Lambda\) inherits a natural \(\mathbb{Z}\)-grading.

If, in addition, \(Q\) is well directed, then \(\Lambda\) has finite global dimension.

In view of the categorical constructions appearing throughout this paper, it is convenient to adopt a functorial perspective on modules. Let \(\Lambda = kQ/I\) be as above. Then \(\Lambda\) admits a complete set of pairwise orthogonal primitive idempotents \(\{e_x \mid x \in Q_0\}\).

A (left) \(\Lambda\)-module \(M\) may be identified with a representation of the bound quiver \((Q,I)\), or equivalently with a \(k\)-linear functor \[M \colon \Lambda \longrightarrow \mathrm{Vect}_k.\] Explicitly, \(M\) assigns to each vertex \(x \in Q_0\) a \(k\)-vector space \(M(x)\cong e_x M\), and to each morphism \(\rho \in \Lambda(x,y)\) a \(k\)-linear map \[M(\rho) \colon M(x) \longrightarrow M(y),\] such that \(M(\rho\sigma)=M(\rho)\circ M(\sigma)\) whenever the composition is defined, and such that the ideal \(I\) acts trivially.

We shall freely use this identification and write \(M(x)\) in place of \(e_x M\).

A morphism \(f\colon M \to N\) of \(\Lambda\)-modules is a natural transformation, that is, a family of \(k\)-linear maps \[f(x)\colon M(x)\longrightarrow N(x), \qquad x\in Q_0,\] such that for every \(\rho \in \Lambda(x,y)\) one has \[f(y)\circ M(\rho)=N(\rho)\circ f(x).\]

In the graded setting, we replace \(\mathrm{Vect}_{k}\) by \(\mathrm{GrVect}_{k}\), the category of \(\mathbb{Z}\)-graded \(k\)-vector spaces and degree-preserving linear maps. A \(\mathbb{Z}\)-graded \(\Lambda\)-module may therefore be regarded as a \(k\)-linear functor \[M\colon \Lambda \longrightarrow \mathrm{GrVect}_{k},\] which assigns to each vertex \(x\in Q_{0}\) a graded vector space \[M(x)=\bigoplus_{n\in\mathbb{Z}} M_{n}(x).\]

For each homogeneous morphism \(\rho\in \Lambda(x,y)\) of degree \(d\), the induced map \[M(\rho)\colon M(x)\longrightarrow M(y)\] is homogeneous of degree \(d\); that is, \[M(\rho)\bigl(M_{n}(x)\bigr)\subseteq M_{n+d}(y) \qquad \text{for all } n\in\mathbb{Z}.\]

In particular, if \(\alpha\colon x\to y\) is an arrow of the quiver \(Q\), then \(\alpha\) has degree \(1\). Consequently the induced morphism \[M(\alpha)\colon M(x)\longrightarrow M(y)\] satisfies \[M(\alpha)\bigl(M_{n}(x)\bigr)\subseteq M_{n+1}(y) \qquad \text{for all } n\in\mathbb{Z}.\]

Thus the arrows of the quiver induce linear maps between the graded components, and by summing over all vertices we obtain a sequence (not necessarily a complex) of finite-dimensional vector spaces and linear maps \[\cdots \longrightarrow \bigoplus_{z\in Q_{0}} M_{n-1}(z) \longrightarrow \bigoplus_{x\in Q_{0}} M_{n}(x) \longrightarrow \bigoplus_{y\in Q_{0}} M_{n+1}(y) \longrightarrow \cdots .\]

This observation will play an important role in the construction of the derived quadratic functor; see Section 3. A morphism \(f\colon M\to N\) of graded \(\Lambda\)-modules is a natural transformation whose components have degree \(0\), equivalently \(f(x)\bigl(M_{n}(x)\bigr)\subseteq N_{n}(x)\) for all \(x\in Q_{0}\) and \(n\in\mathbb{Z}\), and \(f(y)\circ M(\rho)=N(\rho)\circ f(x)\) for all \(\rho\in\Lambda(x,y)\).

We shall use the following notation for categories of modules:

• \(\Lambda\mathrm{-}\mathrm{mod}\): the category of finite-dimensional \(\Lambda\)-modules, that is, modules \(M\) such that each component \(M(x)\) is finite-dimensional and \(M(x)=0\) for all but finitely many vertices \(x\in Q_{0}\);

• \(\Lambda\mathrm{-}\mathrm{gmod}\): the category of finite-dimensional \(\mathbb{Z}\)-graded \(\Lambda\)-modules;

• \(\Lambda\mathrm{-}\mathrm{GMod}\): the category of locally finite-dimensional \(\mathbb{Z}\)-graded \(\Lambda\)-modules; that is, graded modules \(M=\bigoplus_{r\in\mathbb{Z}} M_r\) such that, for each vertex \(x\in Q_0\) and each \(r\in\mathbb{Z}\), the component \(M_r(x)\) is finite-dimensional.

A graded module \(M\in \Lambda\mathrm{-}\mathrm{GMod}\) is said to be right bounded (or bounded above) if there exists \(m\in\mathbb{Z}\) such that \(M_n=0\) for all \(n>m\), and left bounded (or bounded below) if there exists \(m\in\mathbb{Z}\) such that \(M_n=0\) for all \(n<m\). Since objects of \(\Lambda\mathrm{-}\mathrm{GMod}\) are locally finite-dimensional, a graded module \(M\) is finite-dimensional if and only if it is both left bounded and right bounded.

We denote by \(\Lambda\mathrm{-}\mathrm{GMod}^{-}\) (respectively \(\Lambda\mathrm{-}\mathrm{GMod}^{+}\)) the full subcategory of right bounded (respectively left bounded) graded modules.

Both \(\Lambda\mathrm{-}\mathrm{GMod}\) and \(\Lambda\mathrm{-}\mathrm{gmod}\) admit the grading shift autoequivalences \(\langle i\rangle\) for \(i\in\mathbb{Z}\). For a graded module \(M\) and a vertex \(x\in Q_{0}\), the shifted module is given by \[M\langle i\rangle(x)=\bigoplus_{n\in\mathbb{Z}} M_{n+i}(x).\] Note that the grading shift used in [9] is opposite to the convention adopted here.

We fix notation for the graded projective, injective, and simple modules over \(\Lambda\). Let \[\mathbb{D}=\operatorname{Hom}_{k}(-,k)\] denote the \(k\)-linear dual, endowed with the standard grading \((\mathbb{D}V)_{i}=\operatorname{Hom}_{k}(V_{-i},k)\). For each vertex \(x\in Q_{0}\) and each \(r\in\mathbb{Z}\), set \[P_{x}\langle r\rangle:=\Lambda(x,-)\langle r\rangle,\qquad I_{x}\langle r\rangle:=\mathbb{D}\!\bigl(\Lambda^{\mathrm{op}}(x,-)\bigr)\langle r\rangle,\] and let \[S_{x}\langle r\rangle\] denote the degree shift of the simple top of \(P_x\). All these modules are indecomposable.

If \(\Lambda\) is locally finite-dimensional, we denote by \(\Lambda\mathrm{-}\mathrm{Proj}^{\mathbb{Z}}\) (resp. \(\Lambda\mathrm{-}\mathrm{Inj}^{\mathbb{Z}}\)) the full additive subcategory of locally finite-dimensional graded projective (resp.graded injective) \(\Lambda\)-modules; that is, finite direct sums of modules of the form \(P_x\langle r\rangle\) (resp.\(I_x\langle r\rangle\)), where \(x\in Q_0\) and \(r\in\mathbb{Z}\).

If \(\Lambda\) is finite-dimensional, then every graded projective and graded injective \(\Lambda\)-module is finite-dimensional. In this case we write \(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\) and \(\Lambda\mathrm{-}\mathrm{inj}^{\mathbb{Z}}\) for the corresponding full subcategories.

Upon forgetting the grading on \(\Lambda\), we write \(\Lambda\mathrm{-}\mathrm{proj}\) (resp.\(\Lambda\mathrm{-}\mathrm{inj}\)) for the category of finite-dimensional projective (resp.injective) \(\Lambda\)-modules.

Any \(\mathbb{Z}\)-graded \(\Lambda\)-module may be regarded as an (ungraded) \(\Lambda\)-module by forgetting the grading. This defines the canonical forgetful functor \[F\colon \Lambda\mathrm{-}\mathrm{gmod}\longrightarrow \Lambda\mathrm{-}\mathrm{mod}.\]

We record the following standard observation for later use.

2.2 Koszul Algebras and Their Duals↩︎

The aim of this subsection is to recall basic facts on Koszul algebras and the quadratic dual of a quadratic algebra. We then formulate the notion of Koszulity and record the standard homological characterizations used throughout the paper. For further background, we refer the reader to [9], [10], [13], [14].

Let \(\Lambda = kQ/I\) be a locally finite-dimensional graded algebra. We say that \(\Lambda\) is quadratic if \(I\) is generated in degree \(2\), that is, by \(k\)-linear combinations of paths of length \(2\). Write \(V = (kQ)_{2}\) for the \(k\)-vector space spanned by all paths of length \(2\) in \(kQ\), and set \(I_{2} = I \cap V\) for the subspace of quadratic relations.

Let \(Q^{\mathrm{op}}\) be the opposite quiver and \(V^{\mathrm{op}} = (kQ^{\mathrm{op}})_{2}\) the space of paths of length \(2\) in \(Q^{\mathrm{op}}\). Fix a basis \(\{\gamma_{i}\}\) of \(V\) with dual basis \(\{\gamma_{i}^{\mathrm{op}}\} \subset V^{\mathrm{op}}\), and consider the nondegenerate pairing \[\langle - , - \rangle \colon V \times V^{\mathrm{op}} \longrightarrow k, \qquad \langle \gamma_{i}, \gamma_{j}^{\mathrm{op}} \rangle = \delta_{ij}.\]

The quadratic dual is the graded algebra \[\Lambda^{!} = kQ^{\mathrm{op}} / \langle I_{2}^{\perp} \rangle,\] where \(I_{2}^{\perp} \subset V^{\mathrm{op}}\) denotes the orthogonal complement of \(I_{2}\) with respect to the pairing \(\langle - , - \rangle\). Even if \(\Lambda\) is finite-dimensional, the algebra \(\Lambda^{!}\) is in general only locally finite-dimensional, and need not be finite-dimensional.

The next criterion will be invoked in the proof of our main result on graded derived Koszul duality.

Recall that if \(A=\bigoplus_{i\ge 0}A_i\) is a positively graded, locally finite-dimensional algebra with semisimple degree-zero part \(A_0\), then \[\operatorname{gldim}A \;=\; \sup\{\, n\ge 0 \mid \operatorname{Ext}^n_A(A_0,A_0)\neq 0 \,\},\] see, e.g., [16].

