2.1 Tensor algebras

Let $M$ be a $\Lambda ^{{\operatorname {en} }}$-module. Recall that the tensor algebra $\operatorname {T} _{\Lambda} (M)$ of $M$ is the $\mathbb {Z}$-graded vector space

\[T_{\Lambda}(M)=\mathop{\bigoplus}\limits_{i\geq 0}M^{i},\]

where $M^{i}=M\otimes _{\Lambda} \cdots \otimes _{\Lambda} M$ is the tensor product of $i$ copies of $M$ so, in particular, $M^{0}=\Lambda$. There is an obvious graded multiplication map $M^{i}\times M^{j}\to M^{i+j}$ which sends the pair $(\lambda _1\otimes \lambda _2\otimes \cdots \otimes \lambda _i,\lambda _{i+1}\otimes \cdots \otimes \lambda _{i+j})$ of standard basis vectors to the concatenated vector $\lambda _1\otimes \lambda _2\otimes \cdots \otimes \lambda _{i+j}$, and so $\operatorname {T} _{\Lambda} (M)$ is a non-negatively $\mathbb {Z}$-graded algebra. For later use, we prepare the following basic observations, the proofs of which are left to the reader.

Let $\Lambda$ be a basic $\mathbb {F}$-algebra with Jacobson radical $J$. We assume that $S$ is a semisimple subalgebra of $\Lambda$ such that $\Lambda =S\oplus J$. Then we can write $\Lambda =\operatorname {T} _S(V)/I$ for an $S^{{\operatorname {en} }}$-module $V$ and an ideal $I$ of $\operatorname {T} _S(V)$. If $I$ is a homogeneous ideal, then $\Lambda$ inherits a grading from $\operatorname {T} _S(V)$. Any such non-negatively $\mathbb {Z}$-graded algebra $\Lambda$ has a minimal $\mathbb {Z}$-graded projective $\Lambda ^{{\operatorname {en} }}$-module resolution

\[ \cdots\overset{\delta_3}{\rightarrow}P_2\overset{\delta_2}{\rightarrow}P_1\overset{\delta_1}{\rightarrow}P_0\to 0, \]

where each projective module $P_i$ is generated in degrees greater than or equal to $i$. Immediately, we have the following property.

We can write each projective $\Lambda ^{{\operatorname {en} }}$-module in the form $\Lambda \otimes _S K\otimes _S \Lambda$ for some $\mathbb {Z}$-graded projective $S^{{\operatorname {en} }}$-module $K$, where we consider $S$ as a $\mathbb {Z}$-graded algebra concentrated in degree $0$; see [Reference Butler and KingBK99]. In particular, we write $P_i=\Lambda \otimes _S K_i\otimes _S \Lambda$ for $\mathbb {Z}$-graded projective $S^{{\operatorname {en} }}$-modules $K_i$, for $0\leq i$.

In general $K_i\cong \operatorname {Tor} ^{\Lambda }_i(S,S)$, and explicit descriptions for these spaces are known. For $m\geq 0$,

\[ \operatorname{Tor}^{\Lambda}_{2m}(S,S)\cong\frac{I^{m}\cap JI^{m-1}J}{JI^{m}+I^{m}J}\quad \text{and}\quad \operatorname{Tor}^{\Lambda}_{2m+1}(S,S)=\frac{JI^{m}\cap I^{m}J}{I^{m+1}+JI^{m}J}\text{.} \]

For more information and references, see the introduction to [Reference Butler and KingBK99]. For certain kinds of algebras there are nicer descriptions of these spaces: see § 2.2 and the final chapters of [Reference Butler and KingBK99].

As well as our vector-space duality $(-)^{*}$, we have dualities

\begin{align*} (-)^{{\vee}} &:= \operatorname{Hom}_{S^{{\operatorname{en}}}}(-,S^{{\operatorname{en}}}):S^{{\operatorname{en}}}\operatorname{-mod}\overset{\sim}{\rightarrow} S^{{\operatorname{en}}}\operatorname{-mod},\\ (-)^{*\ell} &:= \operatorname{Hom}_S(-,S):S^{{\operatorname{en}}}\operatorname{-mod}\overset{\sim}{\rightarrow} S^{{\operatorname{en}}}\operatorname{-mod},\\ (-)^{*r} &:= \operatorname{Hom}_{S^{{\operatorname{op}}}}(-,S):S^{{\operatorname{en}}}\operatorname{-mod}\overset{\sim}{\rightarrow} S^{{\operatorname{en}}}\operatorname{-mod}. \end{align*}

For $S^{{\operatorname {en} }}$-modules $X$ and $Y$, we have functorial isomorphisms

