2 Preliminaries

2.1 Cluster-like categories

We recall some basic concepts on certain structures in triangulated categories. At the end of this subsection, we state Iyama–Yang’s result (Theorem 2.6), which gives a general framework for the construction of ‘cluster-like’ categories.

Let us start with the following fundamental notion.

Definition 2.1. An object or a subcategory $\mathcal {M}$ in a triangulated category $\mathcal {T}\,{}$ is silting if $\operatorname {\mathrm {Hom}}_{\mathcal {T}}(\mathcal {M},\mathcal {M}[>\!0])=0$ and $\operatorname {\mathrm {thick}}\mathcal {M}=\mathcal {T}\,{}$ .

We assume throughout this paper that our silting subcategory $\mathcal {M}$ satisfies $\mathcal {M}=\operatorname {\mathrm {add}}\mathcal {M}$ : that is, it is closed under direct sums and summands. The standard example of a silting object is $\Lambda \in \operatorname {\mathrm {per}}\Lambda $ for a negative DG algebra $\Lambda $ : that is, a DG algebra with $H^{>0}\Lambda =0$ . The same holds for negative DG categories.

Let $\mathcal {C}$ and $\mathcal {D}$ be subcategories of a triangulated category $\mathcal {T}\,{}$ . We set

$$\begin{align*}\mathcal{C}\ast\mathcal{D}=\{ X\in\mathcal{T} \mid \text{there is a triangle } C \to X \to D \to C[1] \text{ for some } C\in\mathcal{C},\, D\in\mathcal{D} \}. \end{align*}$$

By the octahedral axiom, the operation $\ast $ is associative. One obtains a co-t-structure (or weight structure) [Reference BondarkoBon, Reference PauksztelloP] from a silting subcategory, which is given as follows.

Proposition-Definition 2.2 See, e.g., [Reference Aihara and IyamaAI, Proposition 2.23][Reference Iyama and YangIYa1, Proposition 2.8]

Let $\mathcal {T}\,{}$ be an idempotent complete triangulated category with a silting subcategory $\mathcal {M}$ . Set

$$ \begin{align*} \begin{aligned} t_{\geq0}&=\bigcup_{l\geq0}\mathcal{M}[-l]\ast\cdots\ast\mathcal{M}[-1]\ast\mathcal{M}, \\ t_{\leq0}&=\bigcup_{l\geq0}\mathcal{M}\ast\mathcal{M}[1]\ast\cdots\ast\mathcal{M}[l] \end{aligned}. \end{align*} $$

Then $(t_{\geq 0},t_{\leq 0})$ is a co-t-structure. We call it the $\mathrm{co}\text{-}t\text{-}\mathrm{structure\ associated\ to\ } \mathcal {M}$ .

In what follows, we will simply write $t_{\geq 0}=\cdots \ast \mathcal {M}$ , $t_{\leq 0}=\mathcal {M}\ast \cdots $ and so on. It is important for us that some co-t-structures and t-structures are related.

Definition 2.3 [Reference BondarkoBon]

Let $\mathcal {M}\subset \mathcal {T}\,{}$ be a silting subcategory and $(t_{\geq 0},t_{\leq 0})$ the associated co-t-structure. Let $t=(t^{\leq 0},t^{\geq 0})$ be a t-structure in $\mathcal {T}\,{}$ .

1. 1. We say t is right adjacent to $\mathcal {M}$ if $t_{\leq 0}=t^{\leq 0}$ .

2. 2. We say t is left adjacent to $\mathcal {M}$ if $t_{\geq 0}=t^{\geq 0}$ .

For example, if $\Lambda $ is a negative DG algebra that is homologically smooth such that each cohomology is finite dimensional, then the standard t-structure on $\operatorname {\mathrm {per}}\Lambda $ is right adjacent to a silting object $\Lambda \in \operatorname {\mathrm {per}}\Lambda $ . It follows that its image under the duality $\operatorname {{\mathrm {R}Hom }}_\Lambda (-,\Lambda )\colon \operatorname {\mathrm {per}}\Lambda \leftrightarrow \operatorname {\mathrm {per}}\Lambda ^{\mathrm {op}}$ is left adjacent to a silting object $\Lambda \in \operatorname {\mathrm {per}}\Lambda ^{\mathrm {op}}$ .

Now let us recall the notion of (relative) Serre functors.

Definition 2.4. Let $\mathcal {T}\,{}$ be a k-linear $\operatorname {\mathrm {Hom}}$ -finite triangulated category and $\mathcal {T}^{\mathrm {fd}} \subset \mathcal {T}\,{}$ a thick subcategory.

1. 1. An autoequivalence $S\colon \mathcal {T}\to \mathcal {T}\,{}$ is a Serre functor on $\mathcal {T}\,{}$ if there is a functorial isomorphism

   $$\begin{align*}\operatorname{\mathrm{Hom}}_{\mathcal{T}}(X,Y) \simeq D\operatorname{\mathrm{Hom}}_{\mathcal{T}}(Y,SX) \end{align*}$$
   for all $X, Y \in \mathcal {T}\,{}$ .

2. 2. A triangle autoequivalence $S \colon \mathcal {T} \to \mathcal {T}\,{}$ is a relative Serre functor for $\mathcal {T}^{\mathrm {fd}}\subset \mathcal {T}\,{}$ if it restricts to an autoequivalence on $\mathcal {T}^{\mathrm {fd}}$ and the above functorial isomorphism holds for all $X\in \mathcal {T}^{\mathrm {fd}}$ and $Y \in \mathcal {T}\,{}$ .

3. 3. We say that $(\mathcal {T},\mathcal {T}^{\mathrm {fd}},S,\mathcal {M})$ is a relative Serre quadruple if S is a relative Serre functor for $\mathcal {T}^{\mathrm {fd}}\subset \mathcal {T}\,{}$ and $\mathcal {M}$ is a silting subcategory of $\mathcal {T}\,{}$ .

Note that we require a relative Serre functor S to be an triangle autoequivalence on $\mathcal {T}\,{}$ (as was not the case in [Reference Iyama and YangIYa1]). We need this for a slightly generalised formulation of their results, which we present below in Theorem 2.6. Note also that the natural isomorphism $S\circ [1]\simeq [1]\circ S$ associated to the triangle functor S is not necessarily the identity even when $S=[d+1]$ ; see [Reference KellerKe4, Section 2.5].

We have one more notion to recall.

Definition 2.5. A k-linear idempotent-complete category $\mathcal {C}$ is a dualising variety if $D=\operatorname {\mathrm {Hom}}_k(-,k)$ induces a duality $\operatorname {\mathrm {mod}}\mathcal {C}\leftrightarrow \operatorname {\mathrm {mod}}\mathcal {C}^{\mathrm {op}}$ between the category of finitely presented $\mathcal {C}$ -modules.

For example, the category $\operatorname {\mathrm {proj}}^{\mathbb {Z}}\!\Lambda $ of finitely generated graded projective modules over a finite-dimensional graded algebra $\Lambda $ is a dualising variety.

We are now ready to state the following generalised formulation of Amiot’s cluster category.

Theorem 2.6 [Reference Iyama and YangIYa1, Reference Iyama and YangIYa2]

Let $(\mathcal {T},\mathcal {T}^{\mathrm {fd}}, S, \mathcal {M})$ be a relative Serre quadruple such that $\mathcal {M}$ is a dualising variety, and $(t_{\geq 0},t_{\leq 0})$ be the co-t-structure associated to $\mathcal {M}$ .

1. 1. The silting subcategory $\mathcal {M}$ has a right-adjacent t-structure with $t_{\leq 0}^\perp \subset \mathcal {T}^{\mathrm {fd}}$ if and only if it has a left-adjacent t-structure with ${}^\perp t_{\geq 0} \subset \mathcal {T}^{\mathrm {fd}}$ . Suppose in what follows that the equivalent conditions above are satisfied.

2. 2. The category $\mathcal {T}/\mathcal {T}^{\mathrm {fd}}$ has a Serre functor $S\circ [-1]$ .

3. 3. The quotient functor $\pi \colon \mathcal {T} \to \mathcal {T}/\mathcal {T}^{\mathrm {fd}}$ induces bijections $\operatorname {\mathrm {Hom}}_{\mathcal {T}}(X,Y) \to \operatorname {\mathrm {Hom}}_{\mathcal {T}/\mathcal {T}^{\mathrm {fd}}}(X,Y)$ for all $X \in t_{\leq 0}$ and $Y \in St_{\geq 2}$ . Moreover, the composition $t_{\leq 0}\cap S t_{\geq 2} \subset \mathcal {T} \to \mathcal {T}/\mathcal {T}^{\mathrm {fd}}$ is an additive equivalence.

4. 4. Let $n\geq 1$ , and assume that $\mathcal {M}$ is stable under $S_{n+1}=S\circ [-n-1]$ . Then $\pi (\mathcal {M})\subset \mathcal {T}/\mathcal {T}^{\mathrm {fd}}$ is an n-cluster tilting subcategory.

Proof. (1) is [Reference Iyama and YangIYa1, Theorem 4.10]. For (2) and (3), adapt the proof of [Reference Iyama and YangIYa1, Theorem 5.8(a)(b)]. We include a proof of (4) since it requires a modification from [Reference Iyama and YangIYa1, Theorem 5.8]. Since $\mathcal {M}$ is stable under $S_{n+1}$ , so is $t_{\geq i}$ for each $i\in \mathbb {Z}$ ; thus we have $St_{\geq 2}=S_{n+1}t_{\geq 1-n}=t_{\geq 1-n}=\cdots \ast \mathcal {M}[n-1]$ . Therefore, we deduce that the fundamental domain $t_{\leq 0}\cap S t_{\geq 2}$ is $\mathcal {M}\ast \cdots \ast \mathcal {M}[n-1]$ , hence the result.

If $\Lambda $ is a bimodule $(n+1)$ -CY negative DG algebra with finite-dimensional $H^0\Lambda $ , then $(\operatorname {\mathrm {per}}\Lambda ,\mathcal {D}^b(\Lambda ),[n+1],\Lambda )$ is a relative Serre quadruple. One can apply the above theorem and recover the original results of Amiot and Guo.

2.2 Comparing the derived categories of graded rings

We collect some basic results on derived categories of graded rings based on the comparison of derived categories of an abelian category and its Serre subcategory. Although various conditions on nice behavior for these derived categories are well-known (e.g., [Reference VerdierVe, III, Section 2.2]), we could not find a precise reference for our setting, so we include this section for the convenience of the reader.

Recall that a subcategory $\mathcal {B}$ of an abelian category $\mathcal {A}$ is wide if it is closed under kernels, cokernels and extensions. This ensures that $\mathcal {D}_{\mathcal {B}}^\ast (\mathcal {A})$ , the full subcategory of $\mathcal {D}^\ast (\mathcal {A})$ consisting of complexes with cohomologies in $\mathcal {B}$ , is a thick subcategory of $\mathcal {D}^\ast (\mathcal {A})$ for each boundedness condition $\ast \in \{b,+,-,\{\} \}$ . Similarly, we denote by $\mathcal {C}^\ast _{\mathcal {B}}(\mathcal {A})$ the full subcategory of $\mathcal {C}^\ast (\mathcal {A})$ consisting of complexes with cohomologies in $\mathcal {B}$ .

Proposition 2.7. Let $\mathcal {B}\subset \mathcal {A}$ be a wide subcategory. Suppose that for any morphism $Y\to Y_1$ in $\mathcal {C}^\ast (\mathcal {A})$ with $Y\in \mathcal {C}^\ast (\mathcal {B})$ and $Y_1\in \mathcal {C}^\ast _{\mathcal {B}}(\mathcal {A})$ , there exist quasi-isomorphisms $Y_1\leftarrow Y_2\rightarrow Y_3$ such that

• ○ $Y_2\to Y_1$ is injective (in $\mathcal {C}(\mathcal {A})$ ),

• ○ $Y\to Y_1$ factors through $Y_2\to Y_1$ ,

• ○ $Y_3\in \mathcal {C}(\mathcal {B})$ ,

Then the natural functor $\mathcal {D}^\ast (\mathcal {B})\to \mathcal {D}_{\mathcal {B}}^\ast (\mathcal {A})$ is an equivalence.