We shall use the following standard characterization of finite-dimensional Koszul algebras in terms of the homological dimension of their Koszul duals.

2.3 The Bounded Derived Category of an Exact Category↩︎

The aim of this subsection is to recall the construction of the bounded derived category of an exact category in the sense of Quillen. Our exposition follows [17], [18]; for general background on triangulated categories we refer to [19], [20]. The necessity of this discussion stems from the fact that, in the Koszul dualities considered in this paper, one is naturally led to work with bounded derived categories of certain exact categories which are, in general, not abelian. Nevertheless, all exact categories under consideration admit fully faithful embeddings into abelian categories, thereby allowing the use of the standard tools of homological algebra.

Recall that a category is additive if it is preadditive (that is, each \(\operatorname{Hom}\)-set is an abelian group and composition is bilinear), admits a zero object, and has all finite biproducts. An abelian category is an additive category in which every morphism admits both a kernel and a cokernel, and in which every monomorphism (resp.epimorphism) is the kernel (resp.cokernel) of some morphism. Equivalently, every morphism factors as an epimorphism followed by a monomorphism, and the class of short exact sequences is stable under both pushouts and pullbacks.

From now on, fix an exact category \(\mathcal{B}\) (in the sense of Quillen), viewed as a full subcategory of an ambient abelian category \(\mathcal{C}\) that realizes its exact structure. We assume throughout that \(\mathcal{B}\) satisfies the two–out–of–three property for short exact sequences in \(\mathcal{C}\); we henceforth suppress explicit reference to the chosen exact structure.

Assume now that \(\mathcal{B}\) is an abelian category, and denote by \(\operatorname{Proj}(\mathcal{B})\) (resp.\(\operatorname{Inj}(\mathcal{B})\)) the full subcategory of projective (resp.injective) objects of \(\mathcal{B}\).

\(\bullet\) Assume that \(\mathcal{B}\) has enough projectives. Then every object \(X^\bullet\in \mathsf{D}^{b}(\mathcal{B})\) admits a projective resolution, that is, there exists a quasi-isomorphism \[P^\bullet \longrightarrow X^\bullet\] with \(P^\bullet\) a bounded above complex of projective objects and with bounded cohomology. Choose \(a\in \mathbb{Z}\) such that \[H^{i}(X^\bullet)=0 \qquad \text{for all } i<a.\] Then the induced morphism \[\tau_{\ge a}P^\bullet \longrightarrow \tau_{\ge a}X^\bullet\] is a quasi-isomorphism. In particular, since \(\tau_{\ge a}P^\bullet\cong X^\bullet\) in \(\mathsf{D}^{b}(\mathcal{B})\), this construction, together with the distinguished triangle above, yields a canonical triangulated equivalence \[\mathfrak{T}\colon \mathsf{K}^{-,b}\!\bigl(\operatorname{Proj}(\mathcal{B})\bigr) \xrightarrow{\;\sim\;} \mathsf{D}^{b}(\mathcal{B}).\]

Moreover, if \(P^\bullet\to X^\bullet\) is such a projective resolution and \(Y^\bullet\in \mathsf{D}^{b}(\mathcal{B})\), then there is a natural isomorphism \[\operatorname{Hom}_{\mathsf{K}^{-}(\mathcal{B})}\!\bigl(P^\bullet,Y^\bullet\bigr) \;\cong\; \operatorname{Hom}_{\mathsf{D}^{b}(\mathcal{B})}\!\bigl(X^\bullet,Y^\bullet\bigr).\]

\(\bullet\) Dually, if \(\mathcal{B}\) has enough injectives, then every \(Y^\bullet\in \mathsf{D}^{b}(\mathcal{B})\) admits an injective coresolution \(Y^\bullet\!\to I^\bullet\) with \(I^\bullet\) bounded below and with bounded cohomology. Choose \(b\in\mathbb{Z}\) with \(H^{i}(Y^\bullet)=0\) for all \(i>b\). Then the induced morphism \[\tau_{\le b}Y^\bullet \longrightarrow \tau_{\le b}I^\bullet\] is a quasi-isomorphism, with \(\tau_{\le b}I^\bullet\in \mathsf{D}^{b}\!\bigl(\mathcal{B}\bigr)\). This yields a canonical triangulated equivalence \[\mathfrak{s}:\; \mathsf{K}^{+,b}\!\bigl(\operatorname{Inj}(\mathcal{B})\bigr)\;\xrightarrow{\;\sim\;}\;\mathsf{D}^{b}(\mathcal{B}),\] and, for \(X^\bullet\in \mathsf{D}^{b}(\mathcal{B})\), \[\operatorname{Hom}_{\mathsf{K}^{+}(\mathcal{B})}\!\bigl(X^\bullet,\,I^\bullet\bigr)\;\cong\; \operatorname{Hom}_{\mathsf{D}^{b}(\mathcal{B})}\!\bigl(X^\bullet,\,I^\bullet\bigr).\]

\(\bullet\) If \(\mathcal{B}\) has finite global dimension and enough projectives (resp.enough injectives). Then every object of \(\mathsf{D}^{b}(\mathcal{B})\) admits a representative by a bounded complex of projectives (resp.by a bounded complex of injectives), and we have a triangulated equivalence \[\mathsf{K}^{b}\!\bigl(\operatorname{Proj}(\mathcal{B})\bigr)\;\xrightarrow{\;\sim\;}\; \mathsf{D}^{b}(\mathcal{B}) \qquad \bigl(\text{resp. }\; \mathsf{K}^{b}\!\bigl(\operatorname{Inj}(\mathcal{B})\bigr)\;\xrightarrow{\;\sim\;}\; \mathsf{D}^{b}(\mathcal{B})\bigr)\]

The next proposition and definition record elementary observations tailored to the exact categories that arise throughout this paper.

In what follows, a triple \((\mathcal{A},\mathcal{B},\mathcal{C})\) will denote a chain of full inclusions \[\mathcal{A}\subseteq \mathcal{B}\subseteq \mathcal{C},\] where \(\mathcal{C}\) is an abelian category, \(\mathcal{B}\) is a full exact subcategory of \(\mathcal{C}\) in the sense of Quillen, having enough strongly injective objects and satisfying the two-out-of-three property for short exact sequences in \(\mathcal{C}\), and \(\mathcal{A}\) is a full subcategory of \(\mathcal{B}\) which is a Serre subcategory of \(\mathcal{C}\). For background on the localization of categories we refer to [21], [22].

The exact functor \[Q \colon \mathcal{B} \longrightarrow \mathcal{B}/\mathcal{A}\] extends canonically to a triangulated functor \[Q_* \colon \mathsf{D}^b(\mathcal{B}) \longrightarrow \mathsf{D}^b(\mathcal{B}/\mathcal{A})\] on bounded derived categories. Its kernel is precisely the full thick triangulated subcategory \[\mathsf{D}^b_{\mathcal{A}}(\mathcal{B}) := \left\{ X^\bullet \in \mathsf{D}^b(\mathcal{B}) \,\middle|\, H^n(X^\bullet) \in \mathcal{A} \text{ for all } n \in \mathbb{Z} \right\}.\]

The following result will be used in the sequel in the study of graded singular Koszul duality; compare with [23].

Let \(\Lambda\) be a graded algebra. We use the following notation:

• \(\mathsf{D}^{b}(\Lambda\mathrm{-mod})\) denotes the bounded derived category of finite-dimensional left \(\Lambda\)-modules;

• \(\mathsf{D}^{b}(\Lambda\mathrm{-gmod})\) denotes the bounded derived category of finite-dimensional \(\mathbb{Z}\)-graded left \(\Lambda\)-modules;

• \(\mathsf{D}^{b}(\Lambda\mathrm{-GMod})\) denotes the bounded derived category of locally finite-dimensional \(\mathbb{Z}\)-graded left \(\Lambda\)-modules.

The associated singularity categories are the Verdier quotients \[\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-gmod}) \;:=\; \frac{\mathsf{D}^{b}(\Lambda\mathrm{-gmod})}{\mathsf{K}^{b}(\Lambda\mathrm{-proj}^{\mathbb{Z}})}, \qquad \mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-mod}) \;:=\; \frac{\mathsf{D}^{b}(\Lambda\mathrm{-mod})}{\mathsf{K}^{b}(\Lambda\mathrm{-proj})},\] where \(\mathsf{K}^{b}(\Lambda\mathrm{-proj}^{\mathbb{Z}})\) (resp.\(\mathsf{K}^{b}(\Lambda\mathrm{-proj})\)) denotes the bounded homotopy category of graded (resp.ungraded) projective \(\Lambda\)-modules.

A module \(M\in \Lambda\mathrm{-mod}\) is called Gorenstein–projective if it occurs as the \(0\)-th syzygy of a totally acyclic complex \(P^{\bullet}\) of projective \(\Lambda\)-modules. That is, there exists an exact complex \[P^{\bullet}=\bigl(\cdots \longrightarrow P^{-1} \longrightarrow P^{0} \longrightarrow P^{1} \longrightarrow \cdots\bigr)\] with each \(P^{i}\) projective, such that \(M \cong \ker(P^{0}\to P^{1})\) and the complex \[\operatorname{Hom}_{\Lambda}(P^{\bullet},\Lambda)\] is exact. We denote by \(\Lambda\mathrm{-}\underline{\mathrm{Gproj}}\) the stable category of Gorenstein–projective \(\Lambda\)-modules, and by \(\Lambda\mathrm{-}\underline{\mathrm{Gproj}}^{\mathbb{Z}}\) its graded analogue.

If \(\Lambda\) is Iwanaga–Gorenstein, then the (graded) Gorenstein–projective \(\Lambda\)-modules coincide with the (graded) maximal Cohen–Macaulay \(\Lambda\)-modules.

By a theorem of Buchweitz [24], and its geometric reformulation due to Orlov [25], there are natural triangulated equivalences \[\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-mod}) \;\xrightarrow{\;\sim\;}\; \Lambda\mathrm{-}\underline{\mathrm{Gproj}}, \qquad \mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-gmod}) \;\xrightarrow{\;\sim\;}\; \Lambda\mathrm{-}\underline{\mathrm{Gproj}}^{\mathbb{Z}}.\]

2.4 Differential Graded Categories↩︎

This subsection collects the basic notions on differential graded (dg) categories that will be used in the sequel. Our aim is to realize, for a finite-dimensional Koszul algebra, both the bounded derived category and the singularity category as triangulated hulls of suitably defined dg orbit categories. For completeness, we briefly recall the necessary constructions, including pretriangulated hulls, triangulated hulls, and dg orbit categories. Standard references for this material include [26]–[29].

Throughout, all categories are assumed to be \(k\)-linear, that is, their Hom spaces are \(k\)-vector spaces and composition is \(k\)-bilinear. All functors are assumed to be \(k\)-linear.