(1)\begin{align} & Y^{*\ell}\otimes_SX^{*}\cong(X\otimes_SY)^{*}, \end{align}

(2)\begin{align} & Y^{*\ell}\otimes_SX^{*\ell}\cong(X\otimes_SY)^{*\ell} \end{align}

sending $f\otimes g$ to $(x\otimes y\mapsto g(xf(y))$,

(3)\begin{equation} Y^{*r}\otimes_SX^{*r}\cong(X\otimes_SY)^{*r} \end{equation}

sending $f\otimes g$ to $(x\otimes y\mapsto f(g(x)y))$, and

(4)\begin{equation} Y^{*r}\otimes_SX^{{\vee}}\cong(X\otimes_SY)^{{\vee}} \end{equation}

sending $f\otimes g$ to $(x\otimes y\mapsto \sum _is_i\otimes f(s'_iy))$ for $g(x)=\sum _is_i\otimes s'_i$. For example, (4) can be checked as follows: as $\operatorname {Hom} _{S^{{\operatorname {op} }}}(Y,S^{{\operatorname {en} }})\cong S\otimes _{\mathbb {F}} Y^{*r}\in S^{{\operatorname {en} }}\operatorname {-proj}$, we have

\[ (X\otimes_SY)^{{\vee}}\cong\operatorname{Hom}_{S^{{\operatorname{en}}}}(X,\operatorname{Hom}_{S^{{\operatorname{op}}}}(Y,S^{{\operatorname{en}}})) \cong(S\otimes_{\mathbb {F}} Y^{*r})\otimes_{S^{{\operatorname{en}}}}X^{{\vee}}\cong Y^{*r}\otimes_SX^{{\vee}}. \]

Note the following simple lemma.

Lemma 2.3 Let $L$ be a $\Lambda \otimes _{\mathbb {F}} S^{{\operatorname {op} }}$-module, $X$ be a projective $S^{{\operatorname {en} }}$-module, and $M$ be a $S\otimes _{\mathbb {F}} \Lambda ^{{\operatorname {op} }}$-module. Then there is an isomorphism of $\Lambda ^{{\operatorname {en} }}$-modules that is natural in $L$, $X$, and $M$:

\[ \operatorname{Hom}_{\Lambda^{{\operatorname{en}}}}(L\otimes_S X\otimes_SM,\Lambda^{{\operatorname{en}}})\cong \operatorname{Hom}_{\Lambda}(M,\Lambda)\otimes_S X^{{\vee}}\otimes_S\operatorname{Hom}_{\Lambda^{{\operatorname{op}}}}(L,\Lambda). \]

In particular, for any projective $S^{{\operatorname {en} }}$-module $X$, there is a functorial isomorphism of $\Lambda ^{{\operatorname {en} }}$-modules

\[ \operatorname{Hom}_{\Lambda^{{\operatorname{en}}}}(\Lambda\otimes_S X\otimes_S\Lambda,\Lambda^{{\operatorname{en}}})\cong \Lambda\otimes_S X^{{\vee}}\otimes_S\Lambda. \]

Proof. We include a complete proof for the convenience of the reader. Using the tensor-hom adjunctions, for any $X\in S^{{\operatorname {en} }}\operatorname {-mod}$ we have isomorphisms of $\Lambda ^{{\operatorname {en} }}$-modules

\begin{align*} & \operatorname{Hom}_{\Lambda^{{\operatorname{en}}}}(L\otimes_S X\otimes_SM,\Lambda^{{\operatorname{en}}}) \cong\operatorname{Hom}_{\Lambda^{{\operatorname{en}}}}((L\otimes_{\mathbb {F}} M)\otimes_{S^{{\operatorname{en}}}}X,\Lambda^{{\operatorname{en}}})\\ &\quad \cong\operatorname{Hom}_{S^{{\operatorname{en}}}}(X,\operatorname{Hom}_{\Lambda^{{\operatorname{en}}}}(L\otimes_{\mathbb {F}} M,\Lambda^{{\operatorname{en}}})) \cong X^{{\vee}}\otimes_{S^{{\operatorname{en}}}}\operatorname{Hom}_{\Lambda^{{\operatorname{en}}}}(L\otimes_{\mathbb {F}} M,\Lambda^{{\operatorname{en}}})\\ &\quad \cong X^{{\vee}}\otimes_{S^{{\operatorname{en}}}}(\operatorname{Hom}_{\Lambda}(L,\Lambda) \otimes_{\mathbb {F}}\operatorname{Hom}_{\Lambda^{{\operatorname{op}}}}(M,\Lambda))\cong\operatorname{Hom}_{\Lambda}(M,\Lambda) \otimes_S X^{{\vee}}\otimes_S\operatorname{Hom}_{\Lambda^{{\operatorname{op}}}}(L,\Lambda). \end{align*}

All our isomorphisms are natural.