Proof. We first show that the functor is faithful. Let $X\to Y$ be a morphism in $\mathcal {K}^\ast (\mathcal {B})$ whose image in $\mathcal {D}^\ast (\mathcal {A})$ is $0$ . Then there is a quasi-isomorphism $Y\to Y_1$ in $\mathcal {K}^\ast (\mathcal {A})$ such that the composite $X\to Y\to Y_1$ is $0$ in $\mathcal {K}^\ast (\mathcal {A})$ . Adding to Y a null-homotopic complex $N\in \mathcal {C}^\ast (\mathcal {B})$ , for example the mapping cone of the identity map of X, we have a zero map $X\to Y\oplus N\to Y_1$ in $\mathcal {C}^\ast (\mathcal {A})$ . Now, applying the assumption to the quasi-isomorphism $Y\oplus N\to Y_1$ , there exist quasi-isomorphisms $Y_1\hookleftarrow Y_2\to Y_3$ satisfying the above conditions, giving rise to the following commutative diagram in $\mathcal {C}(\mathcal {A})$ :

Clearly, the composite $Y\oplus N\to Y_2\to Y_3$ is a quasi-isomorphism. Also, since $X\to Y\oplus N\to Y_1$ is $0$ and $Y_2\to Y_1$ is injective, it follows that $X\to Y\oplus N\to Y_3$ is $0$ . By $Y_3\in \mathcal {C}(\mathcal {B})$ , we conclude that $X\to Y$ is $0$ in $\mathcal {D}(\mathcal {B})$ .

We next show that the functor is full. Let $X,Y\in \mathcal {C}^\ast (\mathcal {B})$ , and present a morphism $X\to Y$ in $\mathcal {D}^\ast (\mathcal {A})$ by the diagram $X\rightarrow Y_1\leftarrow Y$ in $\mathcal {C}^\ast (\mathcal {A})$ . Applying the assumption to $X\oplus Y\to Y_1$ , we have quasi-isomorphisms $Y_1\leftarrow Y_2\to Y_3$ , giving rise to the commutative diagram

This shows that the diagram $X\to Y_3\leftarrow Y$ in $\mathcal {C}(\mathcal {B})$ defines the same morphism in $\mathcal {D}(\mathcal {A})$ as $X\to Y_1\leftarrow Y$ .

Finally, noting that the image of $Y_3\in \mathcal {C}(\mathcal {B})$ in $\mathcal {D}(\mathcal {B})$ lies in $\mathcal {D}^\ast (\mathcal {B})$ , we see that the functor is dense by setting $Y=0$ in the assumption.

Applying the above criterion, we can prove the following, which will be implicitly used throughout this paper.

Proposition 2.8. Let $R=\bigoplus _{i\leq 0}R_i$ be a negatively graded algebra such that each $R_i$ is finite dimensional. Then the natural functor $\mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R)\to \mathcal {D}^b_{\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R}(\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R)$ is an equivalence.

Proof. We check the condition in Proposition 2.7 with $\mathcal {B}=\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R$ and $\mathcal {A}=\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R$ . Let $Y\in \mathcal {C}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R)$ , $Y_1\in \mathcal {C}^b_{\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R}(\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R)$ and $Y\to Y_1$ be a morphism in $\mathcal {C}^b(\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R)$ . Since the terms of Y are of finite length, the morphism $Y\to Y_1$ factors through $(Y_1)_{\leq n}\to Y_1$ for sufficiently large n. Also, since the cohomology of $Y_1$ is bounded and each is of finite length, the truncation $(Y_1)_{\leq n}\to Y_1$ with respect to the grading is a quasi-isomorphisms for sufficiently large n. Taking the commonly large n, we obtain an injective quasi-isomorphism $Y_2:=(Y_1)_{\leq n}\to Y_1$ through which $Y\to Y_1$ factors. Similarly, the truncation $Y_2\to (Y_2)_{\geq -m}$ is a quasi-isomorphism for sufficiently large m. Put $Y_3':=(Y_2)_{\geq -m}$ for such m. Then each term of $Y_3^\prime $ is concentrated in degree $[-m,n]$ ; thus it is a module over the finite-dimensional factor algebra $R_{\geq -n-m}$ of R. Now we work over this finite-dimensional algebra. Then one can take quasi-isomorphism $Y^{\prime }_3\to Y_3$ with $Y_3\in \mathcal {C}^{+,b}(\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!R_{\geq -n-m})$ (e.g., with injective terms), and we have $Y_3\in \mathcal {C}(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R)$ .

3 t-structure in $\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$

We will be interested in a negatively graded CY algebra with an a-invariant. Before that, we will place ourselves in a slightly more general setting. Let $R=\bigoplus _{i \leq 0}R_i$ be a negatively graded algebra. We assume the following on R:

• (R1) Each $R_i$ is finite dimensional.

• (R2) $\mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R)$ is contained in $\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$ .

The condition (R2) is clearly satisfied if R is homologically smooth. The aim of this section is to show that there is a t-structure in $\operatorname {\mathrm {per}} ^{\mathbb {Z}}\!R$ in this setting. For each $j\in \mathbb {Z}$ , we denote by $\operatorname {\mathrm {Mod}}^{\leq j}\!R$ (respectively, $\operatorname {\mathrm {Mod}}^{\geq j}\!R$ ) the full subcategory of $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R$ consisting of graded modules concentrated in degree $\leq j$ (respectively, $\geq j$ ).

Theorem 3.1. Let R be a negatively graded algebra satisfying (R1) and (R2). Set

$$ \begin{align*} \begin{aligned} t^{\leq0}&=\{ X \in \operatorname{\mathrm{per}}^{\mathbb{Z}}\!R \mid H^i(X) \in \operatorname{\mathrm{Mod}}^{\leq-i}\!R \text{ for all } i \in \mathbb{Z} \} ,\\ t^{\geq0}&=\{ X \in \operatorname{\mathrm{per}}^{\mathbb{Z}}\!R \mid H^i(X) \in \operatorname{\mathrm{Mod}}^{\geq-i}\!R \text{ for all } i \in \mathbb{Z} \}. \end{aligned} \end{align*} $$

Then $(t^{\leq 0},t^{\geq 0})$ is a t-structure in $\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$ .

Let us collect some basic notions for big triangulated categories. Let $\mathcal {T}\,{}$ be a triangulated category with arbitrary (set-indexed) coproducts. An object $C\in \mathcal {T}\,{}$ is compact if the natural map $\bigoplus _{i\in I}\operatorname {\mathrm {Hom}}_{\mathcal {T}}(C,X_i)\to \operatorname {\mathrm {Hom}}_{\mathcal {T}}(C,\bigoplus _{i\in I}X_i)$ is an isomorphism for every set $\{X_i\mid i\in I\}$ of objects. A subcategory is said to generate $\mathcal {T}\,{}$ if the smallest triangulated subcategory containing it and closed under arbitrary coproducts is the whole $\mathcal {T}\,{}$ . We call a subcategory of $\mathcal {T}\,{}$ a compact set of generators if it is skeletally small, consists of compact objects and generates $\mathcal {T}\,{}$ .

Now recall from [Reference Aihara and IyamaAI, Definition 4.1] that a subcategory $\mathcal {S}$ of $\mathcal {T}\,{}$ is silting if it forms a compact set of generators such that $\operatorname {\mathrm {Hom}}_{\mathcal {T}}(A,B[>\!0])=0$ for all $A,B \in \mathcal {S}$ . Note that this is a modified version of Definition 2.1.

In the remainder of this section, let S be an arbitrary negatively graded algebra. Note that we do not need (R1) or (R2) until the proof of Theorem 3.1. We will simply write $\mathcal {D}$ for $\mathcal {D}(\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!S)$ . Let us start our discussion with the following observation.

We deduce by [Reference KellerKe1, Theorem 4.3] that $\mathcal {D}$ is triangle equivalent to the derived category $\mathcal {D}(\mathcal {A})$ of a negative DG category $\mathcal {A}$ . Then we have the standard t-structure $(\mathcal {D}_{\mathcal {M}}^{\leq 0},\mathcal {D}_{\mathcal {M}}^{\geq 0})$ associated to $\mathcal {M}$ , which is given by

$$ \begin{align*} \begin{aligned} \mathcal{D}_{\mathcal{M}}^{\leq0}&=\{ X \in \mathcal{D} \mid \operatorname{\mathrm{Hom}}_{\mathcal{D}}(M,X[i])=0 \text{ for all } M \in \mathcal{M} \text{ and } i>0. \},\\ \mathcal{D}_{\mathcal{M}}^{\geq0}&=\{ X \in \mathcal{D} \mid \operatorname{\mathrm{Hom}}_{\mathcal{D}}(M,X[i])=0 \text{ for all } M \in \mathcal{M} \text{ and } i<0. \}. \end{aligned} \end{align*} $$

As usual, we put $\mathcal {D}_{\mathcal {M}}^{\leq n}=\mathcal {D}_{\mathcal {M}}^{\leq 0}[-n]$ and $\mathcal {D}_{\mathcal {M}}^{\geq n}=\mathcal {D}_{\mathcal {M}}^{\geq 0}[-n]$ for each $n\in \mathbb {Z}$ .

Now we use the following computation.

Lemma 3.4. We have

$$ \begin{align*} \begin{aligned} \mathcal{D}_{\mathcal{M}}^{\leq0}&=\{ X \in \mathcal{D} \mid H^i(X) \in \operatorname{\mathrm{Mod}}^{\leq-i}\!S \text{ for all } i \in \mathbb{Z} \}, \\ \mathcal{D}_{\mathcal{M}}^{\geq0}&=\{ X \in \mathcal{D} \mid H^i(X) \in \operatorname{\mathrm{Mod}}^{\geq-i}\!S \text{ for all } i \in \mathbb{Z} \}. \end{aligned} \end{align*} $$

Proof. Since $\mathcal {D}_{\mathcal {M}}^{\leq 0}=\{X\in \mathcal {D}\mid \operatorname {\mathrm {Hom}}_{\mathcal {D}}(M[i],X)=0 \text { for all } M\in \mathcal {M} \text { and } i<0\}$ , we have

$$\begin{align*}\begin{aligned} \mathcal{D}_{\mathcal{M}}^{\leq0}&=\{X \in \mathcal{D} \mid \operatorname{\mathrm{Hom}}_{\mathcal{D}}(S(i)[-i][-j],X)=0 \text{ for all } i \in \mathbb{Z} \text{ and } j>0 \} \\ &=\{X \in \mathcal{D} \mid H^{i+j}(X)_{-i}=0 \text{ for all } i \in \mathbb{Z} \text{ and } j>0 \} \\ &=\{X \in \mathcal{D} \mid H^i(X)_{-i+j}=0 \text{ for all } i \in \mathbb{Z} \text{ and } j>0 \}, \end{aligned} \end{align*}$$

thus the first assertion. By $\mathcal {D}_{\mathcal {M}}^{\geq 0}=\{X\in \mathcal {D}\mid \operatorname {\mathrm {Hom}}_{\mathcal {D}}(M[i],X)=0 \text { for all } M\in \mathcal {M} \text { and } i>0\}$ , we similarly have the second equation.

We need one lemma to ensure that the above t-structure in $\mathcal {D}$ restricts to the small derived category.

We are now ready to prove the main result.