The following example provides the basic model for the dg categories considered in this work.

A (left) dg module over \(\mathcal{A}\) is a dg functor \[M\colon \mathcal{A}\longrightarrow \mathsf{C}_{\mathrm{dg}}(k),\] where \(\mathsf{C}_{\mathrm{dg}}(k)\) denotes the dg category of cochain complexes of \(k\)-vector spaces. Morphisms of dg \(\mathcal{A}\)-modules are dg natural transformations, and the resulting dg category is denoted \(\mathcal{A}\mathrm{-}\mathrm{DGMod}\). A dg \(\mathcal{A}\)-module \(M\) is representable if there exists an object \(X\in\mathcal{A}\) and an isomorphism of dg modules \[M \;\cong\; \mathcal{A}(X,-).\]

A morphism \(\eta\colon M\to N\) of dg \(\mathcal{A}\)-modules is a quasi-isomorphism if, for every \(Y\in\mathcal{A}\), the map of complexes \[\eta_{Y}\colon M(Y)\longrightarrow N(Y)\] is a quasi-isomorphism (i.e.induces an isomorphism on cohomology in every degree).

A dg \(\mathcal{A}\)-module \(M\) is quasi-representable if there exists \(X\in\mathcal{A}\) and a quasi-isomorphism of dg modules \[M \xrightarrow{\;\simeq\;} \mathcal{A}(X,-).\]

Following Bondal–Kapranov [26], we recall the explicit construction of the pretriangulated hull of a dg category.

The pretriangulated hull satisfies the following universal property; see [26], [28], [29].

We now recall the notion of dg quotient in the sense of Drinfeld [27], and restrict our attention to those quotients relevant for our purposes; see, for instance, [29].

We shall now recall the notion of Galois coverings for triangulated categories. For subsequent use, we refer to [11] for a thorough discussion in the derived and \(k\)-linear frameworks.

The preceding discussion motivates the introduction of orbit categories and their differential-graded counterparts. We briefly recall these constructions; see [28] for details.

The orbit category is characterized by the following universal property.

The dg orbit category enjoys a universal property parallel to that of Proposition 2.36, most naturally formulated in the language of dg bimodules; see [28]. We briefly fix notation.

Let \(\mathcal{A}\) be a dg category. As above, write \(\mathcal{A}\text{-}\mathrm{DGMod}\) for the dg category of (left) dg \(\mathcal{A}\)-modules, with quasi-isomorphisms defined objectwise. Its Drinfeld quotient \[\mathsf{D}_{\mathrm{dg}}(\mathcal{A})\;:=\;\mathcal{A}\text{-}\mathrm{DGMod}\,\big/\,\mathcal{A}\text{-}\mathrm{Acyc}\] is the dg derived category.

For dg categories \(\mathcal{A}\) and \(\mathcal{B}\), a dg \(\mathcal{A}\text{-}\mathcal{B}\)-bimodule is a dg functor \[X\colon \mathcal{A}^{\mathrm{op}}\otimes \mathcal{B}\longrightarrow \mathsf{C}_{\mathrm{dg}}(k).\] Recall that \(X\) is quasi-representable if, for every \(A\in\mathcal{A}\), the associated right dg \(\mathcal{B}\)-module \(X(A,-)\) is quasi-isomorphic in \(\mathsf{D}_{\mathrm{dg}}(\mathcal{B})\) to a representable dg module \(\mathcal{B}(B,-)\) for some \(B\in\mathcal{B}\).

We write \[\operatorname{rep}(\mathcal{A},\mathcal{B})\] for the full subcategory of \(\mathsf{D}_{\mathrm{dg}}(\mathcal{A}^{\mathrm{op}}\!\otimes\!\mathcal{B})\) whose objects are quasi-representable dg \(\mathcal{A}\text{-}\mathcal{B}\)-bimodules in the dg derived category \(\mathsf{D}_{\mathrm{dg}}(\mathcal{A}^{\mathrm{op}}\!\otimes\!\mathcal{B})\).

We now state the universal property of the dg orbit category.

We now state and prove the main result of this subsection, which follows from [28].

3 Graded Koszul Dualities↩︎

Unless stated otherwise, \(\Lambda\) denotes a finite-dimensional quadratic algebra. Assuming that \(\Lambda\) is Koszul, we establish a triangulated equivalence between the bounded derived category of finite-dimensional graded \(\Lambda\)-modules and the bounded derived category of locally finite-dimensional graded co-perfect \(\Lambda^{!}\)-modules; equivalently, with the bounded derived category of locally finite-dimensional graded perfect \(\Lambda^{!}\)-modules. In particular, this removes the finiteness assumptions (Noetherian or coherence conditions) on the Koszul dual \(\Lambda^{!}\) that appear in the foundational work of Beilinson–Ginzburg–Soergel [9]. A related extension under coherence hypotheses was obtained by Martínez-Villa and Saorín [10]. We further determine when this graded Koszul duality induces both a graded derived equivalence and a derived equivalence between \(\Lambda\) and its Koszul dual \(\Lambda^{!}\).

As applications, we obtain a graded singularity duality for arbitrary finite-dimensional Koszul algebras and, for Iwanaga–Gorenstein Koszul algebras, a general graded version of the BGG correspondence. While the results of Martínez-Villa and Saorín cover the coherent case, including the self-injective situation, the present approach does not require such assumptions and therefore applies in complete generality. Moreover, we establish two dg Koszul dualities: one for the dg bounded derived categories and another for dg singularity categories, thereby extending the BGG correspondence to the dg setting.

3.1 The Derived Quadratic Functor↩︎

In this subsection, we construct explicitly the derived quadratic functor from the bounded derived category of locally finite-dimensional graded \(\Lambda^{!}\)-modules with bounded cohomology to the bounded derived category of finite-dimensional graded \(\Lambda\)-modules.

At the abelian level, this construction goes back to Martínez-Villa and Saorín [10], who established an equivalence between the category of finitely generated graded \(\Lambda^{!}\)-modules and the category of linear complexes of projective \(\Lambda\)-modules by assigning to a graded \(\Lambda^{!}\)-module a complex of projective \(\Lambda\)-modules. See also Mazorchuk–Ovsienko–Stroppel in the context of graded categories; cf. [14].

Bautista and Liu [12] extended this construction to bounded derived categories and obtained a triangulated equivalence for radical-square-zero algebras with gradable infinite quivers (referred to as well-directed quivers in our terminology). More recently, a general Koszul duality framework between \(\mathsf{D}^{\downarrow}(\mathrm{Mod}\,\Lambda^{!})\) and \(\mathsf{D}^{\uparrow}(\mathrm{Mod}\,\Lambda)\) was developed in [13] under the assumption that \(\Lambda\) is Koszul and given by a gradable quiver.

Our approach follows the same general strategy as in [12], [13], but is carried out in the broader setting of quadratic algebras. In particular, we extend the results of [12] from the radical-square-zero case to arbitrary finite-dimensional quadratic algebras; the former case is recovered as a special case of our results. Whenever the arguments coincide with those in [12], [13], we omit the proofs. Throughout, we write \(\otimes=\otimes_k\) for the tensor product over the base field \(k\), and all functors are assumed to be \(k\)-linear.

We begin with the following elementary yet useful observation; see [13].

We now begin the construction of the derived quadratic functor.

Let \(\Lambda^{!}\mathrm{-}\mathrm{GMod}\) denote the abelian category of locally finite-dimensional \(\mathbb{Z}\)-graded left \(\Lambda^{!}\)-modules (as fixed in §1). Let \(\mathsf{C}\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\) be the category of cochain complexes of graded projective \(\Lambda\)-modules whose terms are finite direct sums of indecomposable graded projectives of the form \(P_{x}\langle n\rangle\), where \(x\in Q_{0}\) and \(n\in\mathbb{Z}\).

Following [14] (see also [10], [12], and [13]), we define a \(k\)-linear functor \[K\colon \Lambda^{!}\mathrm{-}\mathrm{GMod}\longrightarrow \mathsf{C}\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\] as follows.

Let \(M\in \Lambda^{!}\mathrm{-}\mathrm{GMod}\). For \(x\in Q_0\) and \(n\in\mathbb{Z}\), we write \(M_x^n\) for the homogeneous component of degree \(n\) at the vertex \(x\); this agrees with the notation used in §1, where we wrote \(M_n(x)\).

We define a graded \(\Lambda\)-module \[K(M)^n \;:=\; \bigoplus_{x\in Q_0} P_x\langle n\rangle \otimes_k M_x^n, \qquad n\in\mathbb{Z}.\]

The differential \(d^n\colon K(M)^n\to K(M)^{n+1}\) is given by its \((y,x)\)-component \[d^n_{yx} \;=\; \sum_{\alpha\colon y\to x} \Bigl(P_{\alpha}\colon P_x\langle n\rangle \longrightarrow P_y\langle n+1\rangle\Bigr) \;\otimes\; \Bigl(M(\alpha^{\mathrm{op}})\colon M_x^n \longrightarrow M_y^{n+1}\Bigr),\] where the sum runs over all arrows \(\alpha\) in \(Q\) with \(s(\alpha)=y\) and \(t(\alpha)=x\), and \(\alpha^{\mathrm{op}}\) denotes the corresponding arrow in the opposite quiver.

Here \(P_{\alpha}\) is given by right multiplication by \(\alpha\); that is, for a path \(q\in P_x\), \[P_{\alpha}(q)=q\,\alpha\in P_y.\] The map \(M(\alpha^{\mathrm{op}})\) is induced by the graded \(\Lambda^{!}\)-module structure.

With these definitions, \(K(M)=(K(M)^n,d^n)\) is a cochain complex of graded projective \(\Lambda\)-modules. Moreover, each \(K(M)^n\) is generated in degree \(-n\).

The assignment \(M\mapsto K(M)\) defines an exact and fully faithful functor \[K\colon \Lambda^{!}\mathrm{-}\mathrm{GMod} \longrightarrow \mathsf{C}\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr),\] whose essential image is the full subcategory \[\mathcal{LC}\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\] consisting of linear complexes, that is, complexes \(\mathbf{P}=(P^n,d^n)\) such that each \(P^n\) is a finite direct sum of modules of the form \(P_x\langle n\rangle\), with \(x\in Q_0\), and hence generated in degree \(n\).

A module \(M \in \Lambda^{!}\mathrm{-}\mathrm{GMod}^{-}\) is said to have bounded cohomology if the complex \(K(M)\) has bounded cohomology. Set \[\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\] to be the full subcategory of \(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-}\) consisting of such modules. The \(k\)-linear functor \[K \colon \Lambda^{!}\mathrm{-}\mathrm{GMod} \longrightarrow \mathsf{C}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\] then restricts to a functor \[K \colon \Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b} \longrightarrow \mathcal{LC}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr),\] where \(\mathcal{LC}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\) denotes the full subcategory of \(\mathcal{LC}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\) consisting of right bounded linear complexes with bounded cohomology.