The four duals $(-)^{*}$, $(-)^{*\ell }$, $(-)^{*r}$, and $(-)^{\vee }$ are isomorphic to each other (see, e.g., [Reference RickardRic02, Section 3, Reference Bocklandt, Schedler and WemyssBSW10, Section 2.1]). In fact, because $S$ is a symmetric $\mathbb {F}$-algebra, there exists an $\mathbb {F}$-linear form $t:S\to \mathbb {F}$ such that $t(xy)=t(yx)$ and the map $S\to S^{*}$ sending $x$ to $(y\mapsto t(xy))$ is an isomorphism. This gives isomorphisms

\[ \alpha:=t\circ(-):(-)^{*\ell}\cong(-)^{*},\quad \beta:=t\circ(-):(-)^{*r}\cong(-)^{*}\enspace \mbox{and}\enspace \gamma:=(t\otimes 1)\circ(-):(-)^{{\vee}}\cong(-)^{*r} \]

of functors.

For later use in § 5, we now show that these isomorphisms are compatible with module structures in the following sense: let $L={\bigoplus} _{i\in \mathbb {Z}}L_i$ be a $\mathbb {Z}$-graded $\operatorname {T} _S(V^{*\ell })^{{{\operatorname {op} }}}$-module, and let

\[ L^{(*)}=\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}L_{{-}i}^{*},\quad L^{(*\ell)}=\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}L_{{-}i}^{*\ell},\quad L^{(*r)}=\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}L_{{-}i}^{*r},\quad \mbox{and}\quad L^{({\vee})}=\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}L_{{-}i}^{{\vee}}. \]

Then $L^{(*)}$ and $L^{(*\ell )}$ are $\mathbb {Z}$-graded $\operatorname {T} _S(V^{*\ell })$-modules, and $L^{(*r)}$ and $L^{(\vee )}$ are $\mathbb {Z}$-graded $\operatorname {T} _S(V^{*r})$-modules as follows: the action of $\operatorname {T} _S(V^{*\ell })^{{{\operatorname {op} }}}$ on $L$ is given by a morphism $a_i:L_i\otimes _SV^{*\ell }\to L_{i+1}$ of $S^{{\operatorname {en} }}$-modules for $i\in \mathbb {Z}$. This corresponds to a morphism $b_i:L_i\to L_{i+1}\otimes _SV$ of $S^{{\operatorname {en} }}$-modules via Hom-tensor adjunction $\operatorname {Hom} _{S^{{\operatorname {en} }}}(A\otimes _SB,C)\cong \operatorname {Hom} _{S^{{\operatorname {en} }}}(A,\operatorname {Hom} _{S^{{\operatorname {op} }}}(B,C))$. Applying $(-)^{{\dagger}ger }$ for ${\dagger}ger =*,*\ell ,*r,\vee$, we obtain morphisms

\begin{align*} & V^{*\ell}\otimes_SL_{i+1}^{{\dagger}}\stackrel{(1)(2)}{\cong}(L_{i+1}\otimes_SV)^{{\dagger}} \xrightarrow{b_i^{{\dagger}}} L_i^{{\dagger}}\quad \mbox{for } \dagger{=}* \hbox{ or } *\ell,\\ & V^{*r}\otimes_SL_{i+1}^{{\dagger}}\stackrel{(3)(4)}{\cong}(L_{i+1}\otimes_SV)^{{\dagger}} \xrightarrow{b_i^{{\dagger}}} L_i^{{\dagger}}\quad \mbox{for } \dagger{=}*\!r \,\hbox{ or } \vee \end{align*}

of $S^{{\operatorname {en} }}$-modules, which give the desired structures on $L^{(*)}$, $L^{(*\ell )}$, $L^{(*r)}$, and $L^{(\vee )}$.

Proof. The assertions follow from the following commutative diagram.

The right squares commute because $\alpha ,\beta ,\gamma$ are morphisms of functors. The top left square commutes because both the north-east composition and the south-west composition send $f\otimes g\in V^{*r}\otimes _SL_{i+1}^{\vee }$ to $(L_{i+1}\otimes _SV\ni x\otimes v\mapsto \sum _it(s_i)f(s'_iv)\in S)$, where $g(x)=\sum _is_i\otimes s'_i$. The bottom left square also commutes because both the north-west composition and the south-east composition send $f\otimes g\in V^{*\ell }\otimes _SL_{i+1}^{*\ell }$ to $(L_{i+1}\otimes _SV\ni x\otimes v\mapsto t(g(xf(v)))\in \mathbb {F})$.

To check that the left middle square commutes, fix $f\otimes g\in V^{*r}\otimes _SL_{i+1}^{*r}$. The north-east composition sends $f\otimes g$ to $(x\otimes v\mapsto t(f(g(x)v)))$. The south-west composition sends $f\otimes g$ to $(x\otimes v\mapsto t(g(xf'(v))))$, where $f'=\alpha _V^{-1}\beta _V(f)\in V^{*\ell }$ satisfies $t\circ f=t\circ f'$. These two elements coincide because $t(f(g(x)v)))=t(f'(g(x)v))=t(g(x)f'(v))=t(g(xf'(v)))$.

2.2 Graded algebras and Koszul algebras

In this section, we give preliminaries on Koszul algebras, which were introduced in [Reference PriddyPri70] and studied extensively in [Reference Beilinson, Ginzburg and SoergelBGS96].