4 Cluster tilting in ${\mathrm {qper}^{\mathbb {Z}}\!} R$ and the ath root of the AR translation

4.1 Cluster tilting

Let us first recall the notion of (twisted) Calabi-Yau algebras, in the graded case, which is of our central interest. Let $\alpha $ be a graded automorphism of a graded ring $\Lambda $ . We say that $\alpha $ is inner if there exists a homogeneous invertible element $a\in \Lambda $ such that $\alpha (m)=ama^{-1}$ for all $m\in \Lambda $ . We denote by $(-)_\alpha $ the twist automorphism on $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!\Lambda $ ; thus for each $M\in \operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!\Lambda $ , the graded module $M_\alpha $ has the same underlying graded vector space as M, with $\Lambda $ action $m\cdot x=m\alpha (x)$ for $m\in M$ and $x\in \Lambda $ . Similarly, given two graded automorphisms $\alpha ,\beta $ of $\Lambda $ and a graded bimodule M, we denote by ${}_\alpha M_\beta $ the twisted graded bimodule that has $\Lambda $ -action $x\cdot m\cdot y=\alpha (x)m\beta (y)$ for $m\in M$ and $x,y\in \Lambda $ . For example, the notation ${}_\alpha M_1$ shows that the action is twisted by $\alpha $ on the left, while it is non-twisted on the right.

Definition 4.1. A graded algebra R is bimodule twisted n-Calabi-Yau of a-invariant a if it satisfies the following conditions:

• ○ R is homologically smooth: that is, $R\in \operatorname {\mathrm {per}}^{\mathbb {Z}}\!R^e$ .

• ○ There exists a graded automorphism $\alpha $ of R such that $\operatorname {{\mathrm {R}Hom }}_{R^e}(R,R^e)(a)[n]\simeq {}_\alpha R_1$ in $\mathcal {D}(\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R^e)$ .

We refer to $\alpha $ as the Nakayama automorphism, which is uniquely determined up to inner automorphism. We say that R is Calabi-Yau if $\alpha $ is inner.

Let R be a negatively graded bimodule twisted n-CY algebra of a-invariant a with Nakayama automorphism $\alpha $ . We moreover assume that each $R_i$ is finite-dimensional over k.

Let us first note a basic fact on the derived categories of R.

Proposition 4.2. $\mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R) \subset \operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$ has a relative Serre functor $(-)_\alpha (a)[n]$ .

Proof. This follows by adapting [Reference KellerKe4, Lemma 4.1]. Note that all the isomorphisms appear in the proof, which therein preserves gradings when the relevant objects are graded, so the argument carries over.

It follows using this relative Serre duality that R is Artin-Schelter regular over $R_0$ of dimension n and a-invariant a in the sense of [Reference Minamoto and MoriMM, Definition 3.1]: that is, we have $\operatorname {\mathrm {gl.dim}} R=n$ , $\operatorname {\mathrm {gl.dim}} R_0<\infty $ and

$$\begin{align*}\operatorname{\mathrm{Ext}}_R^i(R_0,R)=\begin{cases}(DR_0)(-a)& (i=n) \\ 0 & (i\neq n)\end{cases} \end{align*}$$

in $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R$ and in $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R^{\mathrm {op}}$ . Note that we can get rid of the Nakayama automorphism when restricted to one-sided modules. Note also that since R is negatively graded, the condition $\operatorname {\mathrm {gl.dim}} R_0<\infty $ is automatic by the following observation.

Lemma 4.3. Let $R=\bigoplus _{i\leq 0}R_i$ be a negatively graded ring. If R is homologically smooth, then so is $R_0$ .

Proof. We can pick a bimodule projective resolution of R whose terms are concentrated in degree $\leq 0$ . Taking its degree $0$ part yields a bimodule projective resolution of $R_0$ .

Let us also note the following information on the a-invariant of homologically smooth algebras.

Lemma 4.4. Let $R=\bigoplus _{i\leq 0}R_i$ be a negatively graded, bimodule twisted n-CY algebra of a-invariant a such that each $R_i$ is finite dimensional. If $n>0$ , then we have $a>0$ .

Proof. Let $0\to Q_n\to \cdots \to Q_1\to Q_0\to R_0\to 0$ be the minimal projective resolution of $R_0$ in $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R$ . We have $Q_0=R$ , and since R is negatively graded, the remaining terms are generated in degree $<0$ : that is, $Q_l\in \operatorname {\mathrm {add}}\{R(i)\mid i>0\}$ for $l>0$ . On the other hand, we have $Q_n\in \operatorname {\mathrm {add}} R(a)$ by [Reference Minamoto and MoriMM, Proposition 3.4(3)]. This forces $a>0$ for $n>0$ .

Now we return to our original discussion. Let R be a negatively graded bimodule twisted $(d+1)$ -CY algebra of a-invariant a. We need the following reformulation of Proposition 3.3.

Proposition 4.5. $\mathcal {M}=\operatorname {\mathrm {add}}\{R(-i)[i] \mid i \in \mathbb {Z} \}$ is a silting subcategory of $\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$ .

Proof. We have seen in Proposition 3.3 that $\mathcal {M}$ has no positive self-extensions. Also, we clearly have $\operatorname {\mathrm {thick}}\mathcal {M}=\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$ .

Therefore, we obtain a relative Serre quadruple $(\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R, \mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R),(-)_\alpha (a)[d+1],\mathcal {M})$ . The first main result of this paper is that this lies on a context of Theorem 2.6. This also explains the well-known fact (2) on the Serre functor on the derived category of non-commutative projective spaces.

Theorem 4.6. Let $d\geq 0$ , and let R be a negatively graded bimodule twisted $(d+1)$ -CY algebra of a-invariant a such that each $R_i$ is finite dimensional.

1. 1. $\mathcal {M}$ is a dualising variety with left- and right-adjacent t-structures.

2. 2. ${\mathrm {qper}^{\mathbb {Z}}\!} R$ has a Serre functor $(-)_\alpha (a)[d]$ .

3. 3. The quotient functor $\pi \colon \operatorname {\mathrm {per}}^{\mathbb {Z}}\!R \to {\mathrm {qper}^{\mathbb {Z}}\!} R$ induces bijections $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R}(X,Y) \to \operatorname {\mathrm {Hom}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(X,Y)$ for each $X\in \mathcal {M}\ast \cdots $ and $Y\in \cdots \ast \mathcal {M}[d+a-1]$ . Moreover, the composition $\mathcal {M} \ast \cdots \ast \mathcal {M}[d+a-1]\subset \operatorname {\mathrm {per}}^{\mathbb {Z}}\!R\xrightarrow {\pi }{\mathrm {qper}^{\mathbb {Z}}\!} R$ is an equivalence.

4. 4. $\pi (\mathcal {M})=\operatorname {\mathrm {add}}\{R(-i)[i] \mid i \in \mathbb {Z} \} \subset {\mathrm {qper}^{\mathbb {Z}}\!} R$ is a $(d+a)$ -cluster tilting subcategory.

In the proof below, we write $\mathcal {T}=\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R$ and $\mathcal {T}^{\mathrm {fd}}=\mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R)$ .

Proof. (1) Since $\mathcal {M}=\operatorname {\mathrm {add}}\{R(-i)[i]\mid i\in \mathbb {Z} \}$ , we have $\mathcal {M} \simeq \operatorname {\mathrm {proj}}^{\mathbb {Z}}\!\Lambda $ with $\Lambda =\bigoplus _{i \in \mathbb {Z}}\operatorname {\mathrm {Hom}}_{\mathcal {T}}(R,R(-i)[i])=R_0$ , hence $\mathcal {M}$ is a dualising variety.

We next show that the silting subcategory $\mathcal {M}\subset \mathcal {T}\,{}$ has left- and right-adjacent t-structures. By Theorem 2.6(1), it suffices to show the existence of the right-adjacent t-structure with $t_{\leq 0}^\perp \subset \mathcal {T}^{\mathrm {fd}}$ . Set $t^{\leq 0}:=t_{\leq 0}$ and $t^{\geq 0}:=(t^{\leq -1})^\perp $ , where $t^{\leq n}=t^{\leq 0}[-n]$ and so on. Note that $t^{\leq 0}=\mathcal {M}[<\!0]^\perp $ and $t^{\geq 0}=(t^{\leq -1})^\perp =(t_{\leq -1})^\perp =\mathcal {M}[>\!0]^\perp $ . Then as in Lemma 3.4, we have

$$ \begin{align*} \begin{aligned} t^{\leq0}&=\{ X \in \mathcal{T} \mid H^i(X) \in \operatorname{\mathrm{Mod}}^{\leq-i}\!R \text{ for all } i \in \mathbb{Z} \}, \\ t^{\geq0}&=\{ X \in \mathcal{T} \mid H^i(X) \in \operatorname{\mathrm{Mod}}^{\geq-i}\!R \text{ for all } i \in \mathbb{Z} \}. \end{aligned} \end{align*} $$

Now the assertion that $(t^{\leq 0}, t^{\geq 0})$ is a t-structure is precisely what we showed in Theorem 3.1, and clearly $t^{\geq 0} \subset \mathcal {T}^{\mathrm {fd}}$ .

(2) This follows from Theorem 2.6(2).

(3)(4) Let $S=(-)_\alpha (a)[d+1]$ be the relative Serre functor for $\mathcal {T}^{\mathrm {fd}}\subset \mathcal {T}\,{}$ . Then $S_{d+a+1}=(-)_\alpha (a)[-a]$ preserves $\mathcal {M}$ . Therefore, we have $St_{\geq 2}=S_{d+a+1}t_{\geq -d-a+1}=t_{\geq -d-a+1}$ , hence (3) by Theorem 2.6(3), and (4) by Theorem 2.6(4).

We end this subsection with the following computation.

Lemma 4.7. The natural map $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R}(R,R(i))\to \operatorname {\mathrm {Hom}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(R,R(i))$ is an isomorphism whenever $d>0$ or $i<a$ .

Proof. Note that $R(a)[b]\in \mathcal {M}[a+b]$ for each $a,b\in \mathbb {Z}$ . Then, using Theorem 4.6(3), we have the assertion for $i\leq d+a-1$ , so it remains to consider $i\geq d+a$ . Let, more generally, $i>0$ . Then the left-hand side is clearly $0$ . By Serre duality, we have $\operatorname {\mathrm {Hom}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(R,R(i))=D\operatorname {\mathrm {Hom}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(R(i),R(a)[d])$ with $R(i)\in \mathcal {M}[1]\ast \cdots $ and $R(a)[d]\in \mathcal {M}[d+a]$ . Therefore, Theorem 4.6(3) shows that this is $D\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R}(R(i),R(a)[d])$ , which vanishes when $d>0$ or $i<a$ .

4.2 Tilting and the ath root of the AR-translation

In this subsection, we note a result due to Minamoto–Mori [Reference Minamoto and MoriMM] and give a finite-dimensional algebra A, which will play a crucial role in the sequel. Before that, let us recall the following notion.

Definition 4.8 [Reference Herschend, Iyama and OppermannHIO]

A finite-dimensional algebra $\Lambda $ is d-representation infinite if $\operatorname {\mathrm {gl.dim}} \Lambda \leq d$ and

$$\begin{align*}\nu_d^{-i}\Lambda\in\operatorname{\mathrm{mod}}\Lambda \end{align*}$$

holds for all $i\geq 0$ , where $\nu _d$ is the autoequivalence $-\!\otimes^{\mathrm{L}}_\Lambda\!D\Lambda [-d]$ on $\mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ .

Note that we allow $d=0$ and understand ‘0-representation infinite’ algebras as being semisimple.

Proposition 4.9 compare [Reference Minamoto and MoriMM, Theorem 4.12]

Let R be a negatively graded bimodule twisted $(d+1)$ -CY algebra of a-invariant a such that each $R_i$ is finite dimensional.

1. 1. $T=\bigoplus _{l=0}^{a-1}R(-l)$ is a tilting object in ${\mathrm {qper}^{\mathbb {Z}}\!} R$ .