The category \(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\) is exact, closed under extensions, and weakly idempotent complete. Moreover, the class of short exact sequences it inherits from \(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-}\) satisfies the two-out-of-three property, as follows from the long exact sequence in cohomology.

The following result is an immediate consequence of [14].

We now extend \(K\) to bounded cochain complexes in \(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\). In this way we obtain a \(k\)-linear functor \[\mathfrak{K} \colon \mathsf{C}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\bigr) \longrightarrow \mathsf{DC}^{b,-}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr),\] where \(\mathsf{DC}^{b,-}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr)\) denotes the category of double cochain complexes with terms in \(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\), bounded in the horizontal direction and right bounded in the vertical direction.

More precisely, let \[\cdots \longrightarrow M^{p-1} \xrightarrow{\,d^{p-1}\,} M^{p} \xrightarrow{\,d^{p}\,} M^{p+1} \longrightarrow \cdots\] be a bounded cochain complex with each \(M^{p}\in \Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\). For \(x\in Q_{0}\) and \(q\in\mathbb{Z}\), write \(M^{p}_{x,q}\) for the homogeneous component of \(M^{p}\) of internal degree \(q\) at the vertex \(x\). Applying \(K\) degreewise yields a double complex whose term in bidegree \((p,q)\) is \[\mathfrak{K}(M^\bullet)^{p,q} := \bigoplus_{x\in Q_{0}} P_{x}\langle q\rangle \otimes_k M^{p}_{x,q}.\]

A portion of this double complex is displayed in the diagram \[\xymatrix@C=3.4em@R=3.0em{ \cdots & \cdots & \cdots & \cdots \\ \cdots \ar[r] & \displaystyle\bigoplus_{y} P_{y}\langle q+1\rangle \otimes M^{p}_{y,\,q+1} \ar[r]^{\;d^{p,\,q+1}_{1}\;} \ar[u] & \displaystyle\bigoplus_{y} P_{y}\langle q+1\rangle \otimes M^{p+1}_{y,\,q+1} \ar[r] \ar[u] & \cdots \\ \cdots \ar[r] & \displaystyle\bigoplus_{x} P_{x}\langle q\rangle \otimes M^{p}_{x,\,q} \ar[r]^{\;d^{p,\,q}_{1}\;} \ar[u]^{\;d^{p,\,q}_{2}\;} & \displaystyle\bigoplus_{x} P_{x}\langle q\rangle \otimes M^{p+1}_{x,\,q} \ar[r] \ar[u]_{\,d^{p+1,\,q}_{2}} & \cdots \\ \cdots & \cdots \ar[u] & \cdots \ar[u] & \cdots }\]

The horizontal differential \[d^{p,q}_{1}\colon \bigoplus_{x\in Q_{0}} P_{x}\langle q\rangle \otimes M^{p}_{x,q} \longrightarrow \bigoplus_{x\in Q_{0}} P_{x}\langle q\rangle \otimes M^{p+1}_{x,q}\] is defined by \[d^{p,q}_{1} = \sum_{x\in Q_{0}} \mathrm{id}_{P_{x}\langle q\rangle}\otimes d^{p}_{x,q},\] where \(d^{p}_{x,q}\colon M^{p}_{x,q}\to M^{p+1}_{x,q}\) is the \((x,q)\)-component of the cochain differential of \(M^\bullet\).

The vertical differential \[d^{p,q}_{2}\colon \bigoplus_{x\in Q_{0}} P_{x}\langle q\rangle \otimes M^{p}_{x,q} \longrightarrow \bigoplus_{y\in Q_{0}} P_{y}\langle q+1\rangle \otimes M^{p}_{y,q+1}\] is defined by its \((y,x)\)-component \[\bigl(d^{p,q}_{2}\bigr)_{yx} = \sum_{\alpha\colon y\to x} \Bigl(P_{\alpha}\!:\; P_{x}\langle q\rangle \longrightarrow P_{y}\langle q+1\rangle\Bigr) \otimes \Bigl(M^{p}(\alpha^{\mathrm{op}})\!:\; M^{p}_{x,q} \longrightarrow M^{p}_{y,q+1}\Bigr),\] where the sum runs over all arrows \(\alpha\) in \(Q\) with \(s(\alpha)=y\) and \(t(\alpha)=x\). Here \(P_{\alpha}\) denotes right multiplication by \(\alpha\) on \(P_x\), namely \(P_{\alpha}(u)=u\alpha\), and \(M^{p}(\alpha^{\mathrm{op}})\) is the corresponding structure morphism of the graded \(\Lambda^{!}\)-module \(M^{p}\).

By construction, one has \[(d_1)^2=0,\qquad (d_2)^2=0,\qquad d_1d_2=d_2d_1.\] Thus \(\mathfrak{K}(M^\bullet)\) is indeed a double cochain complex.

This construction yields a total complex in the usual way via diagonal summation. More precisely, let \((A^{p,q}, d_1^{p,q}, d_2^{p,q})\) be the double complex with \[A^{p,q} \;=\; \bigoplus_{x \in Q_0} P_{x}\langle q\rangle \otimes M^{p}_{x,q}.\] The total complex \(\operatorname{Tot}(A)\) is defined in degree \(n\) by \[\operatorname{Tot}(A)^{n} \;=\; \bigoplus_{p+q = n}\;\bigoplus_{x \in Q_0} P_{x}\langle q\rangle \otimes M^{p}_{x,q},\] with differential \[d_{\operatorname{Tot}}^{\,n} \;=\; \bigoplus_{p+q = n}\bigl(d_1^{p,q} + (-1)^{p} d_2^{p,q}\bigr).\] This defines a \(k\)-linear functor \[\operatorname{Tot} \colon \mathsf{DC}^{b,-}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr) \longrightarrow \mathsf{C}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr).\]

Composing \(\mathfrak{K}\) with \(\operatorname{Tot}\), we obtain a \(k\)-linear functor \[\mathcal{F} \colon \mathsf{C}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\bigr) \longrightarrow \mathsf{C}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr),\] which we call the complex quadratic functor. By construction, \(\mathcal{F}\) extends \(K\) and restricts to \[K \colon \Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b} \longrightarrow \mathcal{LC}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr).\]

By [12], the functor \(\mathcal{F}\) commutes with the shift functor \([1]\), preserves mapping cones, and sends null-homotopic morphisms to null-homotopic morphisms. Hence it induces a functor on homotopy categories. Since \(\mathcal{F}\) also preserves acyclic complexes, it further descends to the derived categories. In particular, we obtain a commutative diagram \[\xymatrix@C=5.5em@R=4em{ \mathsf{C}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\bigr) \ar[r]^-{\mathcal{F}} \ar[d]_{\scriptstyle \pi} & \mathsf{C}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr) \ar[d]^{\scriptstyle \pi'} \\ \mathsf{K}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\bigr) \ar[r]^-{\mathscr{F}} \ar[d]_{\scriptstyle \epsilon} & \mathsf{K}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr) \ar[d]^{\scriptstyle \mathfrak{T}} \\ \mathsf{D}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\bigr) \ar[r]^-{\mathfrak{F}} & \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-}\mathrm{gmod}\bigr) }\] where:

• \(\mathcal{F}\) is the complex quadratic functor;

• \(\mathscr{F}\) is the induced functor on homotopy categories;

• \(\mathfrak{F}\) is the induced functor on derived categories;

• \(\pi, \pi'\) are the canonical projections;

• \(\epsilon\) is the localization functor;

• \(\mathfrak{T}\) is the standard equivalence \[\mathsf{K}^{-,b}\!\bigl(\Lambda\mathrm{-}\mathrm{proj}^{\mathbb{Z}}\bigr) \xrightarrow{\;\sim\;} \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-}\mathrm{gmod}\bigr),\] see §2.3.

The functor \[\mathfrak{F} \colon \mathsf{D}^{b}\!\bigl(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b}\bigr) \longrightarrow \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-}\mathrm{gmod}\bigr)\] is triangulated and will be referred to as the derived quadratic functor. Explicitly, for a bounded cochain complex \(M^\bullet\), one applies \(K\) degreewise, takes the total complex, and then applies the equivalence \(\mathfrak{T}\): \[\mathfrak{F}(M^\bullet) = \mathfrak{T}\!\left(\operatorname{Tot}\bigl(\mathfrak{K}(M^\bullet)\bigr)\right).\]

In the next subsection, we prove that the functor \(\mathfrak{F}\) is a triangulated equivalence if and only if \(\Lambda\) is a Koszul algebra.

3.2 Graded Derived Koszul duality↩︎

The aim of this subsection is to establish a graded derived Koszul duality for finite-dimensional Koszul algebras. As consequences, we obtain a graded singular Koszul duality for all finite-dimensional Koszul algebras and extend the Bernstein–Gelfand–Gelfand correspondence to the class of finite-dimensional Iwanaga–Gorenstein Koszul algebras.

Before stating the main results, we fix some terminology.

We denote by \(\Lambda^{!}\mathrm{-Proj}\) and \(\Lambda^{!}\mathrm{-Inj}\) the additive categories of graded projective and injective \(\Lambda^{!}\)-modules, respectively. Since \(\Lambda^{!}\) is locally finite-dimensional, all graded modules considered here are assumed to be locally finite-dimensional.

For each vertex \(x\) and each \(n\in\mathbb{Z}\), we denote by \(P_x^{!}\langle n\rangle\), \(I_x^{!}\langle n\rangle\), and \(S_x^{!}\langle n\rangle\) the indecomposable projective, injective, and simple graded \(\Lambda^{!}\)-modules, respectively.

Let \(M\in \Lambda^{!}\mathrm{-}\mathrm{GMod}^{-}\) be a right bounded graded module. The socle of \(M\), denoted \(\operatorname{soc}(M)\), is defined as the largest semisimple graded submodule of \(M\). Concretely, for each vertex \(x\in Q_{0}\) and each degree \(n\in\mathbb{Z}\), one has \[\operatorname{soc}(M)_{n}(x) = \bigcap_{\alpha:y\to x} \ker\!\Bigl(M(\alpha^{\mathrm{op}})\colon M_{n}(x)\longrightarrow M_{n+1}(y)\Bigr),\] where the intersection is taken over all arrows \(\alpha\colon y\to x\) in \(Q\).

A graded \(\Lambda^{!}\)-module \(M\) is said to be finitely cogenerated if \(\operatorname{soc}(M)\) is finite-dimensional and essential in \(M\). Equivalently, there exists a monomorphism \[0 \longrightarrow M \longrightarrow I^{!},\] where \(I^{!}\) is a finitely cogenerated injective \(\Lambda^{!}\)-module.