Let $\Lambda ={\bigoplus} _{i\geq 0}\Lambda _i$ be a positively $\mathbb {Z}$-graded $\mathbb {F}$-algebra satisfying the following conditions:

• • $S:=\Lambda _0$ is a finite-dimensional semisimple $\mathbb {F}$-algebra or, equivalently, the $\mathbb {Z}$-graded radical of $\Lambda$ coincides with $\Lambda _{>0}:={\bigoplus} _{i>0}\Lambda _i$;

• • $\Lambda$ is generated in degree $1$, that is, the multiplication map $\Lambda _1\otimes _{\mathbb {F}} \Lambda _1\otimes _{\mathbb {F}} \cdots \otimes _{\mathbb {F}} \Lambda _1\to \Lambda _j$ is surjective for each $j$.

In this case, we call the grading a radical grading.

We assume, for simplicity, that $\Lambda$ is basic. Our assumptions ensure that $\Lambda$ is a quotient of the tensor algebra $\operatorname {T} _S(V)$ where $V$ is the $S^{{\operatorname {en} }}$-module $\Lambda _1$. When $\Lambda$ is finite-dimensional and $\mathbb {F}$ is algebraically closed, we can identify $S$ with the space $\mathbb {F} Q_0$ of vertices, and $V$ with the space $\mathbb {F} Q_1$ of arrows, of the Gabriel quiver $Q$ of $\Lambda$.

For a $\mathbb {Z}$-graded $\Lambda$-module $M$ and $j\in \mathbb {Z}$, let $M({j})$ denote the shifted $\mathbb {Z}$-graded $\Lambda$-module where $M({j})_i = M_{i+j}$. A complex

\[ \cdots\to M_1\to M_0\to M_{{-}1}\to\cdots \]

of $\mathbb {Z}$-graded $\Lambda$-modules is linear if each module $M_i$ is generated in degree $i$ and each map is homogeneous of degree $0$. The algebra $\Lambda$ is Koszul if each simple module $S_i$ has a linear projective resolution.

All Koszul algebras are quadratic in the sense that they can be written as a quotient of a tensor algebra:

\[ \Lambda\cong\operatorname{T}_S(V)/(R), \]

where $V$ is an $S^{{\operatorname {en} }}$-module, $R$ is a subset of $V\otimes _SV$, and $(R)$ is the ideal in $\operatorname {T} _S(V)$ generated by R. To simplify the proofs, we sometimes assume that $R$ is a sub-$S^{{\operatorname {en} }}$-module of $V\otimes _SV$ instead of just a subset. In particular, it is a vector subspace. This is no real restriction.

We view $S$ as a $\mathbb {Z}$-graded $\mathbb {F}$-algebra concentrated in degree $0$, and $V$ as a $\mathbb {Z}$-graded $S^{{\operatorname {en} }}$-module concentrated in degree $1$. Then the tensor grading and the grading coming from $\Lambda$ coincide, and so we can safely refer to just the grading on $\Lambda$.

We record a useful lemma on quadratic algebras that can be checked easily.

In the rest of this subsection, let $\Lambda \cong \operatorname {T} _S(V)/(R)$ be a quadratic algebra. We have $S^{{\operatorname {en} }}$-modules $K_0=S$, $K_1=V$, $K_2=R$, and

\[ K_j=(V\otimes_SK_{j-1})\cap(K_{j-1}\otimes_SV)=\mathop{\bigcap}\limits_{i=0}^{j-2}V^{i}\otimes_SR\otimes_SV^{j-2-i} \]

for $j\geq 3$. Here, $V^{i}$ denotes the $i$th tensor power $V\otimes _S\cdots \otimes _SV$. Note that $K_i$ is concentrated in degree $i$ and that for $i<0$, we set $K_i=0$. We have obvious inclusions $\iota^\ell_i:K_i\hookrightarrow V\otimes_SK_{i-1}$ and $\iota^r_i:K_i\hookrightarrow K_{i-1}\otimes_SV$ of $S^{\operatorname{en}}$-modules.

Recall [Reference Beilinson, Ginzburg and SoergelBGS96, Section 2.7] that if $U$ is a subset of $V^{i}$, the right orthogonal complement of $U$ is $U^{\perp }=\{f\in (V^{*\ell })^{i}\;|\; f(U)=0\}$, where we identify $(V^{*\ell })^{i}$ with $(V^{i})^{*\ell }$ by (2). The quadratic dual of a quadratic algebra $\Lambda =\operatorname {T} _S(V)/(R)$ is

\[ \Lambda^{!}=\operatorname{T}_S(V^{*\ell})/(R^{{\perp}}). \]

It is again quadratic. If moreover $\Lambda$ is Koszul, then $\Lambda ^{!}$ is also Koszul and it coincides with the opposite ext algebra $\big ({\bigoplus} _{i\geq 0}\operatorname {Ext} ^{i}_{\Lambda} (S,S)\big )^{{\operatorname {op} }}$ (see [Reference Beilinson, Ginzburg and SoergelBGS96, Proposition 2.10.1]). In this case, $\Lambda ^{!}$ has the following description.