2. 2. $A=\operatorname {\mathrm {End}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(T)$ is d-representation infinite.

Therefore, there exists a triangle equivalence ${\mathrm {qper}^{\mathbb {Z}}\!} R\simeq \mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ .

This is a compact version of [Reference Minamoto and MoriMM, Theorem 4.12] as well as a generalization of [Reference Minamoto and MoriMM, Theorem 4.14] to non-coherent algebras. We will give a proof using Theorem 4.6 in Appendix C.

We are now in the position to state the following important consequence. Suppose in what follows that $(-)_\alpha \simeq 1$ on ${\mathrm {qper}^{\mathbb {Z}}\!} R$ : for example, that R is CY. Let A be the d-representation-infinite algebra given in Proposition 4.9, and let F be the autoequivalence on $\mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ , making the diagram below commutative:

(4.1)

Corollary 4.10. We have $F^a=\nu _d$ as autoequivalences of $\mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ .

Proof. Comparing the Serre functors on ${\mathrm {qper}^{\mathbb {Z}}\!} R\simeq \mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ , the autoequivalences $(a)$ on ${\mathrm {qper}^{\mathbb {Z}}\!} R$ and $\nu _d$ on $\mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ are compatible, hence we obtain the desired result.

We can therefore regard F as an ath root of the d-AR translation $\nu _d$ and denote $F=:\nu _d^{1/a}$ and also $F^{-1}=:\nu _d^{-1/a}$ .

Let us give some easy examples of ath roots of the AR translation; see Lemma 10.1 for the CY property of polynomial rings.

Example 4.11. Let $R=k[x,y]$ with $\deg x=\deg y=-1$ , so R is $2$ -CY of a-invariant $2$ . Applying Proposition 4.9, we have a well-known equivalence $\mathcal {D}^b(\operatorname {\mathrm {qgr}} R) \simeq \mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ with A the Kronecker algebra. The AR-quiver of this category has a component that looks like

By diagram (4.1) above, this shows that $\nu _1^{-1/2}$ on $\mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ acts by ‘moving one place to the right’.

We next look at a higher root of the AR translation.

Example 4.12. Let $R=k[x,y]$ with $\deg x=-2$ and $\deg y=-3$ , so R is $2$ -CY of a-invariant $5$ . By Proposition 4.9, there is a triangle equivalence $\mathcal {D}^b(\operatorname {\mathrm {qgr}} R)\simeq \mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ , where A is the path algebra over k of the following quiver of type $\widetilde {A_4}$ :

with the vertex i corresponding to the summand $R(-i)$ . By the triangle equivalence, we see that the AR-quiver of the triangulated category $\mathcal {D}^b(\operatorname {\mathrm {qgr}} R)$ has the following connected component:

where the horizontal ends are identified. We see that $\nu _1^{-1/5}=(-1)$ acts on this component by ‘moving one place down’.

We refer to Section 10 for more general examples for polynomial rings.

The existence of a square root of the AR translation appears in [Reference Keller, Murfet and Van den BerghKMV] for generalised Kronecker quivers. We show in the following example how to recover their context.

Example 4.13. Let $m\geq 2$ , and set

$$\begin{align*}R=k\!\left\langle x_1,\ldots,x_m\right\rangle /(x_1^2+\cdots+x_m^2), \quad \deg x_i=-1. \end{align*}$$

This is a non-Noetherian Artin-Schelter regular algebra of dimension $2$ (see [Reference ZhangZ]) and is graded coherent (see [Reference Minamoto and MoriMM, Theorem 4.16]). We need the following bimodule resolution of R.

Lemma 4.14. The complex

with maps

$$ \begin{align*} \begin{aligned} \mu(1\otimes1)&=1\\ d_1((1\otimes1)_i)&=x_i\otimes1-1\otimes x_i \\ d_2(1\otimes1)&=(x_i\otimes1+1\otimes x_i)_i \end{aligned} \end{align*} $$

gives a bimodule projective resolution of R.

Proof. The multiplication map $\mu $ is clearly surjective, and since R is generated by $x_1,\ldots ,x_m$ , it is exact at $R\otimes R$ . We next prove the exactness at $R\otimes R(2)$ : that is, $d_2$ is injective. For this, we look at each graded component and show that $(R\otimes R)_{-n}\to \bigoplus _{i=1}^m(R\otimes R)_{-n-1}$ is injective for all $n\geq 0$ . Let $\sum _{p+q=n}a_{pq}\in (R\otimes R)_{-n}$ with $a_{pq}\in R_{-p}\otimes R_{-q}$ be a non-zero element and let p be the smallest index such that $a_{pq}\neq 0$ . Note that $R_{-p}\otimes R_{-q}$ is mapped into the direct sum of $(R_{-p}\otimes R_{-q-1}) \oplus (R_{-p-1}\otimes R_{-q})$ ; thus $a_{pq}$ is the component with only possible p, which has non-zero image in $\bigoplus _{i=1}^m R_{-p}\otimes R_{-q-1}$ . Now, this map $R_{-p}\otimes R_{-q}\to \bigoplus _{i=1}^mR_{-p}\otimes R_{-q-1}$ is just $f\otimes g\mapsto (f\otimes x_ig)_i$ , which is injective by [Reference MinamotoMin1, Lemma 2.2]. It follows that the $\bigoplus _{i=1}^m(R_{-p}\otimes R_{-q-1})$ -component of $d_2(a_{pq})$ is non-zero; thus $d_2$ is injective. It remains to prove the acyclicity at $\bigoplus _{i=1}^mR\otimes R(1)$ . It suffices to show that the terms have ‘correct dimensions’: that is, putting $d_n=\dim R_{-n}$ , we have $\sum _{p+q=n}d_pd_q-m\sum _{p+q=n+1}d_pd_q+\sum _{p+q=n+2}d_pd_q-d_{n+2}=0$ . This is easily seen by using $d_{n+2}=md_{n+1}-d_n$ (see [Reference MinamotoMin1, Lemma 2.2]).

Applying $\operatorname {\mathrm {Hom}}_{R^e}(-,R^e)$ to the above complex, we deduce that R is graded bimodule twisted $2$ -CY of a-invariant $2$ with Nakayama automorphism $\sigma \colon x_i\mapsto -x_i$ .

By Proposition 4.9, we have a derived equivalence $\mathcal {D}^b(\operatorname {\mathrm {qgr}} R)\simeq \mathcal {D}^b(\operatorname {\mathrm {mod}} A)$ with $A=\begin {pmatrix}R_0& 0\\R_{-1}&R_0\end {pmatrix}$ , which is the path algebra of the m-Kronecker quiver $Q_m=(\circ \overset {m}{\longrightarrow }\circ )$ . Now the twist automorphism $(-)_\sigma $ is isomorphic to the identity functor on $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R$ , so we have an autoequivalence $\nu _1^{-1/2}$ on $\mathcal {D}^b(\operatorname {\mathrm {mod}} kQ_m)$ . The AR quiver of the derived category has a connected component

We see that $\nu _1^{-1/2}$ acts by ‘moving one place to the right’; compare Example 4.11.

5 CY algebras as DG algebras

We will consider a graded algebra R as a DG algebra with the same underlying graded ring and the vanishing differential. We write $R^{\mathrm {dg}}$ when considering R as a DG algebra.

We first collect some sign conventions that are heavily used in this section. Throughout this section, we denote by $|x|$ the degree of a homogeneous element x in a graded vector space.

We say that a DG algebra $\Lambda $ is twisted bimodule n-CY if it is homologically smooth and there exists a DG automorphism $\sigma $ of $\Lambda $ such that we have an isomorphism $\operatorname {{\mathrm {R}Hom }}_{\Lambda ^e}(\Lambda ,\Lambda ^e)[n] \simeq {}_1\Lambda _\sigma $ in $\mathcal {D}(\Lambda ^e)$ . The aim of this section is to note the following observation. Note that the term ‘CY algebra’ means precisely as in Definition 4.1, and no additional conditions are imposed.

Let us introduce some notations. For a DG algebra $\Lambda $ , we denote by $\mathrm {C}(\Lambda )=Z^0\mathcal {C}_{\mathrm {dg}}(\Lambda )$ the (abelian) category of DG $\Lambda $ -modules. In what follows, we denote by S an arbitrary graded algebra with the corresponding formal DG algebra $S^{\mathrm {dg}}$ . Let X be a graded $(S,S)$ -bimodule. We can view it as a DG $(S^{\mathrm {dg}},S^{\mathrm {dg}})$ -bimodule $X^{\mathrm {dg}}$ with trivial differentials, hence as an $(S^{\mathrm {dg}})^e$ -module, which gives a fully faithful functor

$$\begin{align*}(-)^{\mathrm{dg}} \colon \operatorname{\mathrm{Mod}}^{\mathbb{Z}}\!S^e \longrightarrow \mathrm{C}((S^{\mathrm{dg}})^e), \end{align*}$$

where $S^e=S^{\mathrm {op}}\otimes S$ is the enveloping algebra (as a graded algebra). By our convention, the action of $a\otimes b\in S^e$ on an element $x\otimes y$ in the free $S^e$ -module $S^e$ is given by $(x\otimes y)\cdot (a\otimes b)=ax\otimes yb$ ; thus free modules have outer bimodule structures.

We note the following sign-conventional lemmas. The proofs are left to the reader.

Lemma 5.3. Let $F=S^e(l)$ be a free $S^e$ -module. Then $F^{\mathrm {dg}}$ is isomorphic to the free DG $(S^{\mathrm {dg}})^e$ -module $(S^{\mathrm {dg}})^e[l]$ . The isomorphism is given by

$$\begin{align*}F^{\mathrm{dg}} \longrightarrow (S^{\mathrm{dg}})^e[l], \quad x\otimes y \mapsto (-1)^{l|x|} x\otimes y. \end{align*}$$

We immediately obtain the following relationship for the homological smoothness of graded algebras and the corresponding formal DG algebras.

Let us now recall the notion of a total module of a complex of DG modules. We have two variations: total sum $\operatorname {\mathrm {Tot}}$ and total product $\operatorname {\mathrm {\widehat {Tot }}}$ . Let $X=(\cdots \to X^{i-1}\xrightarrow {\delta _X^{i-1}} X^i\xrightarrow {\delta _X^i} X^{i+1}\to \cdots )$ be a complex of DG $\Lambda $ -modules for a DG algebra $\Lambda $ ; thus each $X^i$ is a DG $\Lambda $ -module, each $\delta _X^i$ is a morphism of DG $\Lambda $ -modules and $\delta _X^{i+1}\circ \delta _X^i=0$ . Then define

$$\begin{align*}\operatorname{\mathrm{Tot}} X=\bigoplus_{i\in\mathbb{Z}}X^i[-i], \qquad \operatorname{\mathrm{\widehat{Tot}}} X=\prod_{i\in\mathbb{Z}}X^i[-i] \end{align*}$$

as graded $\Lambda $ -modules and with differentials $\delta _X+\sum _id_{X^i[-i]}$ . Here, $\delta _X$ is the differential of the complex X and $d_{X^i[-i]}$ is the differential of the DG module $X^i[-i]$ . Then $\operatorname {\mathrm {Tot}} X$ and $\operatorname {\mathrm {\widehat {Tot }}} X$ are DG $\Lambda $ -modules. When a complex X of DG $\Lambda $ -modules is bounded, we have $\operatorname {\mathrm {Tot}} X=\operatorname {\mathrm {\widehat {Tot }}} X$ , which is acyclic whenever X is acyclic.