In particular, each indecomposable graded injective module \(I^{!}_{x}\langle n\rangle\) is finitely cogenerated, and one has \[\operatorname{soc}\!\bigl(I^{!}_{x}\langle n\rangle\bigr) \cong S^{!}_{x}\langle n\rangle.\]

A graded \(\Lambda^{!}\)-module \(M\) is said to be finitely copresented if it admits an injective copresentation \[0 \longrightarrow M \longrightarrow I^{!0} \longrightarrow I^{!1},\] where \(I^{!0}\) and \(I^{!1}\) are finitely cogenerated graded injectives. We denote by \[\Lambda^{!}\mathrm{-}\mathrm{Fcp}^{\mathbb{Z}}\] the full subcategory of \(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-}\) consisting of such modules.

A graded \(\Lambda^{!}\)-module \(M\) is called coperfect if it has finite injective dimension and admits a minimal graded injective coresolution \[0 \longrightarrow M \longrightarrow I^{!0} \longrightarrow I^{!1} \longrightarrow \cdots \longrightarrow I^{!d} \longrightarrow 0,\] in which each \(I^{!i}\) is finitely cogenerated; equivalently, each \(I^{!i}\) is a finite direct sum of indecomposable graded injectives \(I^{!}_{x}\langle n\rangle\) with \(x\in Q_{0}\) and \(n\in\mathbb{Z}\). We denote by \[\Lambda^{!}\mathrm{-}\mathrm{Cop}^{\mathbb{Z}}\] the full subcategory of \(\Lambda^{!}\mathrm{-}\mathrm{GMod}^{-}\) consisting of coperfect graded modules.

Dually, a graded module \(M\in \Lambda^{!}\mathrm{-}\mathrm{GMod}^{+}\) is finitely generated if there exists an epimorphism \(P^{!}\twoheadrightarrow M\), where \(P^{!}\) is a finite direct sum of indecomposable graded projectives \(P^{!}_{x}\langle i\rangle\) (\(x\in Q_{0}\), \(i\in\mathbb{Z}\)). We write \[\Lambda^{!}\mathrm{-}\mathrm{Fg}^{\mathbb{Z}} \qquad (\text{resp.\;}\Lambda^{!}\mathrm{-}\mathrm{gmod})\] for the full subcategory of finitely generated objects in \(\Lambda^{!}\mathrm{-}\mathrm{GMod}\) (resp.the category of finite-dimensional graded \(\Lambda^{!}\)-modules).

A graded module is finitely presented if it admits a projective presentation \[P^{!-1}\longrightarrow P^{!0}\longrightarrow M\longrightarrow 0\] with \(P^{!0}\) and \(P^{!-1}\) finitely generated. The corresponding full subcategory is denoted \[\Lambda^{!}\mathrm{-}\mathrm{Fp}^{\mathbb{Z}}.\]

Finally, a graded module is called perfect if it admits a finite graded projective resolution \[0 \longrightarrow P^{!k} \longrightarrow P^{!k+1} \longrightarrow \cdots \longrightarrow P^{!-1} \longrightarrow P^{!0} \longrightarrow M \longrightarrow 0,\] with each \(P^{!i}\) finitely generated and projective. We denote by \[\Lambda^{!}\mathrm{-}\mathrm{Pe}^{\mathbb{Z}}\] the full subcategory of perfect graded \(\Lambda^{!}\)-modules.

In the sequel we shall show that, if \(\Lambda\) is a finite-dimensional Koszul algebra, then \[\Lambda^{!}\mathrm{-}\mathrm{Cop}^{\mathbb{Z}} \;=\; \Lambda^{!}\mathrm{-}\mathrm{GMod}^{-,b},\] i.e.the coperfect graded modules are precisely the locally finite-dimensional graded modules with bounded cohomology.

For \(x\in Q_{0}\), let \[P^{!\,\mathrm{op}}_{x}\;=\;\Lambda^{!\,\mathrm{op}}(x,-), \qquad I^{!}_{x}\;=\;\mathbb{D}\!\bigl(\Lambda^{!\,\mathrm{op}}(x,-)\bigr) \;=\;\mathbb{D}\,P^{!\,\mathrm{op}}_{x}.\]

If \(\alpha\colon y\to z\) is an arrow in \(Q\), then \(\alpha^{\mathrm{op}}\colon z\to y\) induces (by precomposition) a map \[P^{!\,\mathrm{op}}_{x}(\alpha^{\mathrm{op}})\colon \Lambda^{!\,\mathrm{op}}(x,y)\longrightarrow \Lambda^{!\,\mathrm{op}}(x,z), \qquad f\longmapsto f\circ \alpha^{\mathrm{op}},\] and dualizing gives the structure map \[I^{!}_{x}(\alpha^{\mathrm{op}})\colon \bigl(I^{!}_{x}(z)\bigr)^{m}\longrightarrow \bigl(I^{!}_{x}(y)\bigr)^{m+1}.\]

For each \(x\in Q_{0}\) and \(n\in\mathbb{Z}\), the complex \(K\!\bigl(I^{!}_{x}\langle n\rangle\bigr)\) has \[K\!\bigl(I^{!}_{x}\langle n\rangle\bigr)^{m} \;=\; \bigoplus_{z\in Q_{0}} P_{z}\langle m\rangle \,\otimes_{k}\, \bigl(I^{!}_{x}(z)\bigr)^{m+n}, \qquad m\in\mathbb{Z}.\]

The differential \[d^{m}\colon K\!\bigl(I^{!}_{x}\langle n\rangle\bigr)^{m} \longrightarrow K\!\bigl(I^{!}_{x}\langle n\rangle\bigr)^{m+1}\] has \((y,z)\)-component \[d^{m}_{yz} \;=\; \sum_{\alpha\colon y\to z} \Bigl(P_{\alpha}\!:\; P_{z}\langle m\rangle \longrightarrow P_{y}\langle m+1\rangle\Bigr) \;\otimes\; \Bigl(I^{!}_{x}(\alpha^{\mathrm{op}})\!:\; \bigl(I^{!}_{x}(z)\bigr)^{m+n}\longrightarrow \bigl(I^{!}_{x}(y)\bigr)^{m+1+n}\Bigr).\]

The following lemma extends [12], originally proved for radical square zero algebras, to the class of arbitrary finite-dimensional Koszul algebras. The argument of [13] carries over verbatim to this setting; see also [14] and [9].

The next proposition shows that, for a finite-dimensional Koszul algebra \(\Lambda\), the graded \(\Lambda^{!}\)-modules with bounded cohomology coincide with the coperfect \(\Lambda^{!}\)-modules.

We shall need a few additional preparatory results. We begin with an elementary observation.

We continue with an auxiliary observation which will be used crucially in the proof of the main equivalence.

We are now in a position to state the main result of this section. We shall refer to the equivalence below as the graded derived Koszul duality. It may be viewed as the bounded analogue of [14].

If \(\Lambda^{!}\) is left graded coherent, then \(\Lambda^{!}\mathrm{-}\mathrm{Pe}^{\mathbb{Z}} = \Lambda^{!}\mathrm{-}\mathrm{Fp}^{\mathbb{Z}}\). In this situation, our result recovers that of Martínez-Villa and Saorín (see [32]).

If \(\Lambda^{!}\) is Noetherian, we recover the Koszul duality of Beilinson–Ginzburg–Soergel (see [9]).

3.3 Special Cases of Derived Koszul Duality↩︎

We record two further consequences of the graded derived Koszul duality. In particular, we determine when there exists a triangulated equivalence \[\mathsf{D}^b\bigl(\Lambda^{!}\mathrm{-gmod}\bigr) \;\simeq\; \mathsf{D}^b\bigl(\Lambda\mathrm{-gmod}\bigr).\] The following corollary gives a precise criterion.

Let \(\Lambda\) be a finite-dimensional algebra whose quiver \(Q\) is well directed (see Subsection 2.1). Fix a labelling \(Q_0=\{1,\dots,r\}\) such that every arrow of \(Q\) is oriented from a smaller index to a larger one, and let \(S_1,\dots,S_r\) denote the corresponding simple left \(\Lambda\)-modules.

We say that \(\Lambda\) is Koszul (in the non-graded sense) if, for all \(i,j\in\{1,\dots,r\}\) and all integers \(n\ge 0\), one has \[\operatorname{Ext}^{n}_{\Lambda\mathrm{-mod}}(S_i,S_j)=0 \qquad\text{whenever } j\neq i-n .\]

A second natural question is whether the derived graded equivalence can be upgraded to an equivalence between the ungraded bounded derived categories \[\mathsf{D}^b\!\bigl(\Lambda^{!}\mathrm{-mod}\bigr) \quad\text{and}\quad \mathsf{D}^b\!\bigl(\Lambda\mathrm{-mod}\bigr).\] In general such an equivalence does not exist; we return to this issue in Section 4. Nevertheless, in the well-directed case one obtains the following consequence.

3.4 Graded Singular Koszul Duality and the Graded BGG Correspondence↩︎

In [3], Bernstein, Gelfand, and Gelfand constructed the first example of Koszul duality and deduced the celebrated triangulated equivalence now known as the BGG correspondence. More precisely, they proved that there is a triangulated equivalence between the bounded derived category of coherent sheaves on projective space \(\mathbb{P}^n\) and the stable category of finite-dimensional graded modules over the exterior algebra \(E\) (see [3]): \[\mathsf{D}^b\bigl(\mathrm{Coh}(\mathbb{P}^n)\bigr) \;\xrightarrow{\;\sim\;}\; E\text{-}\underline{\mathrm{grmod}}.\] They also formulated a noncommutative analogue (see [3]) identifying \(\mathsf{D}^b\bigl(\mathrm{Coh}(\mathbb{P}^n)\bigr)\) with a Verdier quotient: \[\mathsf{D}^b\bigl(\mathrm{Coh}(\mathbb{P}^n)\bigr) \;\simeq\; \mathsf{D}^b\bigl(A\mathrm{-}\mathrm{gmod}\bigr)\big/\mathsf{D},\] where \(A\mathrm{-}\mathrm{gmod}\) denotes the category of finitely generated graded modules over the symmetric algebra \(A\), and \(\mathsf{D}\) is the thick triangulated subcategory of \(\mathsf{D}^b\bigl(A\mathrm{-}\mathrm{gmod}\bigr)\) consisting of complexes with finite-dimensional total cohomology (equivalently, bounded complexes of finite-dimensional graded \(A\)-modules). In particular, one obtains an induced equivalence \[\mathsf{D}^b\bigl(A\mathrm{-}\mathrm{gmod}\bigr)\big/\mathsf{D} \;\xrightarrow{\;\sim\;}\; E\text{-}\underline{\mathrm{grmod}}.\] This description follows from the Koszul derived equivalence \[\mathsf{D}^b\bigl(A\mathrm{-}\mathrm{gmod}\bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^b\bigl(E\mathrm{-}\mathrm{gmod}\bigr),\] established in [3].