Lemma 2.6 [BGS96, Section 2.8] For a Koszul algebra $\Lambda$, we have an isomorphism of $\mathbb {Z}$-graded algebras

\[ \Lambda^{!}=\mathop{\bigoplus}\limits_{i\geq 0}(\Lambda^{!})_i\cong\mathop{\bigoplus}\limits_{i\geq 0}K_i^{*\ell}, \]

where the $\mathbb {Z}$-graded algebra structure on ${\bigoplus} _{i\geq 0}K_i^{*\ell }$ is given by the duals $(\iota ^{\ell }_i)^{*\ell }$ and $(\iota ^{r}_i)^{*\ell }$.

Now we assume that $S$ is a separable $\mathbb {F}$-algebra, that is, $S\otimes _{\mathbb {F}}\mathbb {F}'$ is semisimple for all field extensions $\mathbb {F}\subset \mathbb {F}'$ or, equivalently, $S^{{\operatorname {en} }}$ is semisimple [Reference WeibelWei94, Theorem 9.2.11]. Let

\[ P_i=\Lambda\otimes_SK_i\otimes_S\Lambda=\Lambda^{{\operatorname{en}}}\otimes_{S^{{\operatorname{en}}}}K_i. \]

This is a projective $\Lambda ^{{\operatorname {en} }}$-module because $S^{{\operatorname {en} }}$ is semisimple by our assumption. Combining $\iota_i^\ell$ and $\iota_i^r$ with the multiplication for $\Lambda$, they induce maps $\hat {\iota }^{\ell }_i,\hat {\iota }^{r}_i:P_i\to P_{i-1}$. Let

\[ \delta_i=\hat{\iota}^{\ell}_i+({-}1)^{i}\hat{\iota}^{r}_i. \]

One can check that these maps give a chain complex

(5)\begin{equation} \cdots \xrightarrow{\delta_3}P_2\xrightarrow{\delta_2}P_1\xrightarrow{\delta_1}P_0\to 0, \end{equation}

which is called the Koszul bimodule complex. Note that, as $K_i\subseteq V^{i}$ and $V$ is concentrated in degree $1$, each $P_i$ is generated in degree $i$, i.e. the complex is linear.

The next result is an important characterization of Koszul algebras. It can be found as, for example, [Reference Braverman and GaitsgoryBG96, Proposition A.2, Reference Butler and KingBK99, Theorem 9.2].

In this paper, we need separability of $S$ when we consider bimodule resolutions including the Koszul bimodule resolutions. We assume separability in Theorems C and 2.7, §§ 3.1 and 3.3, Corollary 4.3, and § 5.1.

2.3 Higher preprojective algebras

Let

\[ \tau=(-)^{*}\circ\operatorname{Tr}:\Lambda\operatorname{-\underline{mod}}\to\Lambda\operatorname{-\overline{mod}} \]

denote the Auslander–Reiten translation, which is a functor from the stable category of modules over $\Lambda$ to the costable category, and

\[ \tau^{-}=\operatorname{Tr}\circ(-)^{*}:\Lambda\operatorname{-\overline{mod}}\to\Lambda\operatorname{-\underline{mod}} \]

the inverse Auslander–Reiten translation. Note that if $\Lambda$ is hereditary, then the Auslander–Reiten translation in fact can be regarded as an endofunctor of the module category: $\tau = \operatorname{Ext}_\Lambda^1(-,\Lambda )^{*}$ and $\tau ^{-} = \operatorname {Ext} ^{1}_{\Lambda} (\Lambda ^{*},-)$. Moreover, $\tau ^{-}$ is left adjoint to $\tau$.

Recall [Reference IyamaIya07] that the $d$-Auslander–Reiten translation and inverse $d$-Auslander–Reiten translation are defined as

\[ \tau_d=\tau\Omega^{d-1}:\Lambda\operatorname{-\underline{mod}}\to\Lambda\operatorname{-\overline{mod}}\quad \mbox{and}\quad \tau_d^{-}=\tau^{-}\Omega^{-(d-1)}:\Lambda\operatorname{-\overline{mod}}\to\Lambda\operatorname{-\underline{mod}}, \]

where $\Omega :\Lambda \operatorname {-\underline {mod}}\to \Lambda \operatorname {-\underline {mod}}$ denotes the syzygy functor and $\Omega ^{-}:\Lambda \operatorname {-\overline {mod}}\to \Lambda \operatorname {-\overline {mod}}$ the cosyzygy functor. If $\operatorname {gldim}\Lambda \leq d$, then we regard $\tau _d$ and $\tau _d^{-}$ as the endofunctors

\[ \tau_d=\operatorname{Ext}^{d}_{\Lambda}(-,\Lambda)^{*}:\Lambda\operatorname{-mod}\to\Lambda\operatorname{-mod}\quad \mbox{and}\quad \tau_d^{-}=\operatorname{Ext}^{d}_{\Lambda}(\Lambda^{*},-):\Lambda\operatorname{-mod}\to\Lambda\operatorname{-mod} \]

of $\Lambda \operatorname {-mod}$.