For a complex $X=(\cdots \to X^{i-1}\to X^i\to X^{i+1}\to \cdots )$ of DG $\Lambda $ -modules and a DG $\Lambda $ -module Y, we denote by $\operatorname {\mathrm {\mathcal {H}om}}_\Lambda (X,Y)$ the complex

of DG k-modules with $\operatorname {\mathrm {\mathcal {H}om}}_\Lambda (X^i,Y)$ at degree $-i$ , where $\operatorname {\mathrm {\mathcal {H}om}}_\Lambda (-,-)$ is the $\operatorname {\mathrm {Hom}}$ -complex between DG $\Lambda $ -modules defined as follows: for DG $\Lambda $ -modules M and N, the degree n part of the complex $\operatorname {\mathrm {\mathcal {H}om}}_\Lambda (M,N)$ is the space $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!\Lambda }(M,N[n])$ of homogeneous map of graded $\Lambda $ -modules of degree n, with differential $df=d_Nf-(-1)^nfd_M$ for $f\in \operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!\Lambda }(M,N[n])$ .

The following is quite useful for computations.

An important step for the proof of Theorem 5.2 is the following observation. We denote by $\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S^e$ the category of finitely presented graded $S^e$ -modules. Note that we are not assuming that $S^e$ is graded coherent, and $\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S^e$ is just an additive category.

Proof. We first prove that two functors are naturally isomorphic on the category of finitely generated free modules. We will then extend the natural isomorphism to $\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S^e$ .

Let us start by defining an isomorphism $\varphi _P\colon F(P) \to G(P)$ at the free module $P=S^e(l)$ . Clearly, we have $F(P)=S\otimes S(-l)$ , with the action of S given by $b\cdot (x \otimes y)\cdot a=(-1)^{|b|}xa\otimes by$ . On the other hand, using Lemma 5.3, we see $G(P)=(S\otimes S)[-l]$ , with the S-action $b\cdot (x \otimes y)\cdot a=(-1)^{|a|(|b|+|y|)+|b|(l+|x|)}xa\otimes by$ . Now we define an isomorphism $F(P)=S\otimes S(-l) \to (S\otimes S)[-l]=G(P)$ by the formula

$$\begin{align*}\varphi_P \colon x \otimes y \mapsto (-1)^{(|x|+l+1)|y|}x \otimes y. \end{align*}$$

We can readily check that this is $(S,S)$ -bilinear.

We next show that this isomorphism is natural. Let $P \to Q$ be a morphism of finitely generated free modules. We may assume that this is of the form $S^e(l) \to S^e(m)$ with $1\otimes 1 \mapsto \sum _i u_i\otimes v_i$ . Under the functor $(-)^{\mathrm {dg}}$ and the isomorphism in Lemma 5.3, it becomes an $(S^{\mathrm {dg}})^e$ -linear map $(S^{\mathrm {dg}})^e[l]\to (S^{\mathrm {dg}})^e[m]$ sending $1\otimes 1$ to $\sum _i(-1)^{m|u_i|}u_i\otimes v_i$ . Note that we have $|u_i|+|v_i|-m=-l$ . Our task is to show that in the diagram below, the middle square is commutative in $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!S^e$ :

By our sign conventions, the map f is just a left $S^e$ -linear map with $1\otimes 1\mapsto \sum _i u_i \otimes v_i$ , while g is given by $1\otimes 1 \mapsto (-1)^{m(l+1)}\sum _i(-1)^{m|u_i|}u_i\otimes v_i$ . Using these, we can verify the desired commutativity.

It is now routine to extend this natural isomorphism to $\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S^e$ . To define an isomorphism $\varphi _M\colon F(M)\to G(M)$ at $M\in \operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S^e$ , take a presentation $P_1\to P_0\to M\to 0$ by finitely generated free modules. Since F and G are restrictions of left exact functors on $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!S^e$ , we can define $\varphi _M$ to be the unique map, making the following diagram of exact sequences (in $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!S^e$ ) commutative. Note that the right square is commutative by our previous claim:

It is easy to verify that $\varphi _M$ is independent of the presentation, and clearly $\varphi _M$ is an isomorphism.

Now we show that this isomorphism is natural on $\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S$ . Let $f\colon M\to N$ be a morphism in $\operatorname {\mathrm {mod}}^{\mathbb {Z}}\!S$ . Taking presentations $P_1\to P_0\to M\to 0$ and $Q_1\to Q_0\to N\to 0$ by finitely generated free modules, we can lift f to a morphism of presentations. Applying F and G to these, we obtain a commutative diagram

from which we deduce the desired naturality on f.

We are now ready to prove the main theorem of this section.

Proof of Theorem 5.2

We know $R^{\mathrm {dg}}$ is homologically smooth by Proposition 5.5. We compute the inverse dualising complex of $R^{\mathrm {dg}}$ . Let $P=(P_n\to \cdots \to P_1\xrightarrow {} P_0)$ be a projective resolution of R in $\mathcal {C}^b(\operatorname {\mathrm {proj}}^{\mathbb {Z}}\!R^e)$ . Then the cohomology of $P^\vee =\operatorname {\mathrm {Hom}}_{R^e}(P,R^e)$ is concentrated in degree n, where it is ${}_\alpha R(-a)$ . Considering P as a complex $P^{\mathrm {dg}}$ of DG bimodules as above, the total sum of $P^{\mathrm {dg}}$ gives an $(R^{\mathrm {dg}})^e$ -cofibrant resolution of the DG $(R^{\mathrm {dg}})^e$ -module $R^{\mathrm {dg}}$ . Indeed, $\operatorname {\mathrm {Tot}} P^{\mathrm {dg}}\to R^{\mathrm {dg}}$ is a quasi-isomorphism since $\operatorname {\mathrm {Tot}}$ preserves acyclicity, and since each $P_i^{\mathrm {dg}}$ is cofibrant by Lemma 5.3 and $\operatorname {\mathrm {Tot}} P^{\mathrm {dg}}$ is a successive extension of $P_i^{\mathrm {dg}}$ ’s that is split as graded $(R^{\mathrm {dg}})^e$ -modules, it follows that $\operatorname {\mathrm {Tot}} P^{\mathrm {dg}}$ is cofibrant (see [Reference KellerKe1, Section 3.1]). Then $\operatorname {{\mathrm {R}Hom }}_{(R^{\mathrm {dg}})^e}(R^{\mathrm {dg}},(R^{\mathrm {dg}})^e)=\operatorname {\mathrm {\mathcal {H}om}}_{(R^{\mathrm {dg}})^e}(\operatorname {\mathrm {Tot}} P^{\mathrm {dg}},(R^{\mathrm {dg}})^e)$ , which is isomorphic by Lemma 5.6 to the total product (which equals the total sum by boundedness) of the complex

$$ \begin{align*} Q =( \operatorname{\mathrm{\mathcal{H}om}}_{(R^{\mathrm{dg}})^e}(P_0^{\mathrm{dg}},(R^{\mathrm{dg}})^e)\to\operatorname{\mathrm{\mathcal{H}om}}_{(R^{\mathrm{dg}})^e}(P_1^{\mathrm{dg}},(R^{\mathrm{dg}})^e)\to\cdots\to\operatorname{\mathrm{\mathcal{H}om}}_{(R^{\mathrm{dg}})^e}(P_n^{\mathrm{dg}},(R^{\mathrm{dg}})^e)). \end{align*} $$

Now, by Lemma 5.7, this complex is isomorphic to ${}_\sigma (P^\vee )$ as complexes of graded $(R,R)$ -bimodules; thus it has cohomology only at degree n, where it is isomorphic to ${}_{\sigma \alpha } R(-a)$ . Then we have a quasi-isomorphism $Q\rightarrow ({}_{\sigma \alpha } R(-a))^{\mathrm {dg}}[-n]$ of complexes of DG $(R^{\mathrm {dg}})^e$ -modules, so $\operatorname {\mathrm {Tot}} Q$ is quasi-isomorphic to $({}_{\sigma \alpha } R(-a))^{\mathrm {dg}}[-n]$ . We conclude by Lemma 5.4 that it is ${}_\alpha R[-n-a]$ if a is odd, ${}_{\sigma \alpha } R[-n-a]$ if a is even.

We explicitly demonstrate how $R^{\mathrm {dg}}$ fails to be CY.

Example 5.9. Let $R=k[x,y]$ with $\deg x=\deg y=-1$ . Then the graded ring R is bimodule $2$ -CY of a-invariant $2$ and has the Koszul resolution, which we depict in the following way:

where the values on the arrows show the image of $1\otimes 1$ . Now consider this resolution as a complex of DG modules over $S:=(R^{\mathrm {dg}})^e$ . Under the isomorphism in Lemma 5.3, it becomes

Applying $\operatorname {\mathrm {\mathcal {H}om}}_S(-,S)$ , we get the complex of left DG S-modules

whose total module is $\operatorname {{\mathrm {R}Hom }}_S(R,S)$ by Lemma 5.6. We therefore see that $\operatorname {{\mathrm {R}Hom }}_S(R,S)[4]\simeq {}_1R_\sigma $ .

The appearance of the twist automorphism $\sigma $ suggests that we should twist the multiplication of the CY algebra R in order for the DG algebra $R^{\mathrm {dg}}$ to be CY. Let us give an instance where R is twisted CY and $R^{\mathrm {dg}}$ is CY.

Example 5.10. Let $m\geq 2$ and

$$\begin{align*}R=k\!\left\langle x_1,\ldots,x_m\right\rangle /(x_1^2+\cdots+x_m^2). \quad \deg x_i=-1, \end{align*}$$

If $m=2$ , this is a skew polynomial ring with two variables; compare Example 5.9 above. Recall from Example 4.13 that this is twisted bimodule $2$ -CY algebra of a-invariant $2$ with Nakayama automorphism $\sigma \colon x_i\mapsto -x_i$ . Therefore, by Theorem 5.2(2), we deduce that $R^{\mathrm {dg}}$ is (non-twisted) $4$ -CY. Let us explicitly demonstrate this.

Recall that the bimodule projective resolution of R is given by the complex

with maps

$$ \begin{align*} \begin{aligned} d_1((1\otimes1)_i)&=x_i\otimes1-1\otimes x_i \\ d_2(1\otimes1)&=\sum_{i=1}^m(x_i\otimes1+1\otimes x_i). \end{aligned} \end{align*} $$

We follow the computation in Example 5.9 above. Set $S=(R^{\mathrm {dg}})^e$ . Applying the functor $(-)^{\mathrm {dg}}$ and the isomorphism in Lemma 5.3, the above complex becomes

where the maps are right S-linear morphisms such that

$$ \begin{align*} \begin{aligned} d_1((1\otimes1)_i)&=x_i\otimes1-1\otimes x_i \\ d_2(1\otimes1)&=\sum_{i=1}^m(-x_i\otimes1+1\otimes x_i). \end{aligned} \end{align*} $$

Applying $\operatorname {\mathrm {\mathcal {H}om}}_S(-,S)$ , we get an isomorphic complex up to shifts; thus we see that $\operatorname {{\mathrm {R}Hom }}_S(R,S)[4]\simeq R$ in $\mathcal {D}(S)$ .

6 Cluster categories, derived orbit categories and singularity categories

Let R be a CY algebra. We state the main result of this paper, which describes the cluster category of $R^{\mathrm {dg}}$ as an orbit category and a singularity category.

6.1 Cluster categories and orbit categories

Let R be a negatively graded twisted bimodule $(d+1)$ -CY algebra of a-invariant a such that each $R_i$ is finite dimensional. In this subsection, we compare the derived category of R considered as a graded ring and that of R considered as a DG algebra $R^{\mathrm {dg}}$ with vanishing differentials. By Theorem 5.2, we know that $R^{\mathrm {dg}}$ is twisted bimodule $(d+a+1)$ -CY.