In this example both algebras are Koszul: the exterior algebra \(E\) is finite-dimensional, self-injective, and Koszul, whereas its Koszul dual \(A\) has finite global dimension and is (isomorphic to) a polynomial algebra in finitely many variables.

Later, Martı́nez-Villa and Saorı́n generalized the BGG correspondence to the context of finite-dimensional Koszul self-injective algebras whose Koszul dual is coherent. Their results are not phrased in terms of singularity categories. Instead, they proved that if \(\Lambda\) is a finite-dimensional Koszul algebra and its Koszul dual \(\Lambda^!\) is left coherent, then there is a triangulated equivalence \[\mathsf{D}^b\Bigl(\Lambda^!\mathrm{-Fp}^{\mathbb{Z}}\big/\Lambda^!\mathrm{-gmod}\Bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{S}\bigl(\overline{\Lambda\mathrm{-gmod}}\bigr),\] where \(\Lambda^!\mathrm{-Fp}\) denotes the category of finitely presented graded \(\Lambda^!\)-modules, and \(\mathsf{S}\bigl(\overline{\Lambda\mathrm{-gmod}}\bigr)\) denotes the stabilization of the stable category of finitely generated graded \(\Lambda\)-modules modulo injectives. The latter is a triangulated category, canonically equivalent to the Verdier quotient \[\mathsf{D}^b\bigl(\Lambda\mathrm{-gmod}\bigr)\Big/\mathsf{K}^b\bigl(\Lambda\mathrm{-Inj}^{\mathbb{Z}}\bigr).\] We refer to [32] for further details.

Recall from Section 2 that \[\bigl(\Lambda^!\mathrm{-gmod},\;\Lambda^!\mathrm{-Cop}^{\mathbb{Z}},\;\Lambda^!\mathrm{-GMod}^{-}\bigr)\] forms a triple. Moreover, Proposition 2.17 shows that the quotient category \[\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\big/\Lambda^!\mathrm{-gmod}\] is exact and weakly idempotent complete. Furthermore, there is a triangulated equivalence \[\mathsf{D}^b(\Lambda^!\mathrm{-gmod}) \;\cong\; \mathsf{D}^b_{\Lambda^!\mathrm{-gmod}}\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\bigr),\] see the proof of Theorem 3.16. It then follows from Theorem 2.18 that there is a triangulated equivalence \[\mathsf{D}^b\Bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\big/\Lambda^!\mathrm{-gmod}\Bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^b\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\bigr) \Big/ \mathsf{D}^b_{\Lambda^!\mathrm{-gmod}}\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\bigr),\] where \(\mathsf{D}^b_{\Lambda^!\mathrm{-gmod}}\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\bigr)\) denotes the thick triangulated subcategory of \(\mathsf{D}^b\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\bigr)\) consisting of bounded complexes with cohomology in \(\Lambda^!\mathrm{-gmod}\). Consequently, the quotient \[\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\big/\Lambda^!\mathrm{-gmod}\] has finite homological dimension.

We are now ready to prove the second main result of this section, which we call the graded singular Koszul duality.

We now establish the following generalization of the BGG correspondence, valid for all finite-dimensional Gorenstein–Iwanaga Koszul algebras.

We conclude this subsection with a characterization of finite-dimensional Koszul Iwanaga–Gorenstein algebras.

3.5 DG Koszul Duality↩︎

The purpose of this subsection is to construct a differential graded lift \(\mathfrak{F}_{\mathrm{dg}}\) of the graded derived Koszul duality functor \(\mathfrak{F}\), which will play a key role in the proof of the main results of this paper (Theorem 4.3 and Theorem 4.6). Somewhat surprisingly, this construction also yields isomorphisms in the homotopy category of dg categories \(\operatorname{Ho}(\mathrm{dgcat})\), \[\mathsf{D}^{b}_{\mathrm{dg}}\!\bigl(\Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}}\bigr) \;\cong\; \mathsf{D}^{b}_{\mathrm{dg}}\!\bigl(\Lambda\mathrm{-gmod}\bigr),\] and \[\mathsf{D}^{b}_{\mathrm{dg}}\!\Bigl( \Lambda^{!}\mathrm{-Cop}^{\mathbb{Z}} \big/ \Lambda^{!}\mathrm{-gmod} \Bigr) \;\cong\; \mathsf{D}^{\mathrm{dg}}_{\mathrm{sg}}\!\bigl(\Lambda\mathrm{-gmod}\bigr).\] These isomorphisms are compatible with passage to homotopy categories in the sense that applying the functor \(\mathrm{H}^{0}\) recovers, respectively, the derived graded Koszul duality and the graded singular Koszul duality.

We begin by constructing a dg enhancement of the derived Koszul duality.

The dg derived Koszul duality descends to a dg singular Koszul duality.

4 Non-graded Koszul Dualities↩︎

Many categories of interest admit realizations as categories of finite-dimensional modules over suitable Koszul algebras. Prominent examples arise in the representation theory of semisimple Lie algebras, notably the blocks of the Bernstein–Gelfand–Gelfand category \(\mathcal{O}\), as well as in geometric settings such as categories of mixed perverse sheaves on varieties admitting a stratification by affine spaces (see, for example, [9]). It is therefore natural to study bounded derived categories of finite-dimensional modules over finite-dimensional Koszul algebras without imposing a grading, that is, to seek a formulation of Koszul duality in the ungraded setting.

One might expect that a non-graded analogue of Koszul duality would yield an equivalence \[\mathsf{D}^{b}\!\bigl(\Lambda^{!}\mathrm{-mod}\bigr) \;\cong\; \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-mod}\bigr).\] However, such an equivalence does not hold in general. For instance, there exist radical-square-zero algebras which are not piecewise hereditary (see [33]), which already precludes the existence of a derived equivalence of the above form. Moreover, Corollary 3.12 shows that the derived quadratic functor yields a derived equivalence only under a strong orientation assumption, namely when the quivers of the Koszul algebras are well directed.

To overcome this difficulty, we employ techniques from dg theory, notably dg enhancements and dg orbit categories introduced in Section 2, to construct a non-graded version of Koszul duality.

Throughout this section, unless stated otherwise, \(\Lambda\) denotes a finite-dimensional graded algebra.

4.1 Non-graded derived Koszul duality↩︎

In this subsection, we establish a non-graded Koszul duality for the bounded derived category of finite-dimensional modules over an arbitrary finite-dimensional Koszul algebra, as well as for their singularity categories. Our strategy begins by analyzing their derived and singularity categories, namely the graded derived category \(\mathsf{D}^b(\Lambda\mathrm{-gmod})\) and the graded singularity category \(\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-gmod})\), together with their ungraded counterparts \(\mathsf{D}^b(\Lambda\mathrm{-mod})\) and \(\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-mod})\). We then apply the derived and singular Koszul dualities to realize \(\mathsf{D}^b(\Lambda\mathrm{-mod})\) and \(\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-mod})\) as triangulated hulls of suitable dg orbit categories of \[\mathsf{D}^b\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}\bigr) \quad\text{and}\quad \mathsf{D}^b\bigl(\Lambda^!\mathrm{-Cop}^{\mathbb{Z}}/\Lambda^!\mathrm{-gmod}\bigr),\] respectively. As a consequence, we obtain a more general form of the Bernstein–Gelfand–Gelfand correspondence for the stable category of finite-dimensional Gorenstein–projective modules over any finite-dimensional Iwanaga–Gorenstein Koszul algebra.

Recall from Section 2, Proposition 2.2, that there exists an exact functor of abelian categories, often called the forgetful functor, \[F \colon \Lambda\text{-}\mathrm{gmod} \longrightarrow \Lambda\text{-}\mathrm{mod},\] which induces a natural isomorphism of \(k\)-vector spaces \[\bigoplus_{r\in\mathbb{Z}} \mathrm{Hom}_{\Lambda\text{-}\mathrm{gmod}} \bigl(M,\,N\langle r\rangle\bigr) \;\xrightarrow{\sim}\; \mathrm{Hom}_{\Lambda\text{-}\mathrm{mod}}(M,\,N)\] for all finite-dimensional graded modules \(M\) and \(N\).

The functor \(F\) also induces triangulated functors on bounded derived and singularity categories, \[F \colon \mathsf{D}^b(\Lambda\text{-}\mathrm{gmod}) \longrightarrow \mathsf{D}^b(\Lambda\text{-}\mathrm{mod}), \qquad F \colon \mathsf{D}_{\mathrm{sg}}(\Lambda\text{-}\mathrm{gmod}) \longrightarrow \mathsf{D}_{\mathrm{sg}}(\Lambda\text{-}\mathrm{mod}),\] which, by abuse of notation, we continue to denote by \(F\).

The following result describes the relationship between \(\mathsf{D}^b(\Lambda\mathrm{-gmod})\) (respectively \(\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-gmod})\)) and \(\mathsf{D}^b(\Lambda\mathrm{-mod})\) (respectively \(\mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-mod})\)).

We next record a preparatory lemma that will be used throughout the sequel. For closely related statements, compare [14] and [9]; see also [12].

We are now ready to prove the first main result of this paper, which we shall call the non-graded derived Koszul duality.

If \(\Lambda\) has finite global dimension, the non-graded derived Koszul duality specializes to the following form.

4.2 Non-graded Singular Koszul duality and the Non-Graded BGG Correspondance↩︎

The second main result of this paper establishes a non-graded singular Koszul duality. As an application, we obtain a non-graded form of the Bernstein–Gelfand–Gelfand correspondence.

We now present the non-graded version of the BGG correspondence.

5 Applications↩︎

We now apply the non-graded Koszul dualities established above to fill a longstanding gap left open since the work of Beilinson, Ginzburg, and Soergel [9]. In their foundational work, they established a derived Koszul duality for certain graded versions of blocks of the Bernstein–Gelfand–Gelfand category \(\mathcal{O}\). In this section we resolve this gap by establishing a non-graded derived Koszul duality in this setting. We also obtain a non-graded derived Koszul duality for certain categories of perverse sheaves.

5.1 Non-graded derived Koszul duality for the BGG category \(\mathcal{O}\)↩︎

Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra over \(\mathbb{C}\), and fix a triangular decomposition \[\mathfrak{g}=\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+.\] Write \(U(\mathfrak{g})\) for the universal enveloping algebra and \(\mathfrak{b}:=\mathfrak{h}\oplus\mathfrak{n}^+\) for the standard Borel subalgebra. The Bernstein–Gelfand–Gelfand category \(\mathcal{O}\) is the full subcategory of \(U(\mathfrak{g})\)-modules consisting of modules that are

• finitely generated over \(U(\mathfrak{g})\),

• semisimple as \(\mathfrak{h}\)-modules, and

• locally finite under the action of \(\mathfrak{b}\).