Generalizing the classical case, we have two distinguished classes of modules.

Definition 2.8 [HIO14, Definition 4.7] We have the following two full subcategories $\mathscr {P}$ and $\mathscr {I}$ of $\Lambda \operatorname {-mod}$:

\[ \mathscr{P}:=\operatorname{add}\big\{\tau_d^{{-}i}(\Lambda)\,|\, i\ge 0\big\}\quad \mbox{and}\quad \mathscr{I}:=\operatorname{add}\big\{\tau_d^{i}(\Lambda^{*})\,|\, i\ge 0\big\}. \]

Any module in $\mathscr {P}$ is called $d$-preprojective and any module in $\mathscr {I}$ is called $d$-preinjective.

In the rest of this section, we assume that $\Lambda$ has global dimension $d$. The $\Lambda ^{{\operatorname {en} }}$-module

\[ E:=\operatorname{Ext}^{d}_{\Lambda}(\Lambda^{*},\Lambda) \]

plays a central role in this paper. We take this opportunity to record a useful lemma, which makes the $\Lambda ^{{\operatorname {en} }}$-module structure of $E$ clearer.

Lemma 2.9 We have isomorphisms

\[ E\cong\operatorname{Ext}^{d}_{\Lambda^{{\operatorname{en}}}}(\Lambda,\Lambda^{{\operatorname{en}}})\cong\operatorname{Ext}^{d}_{\Lambda^{{\operatorname{op}}}}(\Lambda^{*},\Lambda) \]

of $\Lambda ^{{\operatorname {en} }}$-modules.

Proof. For each finite-dimensional $\Lambda$-module $M$, there is a natural isomorphism $M\cong M^{**}$. Then we use the natural isomorphism of finite-dimensional vector spaces $V^{*}\otimes _{\mathbb {F}} W\cong \operatorname {Hom} _{\mathbb {F}}(V,W)$ to see that we have an isomorphism of $\Lambda ^{{\operatorname {en} }}$-modules

\[ \Lambda^{{\operatorname{en}}}\cong \Lambda\otimes_{\mathbb {F}}\Lambda\cong\Lambda^{**} \otimes_{\mathbb {F}}\Lambda\cong\operatorname{Hom}_{\mathbb{F}}(\Lambda^{*},\Lambda)\text{.} \]

Finally, we use the tensor-hom adjunctions to obtain

\[ \operatorname{Ext}^{d}_{\Lambda^{{\operatorname{en}}}}(\Lambda,\Lambda^{{\operatorname{en}}})\cong\operatorname{Ext}^{d}_{\Lambda^{{\operatorname{en}}}}(\Lambda,\operatorname{Hom}_{\mathbb{F}}(\Lambda^{*},\Lambda)) \cong\operatorname{Ext}^{d}_{\Lambda}(\Lambda\otimes_{\Lambda}\Lambda^{*},\Lambda)\cong\operatorname{Ext}^{d}_{\Lambda}(\Lambda^{*},\Lambda)\text{.} \]

The second isomorphism is shown similarly.

Using $E$, one can describe the functors $\tau _d$ and $\tau _d^{-}$ as follows.

Proposition 2.10 If $\operatorname {gldim}\Lambda \leq d$ then we have isomorphisms of functors

\[ \tau_d\cong\operatorname{Hom}_{\Lambda}(E,-):\Lambda\operatorname{-mod}\to\Lambda\operatorname{-mod}\quad \mbox{and}\quad \tau_d^{-}\cong E\otimes_{\Lambda}-:\Lambda\operatorname{-mod}\to\Lambda\operatorname{-mod}. \]

In particular, $\tau _d^{-}$ is left adjoint to $\tau _d$.

Now we recall the definition of higher preprojective algebras as given in [Reference Iyama and OppermannIO11].

Definition 2.11 The higher preprojective algebra (or, more precisely, the $(d+1)$-preprojective algebra) of $\Lambda$ is the tensor algebra of the $\Lambda ^{{\operatorname {en} }}$-module $E$:

\[ \Pi=\Pi_{d+1}(\Lambda):=\operatorname{T}_{\Lambda}(E). \]

As this is a tensor algebra, it comes with a natural grading that we call the tensor grading, i.e., the degree $i$ part of $\Pi$ is $E^{i}$.

The following result justifies the name of the higher preprojective algebra.

As in the global dimension $1$ case, the preprojective algebra can be identified with

\[ \mathop{\bigoplus}\limits_{i\geq 0}\operatorname{Hom}_{\Lambda}\big(\Lambda,\tau_d^{{-}i}(\Lambda)\big), \]

where the composition of $f:\Lambda \to \tau _d^{-i}(\Lambda )$ and $g:\Lambda \to \tau _d^{-j}(\Lambda )$ is given by

\[ gf=\tau_d^{{-}i}(g)\circ f:\Lambda\to\tau_d^{{-}i-j}(\Lambda). \]

The $i$th part of the tensor grading is just $\operatorname {Hom} _{\Lambda} \big (\Lambda ,\tau _d^{-i}(\Lambda )\big )$.