Recall the notion of total module from the previous section. Consider the DG functor

$$\begin{align*}\operatorname{\mathrm{Tot}} \colon \mathcal{C}^b_{\mathrm{dg}}(\operatorname{\mathrm{Mod}}^{\mathbb{Z}}\!R) \to \mathcal{C}_{\mathrm{dg}}(R^{\mathrm{dg}}) \end{align*}$$

from the DG category of complexes over $\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R$ to the DG category of DG $R^{\mathrm {dg}}$ -modules. This is given on morphisms by taking a map $f\colon X^i\to Y^j$ of graded R-modules considered as an element of $\operatorname {\mathrm {\mathcal {H}om}}_{\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R}(X,Y)$ to itself (without any signs) viewed as an element of $\operatorname {\mathrm {\mathcal {H}om}}_{R^{\mathrm {dg}}}(\operatorname {\mathrm {Tot}} X, \operatorname {\mathrm {Tot}} Y)$ .

Taking the $0$ th cohomology, it induces a triangle functor $\mathcal {K}^b(\operatorname {\mathrm {Mod}}^{\mathbb {Z}}\!R) \to \mathcal {K}(R^{\mathrm {dg}})$ , which clearly takes acyclic complexes to acyclic DG modules. We therefore obtain a triangle functor

$$\begin{align*}\operatorname{\mathrm{Tot}} \colon \operatorname{\mathrm{per}}^{\mathbb{Z}}\!R \to \operatorname{\mathrm{per}} R^{\mathrm{dg}} .\end{align*}$$

Note that this restricts to $\mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R) \to \mathcal {D}^b(R^{\mathrm {dg}})$ ; thus it again induces a triangle functor,

$$\begin{align*}\operatorname{\mathrm{Tot}} \colon {\mathrm{qper}^{\mathbb{Z}}\!} R \to \mathcal{C}(R^{\mathrm{dg}}). \end{align*}$$

Now we have a natural isomorphism $\operatorname {\mathrm {Tot}}\circ (-1)[1]\simeq \operatorname {\mathrm {Tot}}$ , hence a functor

$$\begin{align*}\operatorname{\mathrm{Tot}}\colon {\mathrm{qper}^{\mathbb{Z}}\!} R/(-1)[1]\to\mathcal{C}(R^{\mathrm{dg}}). \end{align*}$$

The following result gives a natural and more concrete description of the cluster category of $R^{\mathrm {dg}}$ . Similar types of results for derived or singularity categories were recently obtained in [Reference Kalck and YangKY, Reference BrightbillBri].

Theorem 6.1. Let R be a negatively graded twisted bimodule $(d+1)$ -CY algebra of a-invariant a with $d\geq 0$ . The functor $\operatorname {\mathrm {Tot}} \colon {\mathrm {qper}^{\mathbb {Z}}\!} R \to \mathcal {C}(R^{\mathrm {dg}})$ induces a fully faithful functor

whose image generates $\mathcal {C}(R^{\mathrm {dg}})$ as a triangulated subcategory.

Proof. Note that the cluster tilting subcategory $\mathcal {M}=\operatorname {\mathrm {add}}\{R(-i)[i] \mid i \in \mathbb {Z} \} \subset {\mathrm {qper}^{\mathbb {Z}}\!} R$ given in Theorem 4.6 is mapped to a cluster tilting object $R \in \mathcal {C}(R^{\mathrm {dg}})$ , and the functor $\operatorname {\mathrm {Tot}}$ induces an equivalence $\mathcal {M}/(-1)[1] \xrightarrow {\simeq }\operatorname {\mathrm {add}} R$ . Indeed, $\mathcal {M}/(-1)[1]$ has an additive generator R, and we have an isomorphism $\operatorname {\mathrm {End}}_{\mathcal {M}/(-1)[1]}(R)=\bigoplus _{i\in \mathbb {Z}}\operatorname {\mathrm {Hom}}_{\mathcal {M}}(R,R(-i)[i]))\xrightarrow {\simeq }\operatorname {\mathrm {End}}_{\mathcal {C}(R^{\mathrm {dg}})}(R)$ of graded rings since the left-hand side vanishes for $i\neq 0$ , and thus both sides are $R_0$ . Our assertion is now a consequence of the covering version of the ‘cluster-Beilinson criterion’ (compare [Reference Keller and ReitenKR2, Lemma 4.5]), which we sketch below.

To save space, we will denote $\mathcal {D}:={\mathrm {qper}^{\mathbb {Z}}\!} R$ and $\mathcal {C}:=\mathcal {C}(R^{\mathrm {dg}})$ and sometimes ${}_{\mathcal {A}}(-,-)$ for the morphism spaces in a category $\mathcal {A}$ . We have to show that $\operatorname {\mathrm {Tot}}$ induces isomorphisms

(6.1)

for all $X,Y\in \mathcal {D}$ . We first prove that the above map is an isomorphism for all $X\in \mathcal {M}$ and $Y\in \mathcal {M}\ast \cdots \ast \mathcal {M}[j]$ with $0\leq j\leq d+a-1$ by induction on j. We have already seen this above for $j=0$ . Let $0\leq j<d+a-1$ and $Y\in \mathcal {M}\ast \cdots \ast \mathcal {M}[j]\ast \mathcal {M}[j+1]$ . Pick a triangle $Y'\to M\to Y\to Y'[1]$ in $\mathcal {D}$ with $M\in \mathcal {M}$ and $Y'\in \mathcal {M}\ast \cdots \ast \mathcal {M}[j]$ . This gives a commutative diagram of exact sequence

in which the left two vertical maps are isomorphisms by induction hypothesis and the right-end terms are $0$ since $\mathcal {M}\subset \mathcal {D}$ and $R\in \mathcal {C}$ are $(d+a)$ -cluster tilting. Therefore, the remaining vertical map is also an isomorphism. Since $\mathcal {D}=\mathcal {M}\ast \cdots \mathcal {M}[d+a-1]$ , this induction shows that equation (6.1) is an isomorphism for all $X\in \mathcal {M}$ and $Y\in \mathcal {D}$ .

We next prove that equation (6.1) is an isomorphism for all $X,Y\in \mathcal {D}$ . We can perform a similar induction on j to show that this is indeed the case for $X\in \mathcal {M}\ast \cdots \ast \mathcal {M}[j]$ , which completes the proof.

Let A be the d-representation infinite algebra given in Proposition 4.9 as the endomorphism ring of a tilting object in ${\mathrm {qper}^{\mathbb {Z}}\!} R$ . Explicitly, we have

$$\begin{align*}A=\left( \begin{array}{cccc} R_0& 0 & \cdots& 0 \\ R_{-1} & R_0&\cdots& 0 \\ \vdots& \vdots&\ddots&\vdots \\ R_{-(a-1)}&R_{-(a-2)}&\cdots&R_0 \end{array} \right). \end{align*}$$

Assuming that the Nakayama automorphism $\alpha $ of R in Theorem 6.1 satisfies $(-)_\alpha \simeq 1$ on ${\mathrm {qper}^{\mathbb {Z}}\!} R$ , we deduce the following by means of the diagram in equation (4.1).

Corollary 6.2. Let R be as in Theorem 6.1, and suppose that $(-)_\alpha \simeq 1$ on ${\mathrm {qper}^{\mathbb {Z}}\!} R$ . Then there exists a fully faithful functor

whose image generates $\mathcal {C}(R^{\mathrm {dg}})$ as a triangulated subcategory.

6.2 Cluster categories and singularity categories

We present another description of the cluster category $\mathcal {C}(R^{\mathrm {dg}})$ . Set

$$\begin{align*}U=\left( \begin{array}{cccc} R_{-1} & R_0&\cdots& 0 \\ \vdots& \vdots&\ddots&\vdots \\ R_{-(a-1)}&R_{-(a-2)}&\cdots&R_0 \\ R_{-a}&R_{-(a-1)}&\cdots&R_{-1} \end{array} \right), \end{align*}$$

which is an $(A,A)$ -bimodule, and let $B=A\oplus U$ be the trivial extension algebra.

Recall that a cotilting bimodule over a ring $\Lambda $ is a $(\Lambda ,\Lambda )$ -bimodule Y satisfying the following:

• ○ $\operatorname {\mathrm {inj.dim}}_\Lambda Y<\infty $ and $\operatorname {\mathrm {inj.dim}}_{\Lambda ^{\mathrm {op}}}Y<\infty $ .

• ○ $\operatorname {\mathrm {Ext}}^i_\Lambda (Y,Y)=0$ and $\operatorname {\mathrm {Ext}}^i_{\Lambda ^{\mathrm {op}}}(Y,Y)=0$ for $i>0$ .

• ○ The natural maps $\Lambda \to \operatorname {\mathrm {End}}_\Lambda (Y)$ and $\Lambda ^{\mathrm {op}}\to \operatorname {\mathrm {End}}_{\Lambda ^{\mathrm {op}}}(Y)$ are isomorphisms.

We have the following basic information on U and B.

Proposition 6.3.

1. 1. U is a cotilting bimodule over A.

2. 2. B is a d-Iwanaga-Gorenstein algebra.

Proof. (1) Since A has finite global dimension, we clearly have $\operatorname {\mathrm {inj.dim}}_AU<\infty $ . Note also that we have $A=\operatorname {\mathrm {End}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(T)$ and $U=\operatorname {{\mathrm {R}Hom }}_{{\mathrm {qper}^{\mathbb {Z}}\!} R}(T,T(-1))$ for a tilting object T given in Proposition 4.9. This gives $\operatorname {\mathrm {Ext}}^i_A(U,U)=0$ for $i>0$ and an isomorphism $A\to \operatorname {\mathrm {End}}_A(U)$ . It remains to verify the conditions on the opposite side. Let $(-)^\ast =\operatorname {{\mathrm {R}Hom }}_{R}(-,R)$ be the duality $\operatorname {\mathrm {per}}^{\mathbb {Z}}\!R\leftrightarrow \operatorname {\mathrm {per}}^{\mathbb {Z}}\!R^{\mathrm {op}}$ . By the Artin-Schelter regularity of R (see the remark after Proposition 4.2), it restricts to $\mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R)\leftrightarrow \mathcal {D}^b(\operatorname {\mathrm {fl}}^{\mathbb {Z}}\!R^{\mathrm {op}})$ and hence induces a duality ${\mathrm {qper}^{\mathbb {Z}}\!} R\leftrightarrow {\mathrm {qper}^{\mathbb {Z}}\!} R^{\mathrm {op}}$ , which we still denote by $(-)^\ast $ . Then $T^\ast \in {\mathrm {qper}^{\mathbb {Z}}\!} R^{\mathrm {op}}$ is a tilting object with $\operatorname {\mathrm {End}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R^{\mathrm {op}}}(T^\ast )=A^{\mathrm {op}}$ , and we have $U\simeq \operatorname {\mathrm {Hom}}_{{\mathrm {qper}^{\mathbb {Z}}\!} R^{\mathrm {op}}}(T^\ast ,T^\ast (-1))$ as $(A,A)$ -bimodules. This gives the desired assertions for the opposite side.

(2) This follows from (1) and [Reference Minamoto and YamauraMY, Theorem 5.3].

Now we state the second main result of this paper.

Theorem 6.4. Let R be as in Corollary 6.2. There exists a triangle equivalence

$$\begin{align*}\mathcal{C}(R^{\mathrm{dg}}) \simeq \mathcal{D}_{\mathrm{sg}}(B). \end{align*}$$

In particular, $\mathcal {D}_{\mathrm {sg}}(B)$ is a twisted $(d+a)$ -CY category with a $(d+a)$ -cluster tilting object.

We postpone the proof of this result to Section 8 since it requires a general result on DG orbit categories, which we give in the following section.

6.3 Examples

Before going on, let us give some demonstrations of our results. See Sections 9, 10 and 11 for systematic examples.