It is an abelian category with enough projectives and admits a block decomposition \[\mathcal{O} \;=\; \bigoplus_{\chi}\, \mathcal{O}_\chi,\] indexed by central characters \(\chi\colon Z(\mathfrak{g})\!\to\!\mathbb{C}\). Among these, the principal block \(\mathcal{O}_0\) (trivial central character) plays a distinguished role; Beilinson–Ginzburg–Soergel [9] showed that \(\mathcal{O}_0\) is equivalent to the category of finitely generated modules over a finite-dimensional Koszul \(\mathbb{C}\)-algebra \(A\).

Let \(\lambda\in\mathfrak{h}^*\) be integral and dominant, and denote by \(\mathcal{O}_\lambda\subset\mathcal{O}\) the block consisting of modules with (generalized) infinitesimal character equal to that of the highest-weight module \(L(\lambda)\). The simple objects of \(\mathcal{O}_\lambda\) are parametrized by a subset \(W_\lambda\subset W\) of the Weyl group, and \(\mathcal{O}_\lambda\) is equivalent to the module category of a finite-dimensional \(\mathbb{C}\)-algebra.

Now let \(\mathfrak{q}\subset\mathfrak{g}\) be a parabolic subalgebra with \(\mathfrak{b}\subset\mathfrak{q}\). The parabolic category \(\mathcal{O}^{\mathfrak{q}}\subset\mathcal{O}_0\) is the full subcategory of \(\mathcal{O}_0\) consisting of objects that are locally finite for the \(\mathfrak{q}\)-action; it is likewise equivalent to the module category of a finite-dimensional \(\mathbb{C}\)-algebra, with simples indexed by a subset \(W_{\mathfrak{q}}\subset W\).

The relationship between \(\mathcal{O}_\lambda\) and \(\mathcal{O}^{\mathfrak{q}}\) was clarified in [9]. Under the hypothesis that the corresponding Weyl–group stabilizers coincide, i.e.\(S_\lambda=S_{\mathfrak{q}}\), one can choose projective generators whose endomorphism algebras are Koszul dual. More precisely, there exist finite-dimensional Koszul algebras \(A_Q\) and \(A^Q\) such that \[\mathcal{O}_Q := A_Q\mathrm{-}\operatorname{gmod}, \qquad \mathcal{O}^Q := A^Q\mathrm{-}\operatorname{gmod}\] are graded models for \(\mathcal{O}_\lambda\) and \(\mathcal{O}^{\mathfrak{q}}\), respectively. In this situation Koszul duality yields a triangulated equivalence (see [9]) \[\mathsf{D}^b(\mathcal{O}_Q)\;\xrightarrow{\sim}\;\mathsf{D}^b(\mathcal{O}^Q),\] in complete analogy with Corollary 3.12.

Since \(A_Q\) and \(A^Q\) have finite global dimension, forgetting the gradings interacts well with our non-graded Koszul duality machinery. In particular, using Corollary 4.5 we obtain the following result, which furnishes a non-graded counterpart to the Beilinson–Ginzburg–Soergel equivalences and thus extends the graded framework of [9] to the ungraded setting.

5.2 Non-graded derived Koszul duality for Perverse Sheaves↩︎

Let \(X\) be a complex algebraic variety equipped with a Whitney stratification \[X = \bigsqcup_{w \in W} X_w,\] where each stratum \(X_w\) is a locally closed algebraic subvariety isomorphic to an affine space. Denote by \(\mathsf{P}(X, W)\) the abelian category of perverse sheaves on \(X\) that are constructible with respect to the given stratification. This category forms the heart of a natural \(t\)-structure on the full triangulated subcategory \(\mathsf{D}^b(X, W) \subset \mathsf{D}^b(X)\) of algebraically constructible complexes, and is known to have enough projective objects.

Assume now that for every \(w \in W\), the closure \(\overline{X}_w\) admits a resolution of singularities \[\widehat{X}_w \longrightarrow \overline{X}_w\] whose rational homology is spanned by classes of algebraic cycles. Under this geometric assumption, it follows from [9] that the category \(\mathsf{P}(X, W)\) is equivalent to the category of finite-dimensional modules over a finite-dimensional \(\mathbb{N}\)-graded Koszul \(\mathbb{C}\)-algebra \[A = \bigoplus_{i \geq 0} A_i, \quad \text{with } A_0 \cong \mathbb{C}^{\# W}.\] Let \(L := \bigoplus_{w \in W} L_w\) be the direct sum of the simple perverse sheaves associated to the strata, where each \(L_w = \operatorname{IC}(\mathbb{C}_{X_w})\) denotes the intersection cohomology complex extended by zero. Then the Ext-algebra \[A^! := \operatorname{Ext}^\bullet_{\mathsf{D}^b(X)}(L, L)\] is Koszul and Koszul dual to \(A\).

Since \(\mathsf{P}(X, W) \simeq A\mathrm{-mod}\) and has finite homological dimension (see [9]), our main duality result yields the following triangulated equivalence.

This provides a non-graded Koszul duality realization of the bounded derived category of perverse sheaves on \(X\). For a graded derived Koszul duality, one instead works with the mixed version of \(\mathsf{P}(X, W)\), as discussed in [9].

6 Classical Examples↩︎

The purpose of this section is to examine the behavior of the Koszul dualities established in the previous sections for several important classes of Koszul algebras. We consider, in particular, the hereditary and radical-square-zero cases, as well as the exterior and symmetric algebras, which are Koszul dual to one another. We show that our constructions recover the Bernstein–Gelfand–Gelfand correspondence. In addition, we establish that Beilinson algebras admit a well-structured derived non-graded Koszul duality.

6.1 Algebras with Radical Square Zero and Hereditary Algebras↩︎

In this subsection we investigate our Koszul dualities in the setting of hereditary algebras and algebras with radical square zero. A related graded derived Koszul duality was established by Bautista and Liu [12] in the context of locally finite gradable quivers. However, their construction applies only to a single connected component of the quiver of the Koszul dual and is not formulated in terms of graded modules. Moreover, they construct a Galois covering functor for \(\mathsf{D}^b(\Lambda\mathrm{-mod})\); see [12]. Even in these cases, the method of proof we employ here differs substantially from theirs. At the level of singularity categories, Smith [35] obtained a related result using a different approach.

In forthcoming work [36], we shall further develop these ideas in the context of quadratic monomial algebras.

Recall that if \(\Lambda = kQ/J^2\) is an algebra with radical square zero, then its Koszul dual \(\Lambda^!\) is the opposite path algebra, \(\Lambda^! = kQ^{\mathrm{op}}\). In particular, \(\Lambda^!\) is hereditary.

We begin with the following result, which is a graded reformulation of [37].

When \(\Lambda^!\) is hereditary, the category of coperfect modules admits a particularly explicit description, as does its associated tails category.

We are now ready to state our graded derived and singular Koszul dualities for finite-dimensional algebras with radical square zero.

Recall that any abelian semisimple category equipped with an autoequivalence admits a canonical triangulated structure; see [38]. More precisely, if \(T\) is an autoequivalence of an abelian semisimple category \(\mathcal{A}\), then one can endow \(\mathcal{A}\) with a triangulated structure whose translation functor is \(T\), and whose distinguished triangles are, up to isomorphism, finite direct sums of triangles of the form \[X \to 0 \to T X \xrightarrow{\mathrm{id}} T X,\quad X \xrightarrow{\mathrm{id}} X \to 0 \to T X,\quad 0 \to X \xrightarrow{\mathrm{id}} X \to 0.\] Applying this observation in our context, we obtain the following result.

Recall from Section 3 that there exists a functor \[K \colon \Lambda^!\mathrm{-GMod} \longrightarrow \mathsf{C}\bigl(\Lambda\mathrm{-proj}^{\mathbb{Z}}\bigr),\] which restricts on finitely co-presented modules to \[K \colon \Lambda^!\mathrm{-Fcp}^{\mathbb{Z}} \longrightarrow \mathsf{C}^{-,b}\bigl(\Lambda\mathrm{-proj}^{\mathbb{Z}}\bigr).\] By construction of the functor \(K\), there exists an additive functor \(\overline{K}\) making the following diagram commute: \[\begin{tikzcd}[column sep=small] \Lambda^!\mathrm{-Fcp}^{\mathbb{Z}} \ar[d, two heads, "\pi"'] \ar[r, "K"] & \mathsf{C}^{-,b}\bigl(\Lambda^{\mathbb{Z}}\mathrm{-proj}\bigr) \ar[r] & \mathsf{K}^{-,b}\bigl(\Lambda^{\mathbb{Z}}\mathrm{-proj}\bigr) \ar[r, "\sim"] & \mathsf{D}^b(\Lambda\mathrm{-gmod}) \ar[r] & \mathsf{D}_{\mathrm{sg}}(\Lambda\mathrm{-gmod}) \\ \Lambda^!\mathrm{-Fcp}^{\mathbb{Z}} / \Lambda^!\mathrm{-gmod} \ar[rrrru, dashed, "\overline{K}"'] \end{tikzcd}\]

We now obtain a simplified description of the graded singularity category of a finite-dimensional algebra with radical square zero, to be compared with [35].

For algebras with radical square zero, Theorem 4.1 can be improved as follows:

The non-graded derived and singular Koszul dualities can also be refined as follows:

Recall that, by Gabriel’s theorem [39], a finite-dimensional hereditary algebra \(\Lambda\) is of finite representation type if and only if the underlying graph of its quiver is a Dynkin diagram of type \[A_n,\quad D_n,\quad E_6,\quad E_7,\quad \text{or } E_8.\]

It is well known that if a finite-dimensional graded algebra is of finite representation type, then every module is gradable; see [40]. In particular, the forgetful functor \[F\colon \Lambda\mathrm{-gmod}\longrightarrow \Lambda\mathrm{-mod}\] is a \(\mathsf{G}\)-Galois covering.

Before describing the bounded derived category of finite-dimensional hereditary algebras of finite representation type, we recall the following well-known characterization of objects in \(\mathsf{D}^b(\Lambda\mathrm{-mod})\); see, for example, [41].

We now give a new description of the bounded derived category of a finite-dimensional hereditary algebra of finite representation type.

We now turn to the final example of this subsection and present both graded and ungraded versions of the Bernstein–Gelfand–Gelfand correspondence for algebras with radical square zero. Recall from [42] that any finite-dimensional algebra with radical square zero is either selfinjective or Cohen–Macaulay free, that is, every Gorenstein-projective module is projective. Invoking this result, we obtain the following triangulated equivalences.