3.1 Preliminaries

In this subsection, let $\Lambda$ be a finite-dimensional $\mathbb {F}$-algebra $\Lambda$ with global dimension at most $d$, where $d$ is a positive integer. Moreover, we assume that

\[ \Lambda=\operatorname{T}_S(V)/I \]

for a separable $\mathbb {F}$-algebra $S$. Thus, $S^{{\operatorname {en} }}$ is semisimple, and the projective dimension of the $\Lambda ^{{\operatorname {en} }}$-module $\Lambda$ coincides with the global dimension of $\Lambda$, which is at most $d$. As before, let

\[ E=\operatorname{Ext}^{d}_{\Lambda}(\Lambda^{*},\Lambda). \]

We take a minimal projective resolution of the $\Lambda ^{{{\operatorname {en} }}}$-module $\Lambda$:

(6)\begin{equation} 0\to P_d\xrightarrow{\delta_d}\cdots\xrightarrow{\delta_3}P_2 \xrightarrow{\delta_2}P_1\xrightarrow{\delta_1}P_0\to 0\quad \mbox{with}\ P_i=\Lambda\otimes_SK_i\otimes_S\Lambda, \end{equation}

where $K_i$ is an $S^{{\operatorname {en} }}$-module. For each $i\ge 1$, we define a map $\delta '_i$ by the commutative diagram

(7)

where the vertical maps are given by Lemma 2.3.

Let $\bar {V}$ be the $S^{{\operatorname {en} }}$-module

\[ \bar{V}:=V\oplus K_d^{{\vee}}. \]

This notation is meant to be reminiscent of $\bar {Q}$, which, in the global dimension $1$ case, is used to denote the doubled quiver of the underlying quiver $Q$. For $T:=\operatorname {T} _S(V)$, we have an isomorphism

\[ \operatorname{T}_S(\bar{V})\cong\operatorname{T}_T\big(T\otimes_SK_d^{{\vee}}\otimes_ST\big) \]

by Lemma 2.1(a). Regarding $T\otimes _SK_d^{\vee }\otimes _ST$ as a subspace of $\operatorname {T} _S(\bar {V})$, we have the following description of $\Pi$.

Proof. We only need to prove part (b) of the proposition, from which part (a) follows. By Proposition 3.1, we have

\begin{align*} E &\cong \operatorname{Cok}\delta_d' = \big(\Lambda\otimes_SK_d^{{\vee}}\otimes_S\Lambda\big)/\Lambda\delta_d'\big(K_{d-1}^{{\vee}}\big)\Lambda\\ &\cong \big(T\otimes_SK_d^{{\vee}}\otimes_ST\big)/\big(I\otimes_SK_d^{{\vee}}\otimes_ST+T\otimes_SK_d^{{\vee}}\otimes_SI+TLT\big). \end{align*}

Thus, applying Lemma 2.1(a) and (b), we have

\begin{align*} \operatorname{T}_S(\bar{V})/(I+L) &\cong\operatorname{T}_T\big(T\otimes_SK_d^{{\vee}}\otimes_ST\big)/(I+TLT)\\ &\cong\operatorname{T}_{\Lambda}\big(\big(T\otimes_SK_d^{{\vee}}\otimes_ST\big)/\big(I\otimes_SK_d^{{\vee}} \otimes_ST+T\otimes_SK_d^{{\vee}}\otimes_SI+TLT\big)\big)\\ &\cong\operatorname{T}_{\Lambda}(E)=\Pi \end{align*}

as desired.

Consider the case where $\mathbb {F}$ is algebraically closed, so we can describe $\Lambda$ as $\mathbb {F} Q/I$. Let $\{k_1,\ldots ,k_r\}$ be a basis of $K_d$, each with a unique source and target $s(k_i)$ and $t(k_i)$, and let $\bar {Q}$ be the quiver obtained by adding $r$ arrows $k_i^{*}:t(k_i)\to s(k_i)$ to $Q$. Then, just as $V$ is the arrow space of $Q$, $\bar {V}$ is the arrow space of $\bar {Q}$, and Proposition 3.2 says that $\bar {Q}$ is the Gabriel quiver of $\Pi$.

We can therefore calculate the Gabriel quiver $\bar {Q}$ of $\Pi$ as follows. First, for each vertex $i$ of $Q$, compute the projective resolution

\[ 0\to P_{i,n}\to\cdots\to P_{i,0}\to 0 \]

of the simple left $\Lambda$-module $S_i$, where some projective modules $P_{i,h}$ may be zero. Then, for each $i$ and for each summand of the projective module $P_{i,n}$ that is isomorphic to the projective cover of $S_j$, add an arrow $i\to j$ to the quiver $Q$. The resulting quiver is $\bar {Q}$.