Example 6.5. Let us start with an almost trivial example. Let

$$\begin{align*}R=k[x], \quad \deg x=-1. \end{align*}$$

This is bimodule $1$ -CY of a-invariant $1$ ; thus $R^{\mathrm {dg}}$ is bimodule $2$ -CY by Theorem 5.2. We have $A=k$ (which is understood to be ‘ $0$ -representation infinite’) and $U=k$ ; thus $B=A\oplus U=k[t]/(t^2)$ . By Corollary 6.2 and Theorem 6.4, we have equivalences of triangulated categories

$$\begin{align*}\mathcal{D}^b(\operatorname{\mathrm{mod}} k)/[1]\simeq\mathcal{C}(R^{\mathrm{dg}})\simeq \mathcal{D}_{\mathrm{sg}}(k[t]/(t^2)), \end{align*}$$

which are triangulated categories with $1$ point. See Example A.5 for a generalization, where the case $\deg x=-n$ for arbitrary $n\geq 1$ is discussed.

Example 6.6. This is a continuation of Examples 4.13 and 5.10. Let $m\geq 2$ , and set

$$\begin{align*}R=k\!\left\langle x_1,\ldots,x_m\right\rangle /(x_1^2+\cdots+x_m^2), \quad \deg x_i=-1, \end{align*}$$

which is twisted $2$ -CY of a-invariant $2$ (Example 4.13), and $R^{\mathrm {dg}}$ is $4$ -CY (Example 5.10). The $1$ -representation infinite algebra A is the path algebra of the m-Kronecker quiver $Q_m\colon \circ \overset {m}{\longrightarrow }\circ $ , and the autoequivalence $\nu _1$ of $\mathcal {D}^b(\operatorname {\mathrm {mod}} kQ_m)$ has a square root (Example 4.13). Also, it is not difficult to see that the $1$ -Iwanaga-Gorenstein algebra B is presented by the following quiver with relations:

By Corollary 6.2 and Theorem 6.4, there exist triangle equivalences

$$\begin{align*}\mathcal{D}^b(\operatorname{\mathrm{mod}} kQ_m)/\nu_1^{-1/2}[1] \simeq \mathcal{D}_{\mathrm{sg}}(B)\simeq \mathcal{C}(R^{\mathrm{dg}}). \end{align*}$$

Remark 6.7. In [Reference Keller, Murfet and Van den BerghKMV, Theorem 1.4], Keller–Murfet–Van den Bergh proved that any algebraic $3$ -CY triangulated category $\mathcal {T}\,{}$ with a $3$ -cluster tilting object T such that $\operatorname {\mathrm {End}}_{\mathcal {T}}(T)=k$ and $\operatorname {\mathrm {Hom}}_{\mathcal {T}}(T,T[-1])=k^m$ is triangle equivalent to $\mathcal {D}^b(\operatorname {\mathrm {mod}} kQ_m)/\tau ^{-1/2}[1]$ , where $\tau ^{-1/2}$ is the square root of the AR translation defined in [Reference Keller, Murfet and Van den BerghKMV] using reflection functors.

Now, since $\mathcal {C}=\mathcal {C}(R^{\mathrm {dg}})$ for R in Example 6.6 above is a $3$ -CY category with a $3$ -cluster tilting object R satisfying $\operatorname {\mathrm {End}}_{\mathcal {C}}(R)=k$ and $\operatorname {\mathrm {Hom}}_{\mathcal {C}}(R,R[-1])=k^m$ , the above equivalent triangulated categories are precisely the $3$ -CY category in [Reference Keller, Murfet and Van den BerghKMV, Theorem 1.4]. Our result shows that the $3$ -CY category in [Reference Keller, Murfet and Van den BerghKMV] is also the singularity category of B and can be realised as the cluster category of the DG algebra $R^{\mathrm {dg}}$ . We refer to Example A.6 for a generalization, which covers the situation in [Reference Keller, Murfet and Van den BerghKMV, Remark 3.4.5].

7 Quasi-equivalence of DG orbit categories

Let $\mathcal {T}\,{}$ be a triangulated category and $F \colon \mathcal {T} \to \mathcal {T}\,{}$ an autoequivalence. In order to discuss when the orbit category $\mathcal {T}/F$ is triangulated, a triangulated hull of this orbit category was constructed by Keller [Reference KellerKe2]. The idea was to take an orbit at the level of enhancement of $\mathcal {T}\,{}$ .

Let $\mathcal {A}$ be a pretriangulated DG category and F a DG endofunctor on $\mathcal {A}$ inducing an equivalence on $H^0\mathcal {A}$ . Then the DG orbit category of $\mathcal {A}$ with respect to F, which we denote by $\mathcal {B}=\mathcal {A}/F$ , is the DG category with the same objects as $\mathcal {A}$ and with the morphism complex

$$\begin{align*}\mathcal{B}(L,M)=\operatorname*{\mathrm{colim}} \left( \bigoplus_{n\geq0}\mathcal{A}(F^nL,M) \xrightarrow{F} \bigoplus_{n\geq0}\mathcal{A}(F^nL,FM) \xrightarrow{F} \bigoplus_{n\geq0}\mathcal{A}(F^nL,F^2M) \xrightarrow{F} \cdots \right) \end{align*}$$

for each $L, M \in \mathcal {B}$ . It follows that we have $H^0\mathcal {A}/H^0F=H^0\mathcal {B}$ ; hence $\operatorname {\mathrm {tria}}\mathcal {B}$ , the full triangulated subcategory of $\mathcal {D}(\mathcal {B})$ generated by the representable $\mathcal {B}$ -modules, is a triangulated hull of $H^0\mathcal {A}/H^0F$ in the sense that there is a fully faithful functor $H^0\mathcal {A}/H^0F\hookrightarrow \operatorname {\mathrm {tria}}\mathcal {B}$ whose image generates $\operatorname {\mathrm {tria}}\mathcal {B}$ as a triangulated subcategory.

Observe that the above embedding is not dense in general, and thus it is not clear a naive expectation for orbit categories carries over to triangulated hulls. We give an answer to one such problem.

Let us briefly recall some relevant notions; see [Reference KellerKe1, Section 7] for details. Let $\mathcal {B}$ and $\mathcal {C}$ be DG categories. A quasi-functor $\mathcal {C} \to \mathcal {B}$ is a $(\mathcal {C},\mathcal {B})$ -bimodule X whose value we denote by $X(B,C)$ for $B \in \mathcal {B}$ and $C \in \mathcal {C}$ , such that the DG $\mathcal {B}$ -module $X(-,C)$ is isomorphic in $\mathcal {D}(\mathcal {B})$ to a representable DG $\mathcal {B}$ -module for each $C \in \mathcal {C}$ . A quasi-functor $X \colon \mathcal {C} \to \mathcal {B}$ is a quasi-equivalence if $-\!\otimes^{\mathrm{L}}_{\mathcal {C}}\!X \colon \mathcal {D}(\mathcal {C}) \to \mathcal {D}(\mathcal {B})$ is an equivalence and restricts to an equivalence $H^0\mathcal {C}\xrightarrow {\simeq } H^0\mathcal {B}$ . In this case, we deduce that there is a triangle equivalence $-\!\otimes^{\mathrm{L}}_{\mathcal {C}}\!X\colon \operatorname {\mathrm {tria}}\mathcal {C} \xrightarrow {\simeq }\operatorname {\mathrm {tria}}\mathcal {B}$ . We say two DG categories $\mathcal {B}$ and $\mathcal {C}$ are quasi-equivalent if there exists a quasi-equivalence $\mathcal {C}\to \mathcal {B}$ .

In view of relating the categories $\mathcal {B}$ and $\mathcal {C}$ , consider the following direct system of complexes indexed by $\mathbb {N}\times \mathbb {N}$ :

(7.1)

where the vertical transition maps are induced by G and the horizontal ones by $G^{p+1}F^{1+n}L \to G^pF^nL$ .

We first fix $q\geq 0$ and consider the colimit

$$\begin{align*}U_q(L,M):=\operatorname*{\mathrm{colim}}_{p\geq0}\bigoplus_{n\geq0}\mathcal{A}(G^{p+q}F^nL,G^qM) \end{align*}$$

of the $(q+1)$ -st row. We can regard $U_q(-,M)$ as a DG $\mathcal {B}$ -module for each $M \in \mathcal {A}$ as follows: Let $b \in \mathcal {B}(K,L)$ be a morphism represented by $F^mK \to F^rL$ and $u\in U_q(L,M)$ an element represented by $G^{p+q}F^nL \to G^qM$ . Enlarging n if necessary, we may assume $r\leq n$ . Then define $u\cdot b$ by the composition $G^{p+q}F^{m+n-r}K \xrightarrow {G^{p+q}F^{n-r}b}G^{p+q}F^nL \xrightarrow {u}G^qM$ . Since there is a commutative diagram

for each $r\leq n$ , we see that this action is well-defined.

Let us note the following property of $U_0$ .

Lemma 7.3. The maps $\mathcal {A}(F^n(-),F^pM)\xrightarrow {G^p}\mathcal {A}(G^pF^n(-),G^pF^pM)\to \mathcal {A}(G^pF^n(-),M)$ induce a quasi-isomorphism

of DG $\mathcal {B}$ -modules.

Proof. The commutative diagram

shows the existence of the morphism on the colimits.

It is clear that the induced morphism is a quasi-isomorphism since F and G are mutually inverse equivalences on $H^0\mathcal {A}$ .

Note that $U_q(-,-)$ constructed above does not have a $\mathcal {C}$ -action. The next step toward relating $\mathcal {B}$ and $\mathcal {C}$ is constructing a DG $(\mathcal {C},\mathcal {B})$ -bimodule that is quasi-isomorphic over $\mathcal {B}$ to $U_0$ .

Lemma 7.4. The vertical maps in equation (7.1) induce a sequence of quasi-isomorphisms

of DG $\mathcal {B}$ -modules for each $M \in \mathcal {A}$ .

Now define the DG $\mathcal {B}$ -module $U(-,M)$ by the colimit

$$\begin{align*}U(-,M):=\operatorname*{\mathrm{colim}}_{q\geq0}U_q(-,M). \end{align*}$$

For each $L \in \mathcal {B}$ , we have $U(L,M)=\operatorname *{\mathrm {colim}}_{p,q\geq 0}\bigoplus _{n\geq 0}\mathcal {A}(G^{p+q}F^nL,G^qM)$ .

We observe that $U(L,M)$ has a left $\mathcal {C}$ -action as follows: Let $c\in \mathcal {C}(M,N)$ be a morphism presented by $G^mM \to G^rN$ and $u\in U(L,M)$ an element presented by $G^{p+q}F^nL\to G^qM$ . Enlarging m and p if necessarily we may assume $q\leq m$ and $r\leq p$ . Then define $c\cdot u$ by the composition $G^{p+m}F^nL\xrightarrow {G^{m-q}u} G^mM \xrightarrow {c}G^rN$ . It is clear that this is well-defined and compatible with the right $\mathcal {B}$ -action. We have therefore obtained a $(\mathcal {C},\mathcal {B})$ -bimodule U.

As a final preparation, we describe an equivalence between the orbit categories $H^0\mathcal {B}$ and $H^0\mathcal {C}$ .

Proof. We have to show that the following diagram of isomorphisms is commutative, where we write ${}_h(G^aF^b,G^cF^d)$ for $H^0\mathcal {A}(G^aF^bL,G^cF^dM)$ , and the unlabeled maps are induced by $G\circ F\to 1_{\mathcal {A}}$ :

We see this by filling and lifting the above diagram to a diagram of quasi-isomorphism in $\mathcal {A}$ as below. Here again we omit the objects L and M:

It is now easy to verify that this diagram commutative.

Now we are ready to prove the main theorem of this section.

Proof of Theorem 7.1

We show that the $(\mathcal {C},\mathcal {B})$ -bimodule

$$\begin{align*}U(L,M)=\operatorname*{\mathrm{colim}}_{p,q\geq0}\bigoplus_{n\geq0}\mathcal{A}(G^{p+q}F^nL,G^qM) \end{align*}$$

constructed above gives a quasi-equivalence. Since the DG categories $\mathcal {B}$ and $\mathcal {C}$ have shifts (that is, the Yoneda embedding $\mathcal {B}\hookrightarrow \mathcal {C}_{\mathrm {dg}}(\mathcal {B})$ is closed under $[\pm 1]$ and the same for $\mathcal {C}$ ), we only have to show that $-\!\otimes^{\mathrm{L}}_{\mathcal {C}} U$ induces an equivalence $H^0\mathcal {C}\xrightarrow {\simeq }H^0\mathcal {B}$ .