6.2 Exterior Algebras and Symmetric Algebras↩︎

In this subsection we present a second example illustrating the structure of our Koszul-type dualities in the context of the exterior algebra and its Koszul dual, the symmetric algebra. In particular, we establish an ungraded version of the Bernstein–Gelfand–Gelfand (BGG) correspondence for the exterior algebra, thereby allowing a direct comparison with the classical construction in [3].

Recall that Theorem 3.8 establishes a triangulated equivalence \[\mathfrak{F}\colon \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-gmod}\bigr) \longrightarrow \mathsf{D}^{b}\!\bigl(\Lambda^{!}\mathrm{-Pe}^{\mathbb{Z}}\bigr).\]

We now turn our attention to the exterior algebra. We begin by recalling some classical definitions and structural results. Let \(V\) be a finite-dimensional vector space over a field \(k\), with basis \(\{e_1, e_2, \dots, e_n\}\). The exterior algebra on \(V\), denoted \(\bigwedge V\), is the associative finite-dimensional graded \(k\)-algebra generated by the elements \(e_1, e_2, \dots, e_n\) in degree one, subject to the relations \[e_i e_j + e_j e_i = 0 \qquad \text{for all } 1 \leq i, j \leq n.\] In particular, \(e_i^2 = 0\) for all \(i\). These relations imply that all monomials of the form \(e_{i_1} e_{i_2} \cdots e_{i_r}\) are alternating in the indices, and hence \(\bigwedge V\) is naturally graded: \[\bigwedge V = \bigoplus_{i=0}^{n} \bigwedge\nolimits^i V,\] where \(\bigwedge^i V\) is the \(k\)-vector space spanned by the wedge products \(e_{j_1} \wedge \cdots \wedge e_{j_i}\) with \(j_1 < \cdots < j_i\). It is well known that \(\bigwedge V\) is a self-injective algebra.

Alternatively, \(\bigwedge V\) can be realized as a quotient of the path algebra of the quiver \(Q\) with a single vertex and \(n\) loops: \[Q_0 = \{*\}, \qquad Q_1 = \{\alpha_1, \alpha_2, \dots, \alpha_n\},\] by the two-sided ideal generated by the quadratic relations \[\alpha_i \alpha_j + \alpha_j \alpha_i = 0 \qquad \text{for all } 1 \leq i, j \leq n.\] That is, \[\bigwedge V \cong kQ \big/ \left( \alpha_i \alpha_j + \alpha_j \alpha_i \;\middle|\; 1 \leq i, j \leq n \right),\] so that, in particular, \(\alpha_i^2 = 0\) for all \(i\).

The exterior algebra \(\bigwedge V\) is known to be Koszul and self-injective. Its Koszul dual is the symmetric algebra \(\mathrm{S}V\), which is canonically isomorphic to the polynomial algebra \(k[X_1, X_2, \dots, X_n]\). This algebra has finite global dimension and admits a quiver-theoretic realization as a bound path algebra: \[kQ \big/ \left( \alpha_i \alpha_j - \alpha_j \alpha_i \;\middle|\; 1 \leq i < j \leq n \right),\] where the quiver \(Q\) again consists of a single vertex with \(n\) loops \(\alpha_1, \dots, \alpha_n\).

We now turn to our first application. Theorem 3.8 induces the following triangulated equivalence, which is the foundational example of Koszul-type duality.

Since \(\mathrm{S}(V)\) is Noetherian, the subcategory \(\mathrm{S}(V)\mathrm{-Pe}^{\mathbb{Z}}\) is abelian and coincides with \(\mathrm{S}(V)\mathrm{-Fg}^{\mathbb{Z}}\), the category of finitely generated graded \(\mathrm{S}(V)\)-modules.

Since \(\bigwedge V\) is self-injective, every injective module is also projective, and the category of finite-dimensional graded Gorenstein projective \(\bigwedge V\)-modules coincides with the category of finite-dimensional graded \(\bigwedge V\)-modules. This yields the well-known Bernstein–Gelfand–Gelfand correspondence.

We are now ready to establish a pair of new non-graded triangulated equivalences in the context of the exterior algebra and its Koszul dual, the symmetric algebra.

6.3 Beilinson Algebras↩︎

Recall that the \(d\)-th Beilinson algebra \(\mathcal{B}_{d}\) is finite-dimensional and admits a presentation \(\mathcal{B}_{d}\cong kQ/I\), where \(Q\) has vertices \(\{0,1,\dots,d\}\) and, for each \(r=0,\dots,d-1\), has \(d+1\) parallel arrows from \(r\) to \(r+1\). Schematically, one may depict \(Q\) as follows: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/jixzcpue.png}\label{chpkljam}\end{figure}\tag{1}\] The ideal \(I\) is generated by the commutativity relations on length-two paths: for every \(r=0,\dots,d-2\) and every \(0\le a,b\le d\), \[x^{(r+1)}_{a}\,x^{(r)}_{b}=x^{(r+1)}_{b}\,x^{(r)}_{a}.\] In particular, \(Q\) is well directed.

Beilinson’s theorem yields a triangulated equivalence \[\mathsf{D}^{b}\!\bigl(\mathrm{coh}\,\mathbb{P}^{d}\bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^{b}\!\bigl(\mathcal{B}_{d}\mathrm{-mod}\bigr), \qquad\text{see \cite{5}.}\]

Moreover, \(\mathcal{B}_{d}\) is Koszul. Its Koszul dual \(\mathcal{B}_{d}^{!}\) is given by the opposite quiver together with quadratic anti-commutativity relations.

Therefore, by Corollary 3.11 the derived Koszul duality induces a triangulated equivalence \[\mathfrak{F}\colon \mathsf{D}^{b}\!\bigl(\mathcal{B}_{d}^{!}\mathrm{-mod}\bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^{b}\!\bigl(\mathcal{B}_{d}\mathrm{-mod}\bigr).\]

7 Bounded Derived Categories of Finite-Dimensional Graded Algebras↩︎

7.1 Two Conjectures↩︎

We conclude by formulating two conjectures concerning the bounded derived category and the singularity category of finite-dimensional graded algebras. Motivated by the work of Beilinson–Ginzburg–Soergel on realizations of categories of (mixed) perverse sheaves \(\mathsf{P}(X,W)\) (see [9]), and by the results established above, it is natural to expect that analogous realization results, and hence more general duality phenomena, should hold for arbitrary finite-dimensional graded algebras. The following conjectures record directions that originally inspired this work.

Both conjectures are now established for finite-dimensional Koszul algebras. Moreover, Conjecture 2 is known to hold when \(\Lambda\) is self-injective with Gorenstein parameter \(\ell\); see [43]. These conjectures may also be viewed as being in the same general spirit as Kontsevich’s homological mirror symmetry conjecture [44], where triangulated hulls arise naturally in the setting of \(A_{\infty}\)-categories.

7.2 DG enhancement for the bounded derived category of a finite-dimensional graded algebra↩︎

We next provide evidence for Conjecture 1 by exhibiting an explicit candidate for the exact category \(\mathcal{A}\) and for the acting group \(G\) that realize the proposed orbit–category description. The construction is largely dictated by the framework developed in the preceding sections and may be viewed as a natural application of it. In our setting we take \[\mathcal{A}=\mathcal{PLC}^{b}(\Lambda^{!}),\] the exact (indeed abelian) category of perfectly linear bounded complexes, and we specify the group \(G\) explicitly below.

We stress, however, that the finiteness of the homological dimension required in Conjecture 1 is not addressed here: the category \(\mathcal{PLC}^{b}(\Lambda^{!})\) need not have finite homological dimension in general. In particular, when \(\Lambda\) has infinite global dimension, the homological dimension of \(\mathcal{PLC}^{b}(\Lambda^{!})\) is typically infinite. Before stating the results, we recall the notion of the quadratic dual of a finite-dimensional graded algebra \(\Lambda\), following Martı́nez–Villa and Saorı́n [10].

Let \(\Lambda=kQ/I\) be a finite-dimensional graded algebra. Denote by \(\widetilde{\Lambda}=kQ/\widetilde{I}\) the quadratic algebra associated to \(\Lambda\), where \(\widetilde{I}\) is the ideal generated by the quadratic relations in \(I\). The quadratic dual of \(\Lambda\) is defined to be the quadratic dual of \(\widetilde{\Lambda}\) in the sense of Subsection 2.2, namely \[\Lambda^{!}:=\widetilde{\Lambda}^{!}.\]

Moreover, by [10] (see also [14]), there is a fully faithful functor \[K\colon \Lambda\mathrm{-gmod}\longrightarrow \mathcal{LC}^{b}\!\bigl(\Lambda^{!}\mathrm{-proj}^{\mathbb{Z}}\bigr),\] which sends a graded \(\Lambda\)-module \(M\) to the associated linear complex of finite-dimensional graded projective \(\Lambda^{!}\)-modules (cf.Subsection 3.1). More precisely, \(K\) can be chosen as the composite of the canonical embedding \[\Lambda\mathrm{-gmod}\longrightarrow \widetilde{\Lambda}\mathrm{-gmod}\] with the equivalence \[\widetilde{\Lambda}\mathrm{-gmod}\;\cong\; \mathcal{LC}^{b}\!\bigl(\Lambda^{!}\mathrm{-proj}^{\mathbb{Z}}\bigr).\]

In particular, if \(\Lambda\) is quadratic, then \(K\) is an equivalence.

We denote by \(\mathcal{PLC}^{b}(\Lambda^{!})\) the full subcategory of \(\mathcal{LC}^{b}\!\bigl(\Lambda^{!}\mathrm{-proj}^{\mathbb{Z}}\bigr)\) consisting of perfectly linear bounded complexes, that is, the essential image of \(K\). In particular, the functor \(K\) restricts to an equivalence \[\Lambda\mathrm{-gmod}\;\xrightarrow{\;\sim\;}\;\mathcal{PLC}^{b}(\Lambda^{!}),\] and we fix a quasi-inverse \[K'\colon \mathcal{PLC}^{b}(\Lambda^{!})\longrightarrow \Lambda\mathrm{-gmod}.\] Consequently, \(K'\) induces a triangulated equivalence \[\mathsf{D}^{b}\!\bigl(\mathcal{PLC}^{b}(\Lambda^{!})\bigr) \;\xrightarrow{\;\sim\;}\; \mathsf{D}^{b}\!\bigl(\Lambda\mathrm{-gmod}\bigr),\] which admits a dg enhancement, see for instance [28].

By Lemma 4.2, for every bounded complex \(X^{\bullet}\) and every \(i\in\mathbb{Z}\) there is a natural isomorphism \[K'\bigl(X^{\bullet}\langle -i\rangle[i]\bigr)\;\cong\;K'(X^{\bullet})\langle i\rangle.\] Combining Theorem 4.1 with the above triangulated equivalence and invoking Theorem 2.42, we obtain the following dg-enhanced orbit descriptions.

E-mail address: alesm.bouhada@gmail.com

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