For each $M \in \mathcal {C}$ , we have quasi-isomorphisms

$$\begin{align*}u_M \colon \mathcal{B}(-,M) \to U_0(-,M) \to U(-,M) \end{align*}$$

of DG $\mathcal {B}$ -modules by Lemma 7.3 and Lemma 7.4; thus U is a quasi-functor.

It remains to show that the induced map

$$\begin{align*}\operatorname{\mathrm{Hom}}_{\mathcal{D}(\mathcal{C})}(\mathcal{C}(-,L),\mathcal{C}(-,M)) \to \operatorname{\mathrm{Hom}}_{\mathcal{D}(\mathcal{B})}(U(-,L),U(-,M)) \end{align*}$$

is an isomorphism for each $L, M \in \mathcal {A}$ . It suffices to show that the following diagram is commutative:

The equivalence $H^0\mathcal {B} \to H^0\mathcal {C}$ given in Lemma 7.5 shows that if $f \in H^0\mathcal {B}(L,M)$ is presented by a morphism $F^nL \to F^pM$ in $Z^0\mathcal {A}$ , then the corresponding morphism $g \in H^0\mathcal {C}(L,M)$ is presented by $G^pL \to G^nM$ in $Z^0\mathcal {A}$ so that the diagram

(7.2)

is commutative in $H^0\mathcal {A}$ .

Let f and g be of the form above. Then the commutativity we want amounts to saying that $u_M\cdot f=g\cdot u_L$ in $H^0U(L,M)$ , where we may regard $\cdot $ as the right action of $\mathcal {B}$ (respectively, left action of $\mathcal {C}$ ) on U. Since $u_L \in U(L,L)$ is presented by the identity morphism in $\mathcal {A}$ the element $g \cdot u_L \in U(L,M)$ is presented by $G^{p+n}F^nL \to G^pL \xrightarrow {1} G^pL \xrightarrow {g} G^nM$ . Similarly, since $u_M \in U(M,M)$ is presented by the identity morphism in $\mathcal {A}$ , the element $u_M\cdot f\in U(L,M)$ is presented by $G^pF^nL \xrightarrow {G^pf} G^pF^pM \to M$ , hence by $G^{n+p}F^nL \xrightarrow {G^{n+p}f} G^{n+p}F^pM \to G^nM$ obtained by applying $G^n$ . Then the commutativity of the diagram (7.2) in $H^0\mathcal {A}$ implies that we have $u_M\cdot f=g\cdot u_L$ in $H^0U(L,M)$ .

Let us apply our general result to the setting of finite-dimensional algebras. Let $\Lambda $ be a finite-dimensional algebra of finite global dimension, $\mathcal {A}=\mathcal {C}^{-,b}(\operatorname {\mathrm {proj}}\Lambda )$ , and X a complex of projective $(\Lambda ,\Lambda )$ -bimodules such that $F=-\otimes _\Lambda X\colon \mathcal {A}\to \mathcal {A}$ induces an autoequivalence on $H^0\mathcal {A}=\mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ . Suppose that for each $L, M\in \mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ , we have $\operatorname {\mathrm {Hom}}_{\mathcal {D}(\Lambda )}(L,F^iM)=0$ for almost all $i\in \mathbb {Z}$ . Let $\mathcal {B}=\mathcal {A}/F$ be the DG orbit category and $\Gamma =\Lambda \oplus X[-1]$ the trivial extension DG algebra. Note that our assumptions imply that the orbit category $H^0\mathcal {B}$ is idempotent complete since we have $J_{H^0\mathcal {B}}(L,L)=J_{H^0\mathcal {A}}(L,L)\oplus \bigoplus _{i\neq 0}H^0\mathcal {A}(L,F^iL)$ for each L, that is indecomposable in $H^0\mathcal {A}$ , and where J is the Jacobson radical of each category. It follows that the triangulated hull $\operatorname {\mathrm {tria}}\mathcal {B}$ equals the perfect derived category $\operatorname {\mathrm {per}}\mathcal {B}$ . Then Keller’s theorem [Reference KellerKe2, Theorem 2, also correction] gives an equivalence

which is compatible with the natural functors from $\mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ .

We have the same equivalence arising from the inverse of F. Let Y be a bimodule projective resolution of $\operatorname {{\mathrm {R}Hom }}_\Lambda (X,\Lambda )$ and $G=-\otimes _\Lambda Y \colon \mathcal {A} \to \mathcal {A}$ , which induces a quasi-inverse to $F=-\otimes _\Lambda X$ on $\mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ . Let $\mathcal {C}=\mathcal {A}/G$ be the DG orbit category and $\Delta =\Lambda \oplus Y[-1]$ be the trivial extension DG algebra so that we have an equivalence

By the above equivalences and Theorem 7.1, we deduce the following consequence.

Corollary 7.6. There exists a triangle equivalence

which is compatible with the projection functors from $\mathcal {D}^b(\operatorname {\mathrm {mod}} \Lambda )$ .

8 Proof of Theorem 6.4

We now give a proof of a main result, Theorem 6.4, of this paper using the result from the previous section. In fact, the essential part of the proof does not depend on our specific setup, so we first state the intermediate result in Proposition 8.1 below, which can also be viewed as a DG version of Theorem 6.4.

Let $\Lambda $ be a finite-dimensional algebra that is homologically smooth and X a complex of $(\Lambda ,\Lambda )$ -bimodules such that $F=-\!\otimes^{\mathrm{L}}_\Lambda\!X$ gives an autoequivalence on $\mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ . We assume the following on the tilting complex X:

• (X1) For each $L, M\in \mathcal {D}^b(\operatorname {\mathrm {mod}}\Lambda )$ , we have $\operatorname {\mathrm {Hom}}_{\mathcal {D}(\Lambda )}(L,F^iM)=0$ for almost all $i\in \mathbb {Z}$ .

• (X2) X is concentrated in degree $\leq 0$ .

Let

$$\begin{align*}\Gamma=\Lambda\oplus X[-1], \quad \Sigma=T^{\mathrm{L}}_\Lambda X \end{align*}$$

be the trivial extension DG algebra and, respectively, the derived tensor algebra: that is, the tensor algebra of a bimodule projective resolution of X. Then the condition (X2) shows that $\Sigma $ is a negative DG algebra, and (X1) implies that its cohomology $H^0\Sigma =\bigoplus _{i\geq 0}\operatorname {\mathrm {Hom}}_{\mathcal {D}(\Lambda )}(\Lambda ,F^i\Lambda )$ is finite dimensional.

Recall from the introduction that we have denoted by

$$\begin{align*}\mathcal{C}(\Pi)=\operatorname{\mathrm{per}}\Pi/\mathcal{D}^b(\Pi) \end{align*}$$

and called it the cluster category for any DG algebra $\Pi $ satisfying $\operatorname {\mathrm {per}}\Pi \supset \mathcal {D}^b(\Pi )$ . Our general intermediate result is an equivalence between the cluster category of the tensor algebra and the singularity category of the trivial extension algebra.

Proposition 8.1. The DG algebra $\Sigma $ satisfies $\operatorname {\mathrm {per}}\Sigma \supset \mathcal {D}^b(\Sigma )$ , and there exists a triangle equivalence

$$\begin{align*}\mathcal{C}(\Sigma)\simeq\operatorname{\mathrm{thick}}_{\mathcal{D}(\Gamma)}\Lambda/\operatorname{\mathrm{per}}\Gamma. \end{align*}$$

The first step is to apply Corollary 7.6. Let Y be a bimodule projective resolution of $\operatorname {{\mathrm {R}Hom }}_\Lambda (X,\Lambda )$ , and set

$$\begin{align*}\Delta=\Lambda\oplus Y[-1]. \end{align*}$$

Then we have a triangle equivalence

(8.1) $$ \begin{align} \operatorname{\mathrm{thick}}_{\mathcal{D}(\Gamma)}\Lambda/\operatorname{\mathrm{per}}\Gamma \simeq \operatorname{\mathrm{thick}}_{\mathcal{D}(\Delta)}\Lambda/\operatorname{\mathrm{per}}\Delta. \end{align} $$

We next use the following computation of a DG endomorphism algebra.

We also need the equivalence by relative Koszul dual.

We now return to our setup from Section 6. Recall that R is a negatively graded bimodule $(d+1)$ -CY algebra of a-invariant a such that each $R_i$ is finite dimensional and that A is a d-representation infinite algebra in Proposition 4.9 given by

$$\begin{align*}A=\left( \begin{array}{cccc} R_0& 0 & \cdots& 0 \\ R_{-1} & R_0&\cdots& 0 \\ \vdots& \vdots&\ddots&\vdots \\ R_{-(a-1)}&R_{-(a-2)}&\cdots&R_0 \end{array} \right), \end{align*}$$

whose d-AR-translation $\nu _d$ has an ath root defined by diagram (4.1). We have a cotilting bimodule $U=\nu _d^{-1/a}A$ and a d-Iwanaga-Gorenstein algebra $B=A\oplus U$ (see Proposition 6.3).

Let us first apply Proposition 8.1. Set $\Lambda =A$ and $X=U[1]$ . Since U is a preprojective module over a d-representation infinite algebra, it clearly satisfies (X1) and (X2). Then the DG algebra $\Gamma =\Lambda \oplus X[-1]$ concentrates in degree $0$ and is nothing but our Iwanaga-Gorenstein algebra $B=A\oplus U$ . Since A has finite global dimension, the right-hand side of Proposition 8.1 is precisely the singularity category $\mathcal {D}_{\mathrm {sg}}(B)$ of B. Also, we see that the DG algebra $\Sigma $ is the tensor algebra

$$\begin{align*}S:=T_A(U[1]) \end{align*}$$

with trivial differentials. Proposition 8.1 for this case then gives the following equivalence.

Corollary 8.4. There exists a triangle equivalence

$$\begin{align*}\mathcal{C}(S)\simeq\mathcal{D}_{\mathrm{sg}}(B). \end{align*}$$

To compare the cluster categories of S and $R^{\mathrm {dg}}$ , we need another intermediate DG algebra, which is

$$\begin{align*}\widetilde{S}:=\operatorname{\mathrm{\mathcal{E}nd}}_{R^{\mathrm{dg}}}(\widetilde{T}), \text{ with } \widetilde{T}=R \oplus R[-1] \oplus \cdots\oplus R[-(a-1)], \end{align*}$$

where $\operatorname {\mathrm {\mathcal {E}nd}}_{R^{\mathrm {dg}}}(-)=\operatorname {\mathrm {\mathcal {H}om}}_{R^{\mathrm {dg}}}(-,-)$ is the endomorphism DG algebra.

Let us first state an easy relationship between $R^{\mathrm {dg}}$ and $\widetilde {S}$ . We say that DG algebras $\Pi _1$ and $\Pi _2$ are DG Morita equivalent if there is a $(\Pi _1,\Pi _2)$ -bimodule X such that $-\!\otimes^{\mathrm{L}}_{\Pi _1}\!X$ induces an equivalence $\mathcal {D}(\Pi _1) \xrightarrow {\simeq }\mathcal {D}(\Pi _2)$ , or equivalently there exists a compact generator $M \in \mathcal {D}(\Pi _2)$ whose DG endomorphism algebra $\operatorname {{\mathrm {R}Hom }}_{\Pi _2}(M,M)$ is quasi-equivalent to $\Pi _1$ [Reference KellerKe1, Theorem 8.2]; see also [Reference KellerKe3, Section 3.8].

We immediately have the following lemma.