Example 3.3 Let

\[ Q=\big[1\overset{\alpha}{\rightarrow}2\overset{\beta}{\rightarrow}3\overset{\gamma}{\rightarrow}4\overset{\delta}{\rightarrow}5\overset{\varepsilon}{\rightarrow}6\big] \]

and $\Lambda =\mathbb {F} Q/(\beta \gamma \delta ,\gamma \delta \varepsilon )$. Let $S_i$ denote the simple left $\Lambda$-module associated to the vertex $i$, and $P(S_i)$ its projective cover. One can check that $\Lambda$ has global dimension $3$ and the only simple module with projective dimension $3$ is $S_6$. Its projective resolution is

\[ 0\to P(S_2)\overset{\cdot \beta}{\rightarrow} P(S_3)\overset{\cdot {\gamma\delta}}{\rightarrow} P(S_5)\overset{\cdot \varepsilon}{\rightarrow} P(S_6)\to 0, \]

where $\cdot a$ denotes right multiplication by $a$. Thus, the quiver $\bar {Q}$ of $\Pi$ is just $Q$ with an extra arrow from $6$ to $2$, which we label $(\beta \gamma \delta \varepsilon )^{*}$.

3.2 Superpotentials and higher Jacobi algebras

To introduce our main notions of superpotentials, we need preparations. For an $\mathbb {F}$-algebra $A$ and an $A^{{\operatorname {en} }}$-module $M$, we write

\[ c(M):=A\otimes_{A^{{\operatorname{en}}}}M \]

for the zeroth Hochschild homology $H_0(A,M)$ of $A$. This can be naturally identified with the quotient of $M$ modulo the subgroup generated by $am-ma$ with $a\in A$ and $m\in M$. Therefore, we have a natural surjective map $\pi :M\twoheadrightarrow A\otimes _{A^{{\operatorname {en} }}}M$ of $\mathbb {F}$-modules.

For $A^{{\operatorname {en} }}$-modules $M_1,\ldots ,M_{\ell}$, we clearly have functorial isomorphisms

(8)\begin{equation} c(M_1\otimes_A\cdots\otimes_AM_{\ell})\cong c(M_2\otimes_A\cdots \otimes_AM_{\ell}\otimes_AM_1)\cong\cdots\cong c(M_{\ell}\otimes_AM_1\otimes_A\cdots\otimes_AM_{\ell-1}) \end{equation}

given by $m_1\otimes\cdots\otimes m_\ell\mapsto m_2\otimes\cdots\otimes m_\ell\otimes m_1\mapsto\cdots\mapsto m_\ell\otimes m_1\otimes\cdots\otimes m_{\ell-1}$. For $M,N\in A^{{\operatorname {en} }}\operatorname {-mod}$, there is a functorial isomorphism

(9)\begin{equation} c(M\otimes_AN)\cong M\otimes_{A^{{\operatorname{en}}}}N \end{equation}

given by $a\otimes (m\otimes n)\mapsto am\otimes n= m\otimes na$, whose inverse is $m\otimes n\mapsto 1\otimes (m\otimes n)$. It gives a functorial morphism

(10)\begin{equation} c(M\otimes_AN)\to\operatorname{Hom}_{A^{{\operatorname{en}}}}(M^{{\vee}},N), \end{equation}

which is an isomorphism if $M$ is a finitely generated projective $A^{{\operatorname {en} }}$-module.

Setting $M=A^{{\operatorname {en} }}$ in (9), we have a functorial isomorphism of $A^{{\operatorname {en} }}$-modules

(11)\begin{equation} c(A^{{\operatorname{en}}}\otimes_A N)\cong N. \end{equation}

For $M,N\in A^{{\operatorname {en} }}\operatorname {-mod}$, we have a well-defined pairing

(12)\begin{equation} \operatorname{ev}_M\otimes 1_N:M^{{\vee}}\otimes_{\mathbb{F}}c(M\otimes_AN)\to N \end{equation}

given as the composition $M^{\vee }\otimes _{\mathbb {F}} c(M\otimes _AN)\to c(A^{{\operatorname {en} }}\otimes _AN)\xrightarrow {(11)} N$, where the first map sends $f\otimes (1_A\otimes (m\otimes n))$ to $f(m) n$.

Now we are ready to introduce the following, which is a central notion in this paper.

By (8), we have a well-defined automorphism

\[ \rho:S\otimes_{S^{{\operatorname{en}}}}U^{\ell}\to S\otimes_{S^{{\operatorname{en}}}}U^{\ell},\quad (x_1\otimes x_2\otimes\cdots\otimes x_{\ell}) \mapsto (x_2\otimes\cdots\otimes x_{\ell}\otimes x_1). \]

Using $\rho$, we define $\varphi$ by

\[ \varphi:=\sum_{i=0}^{\ell-1}({-}1)^{(\ell-1)i}\rho^{i}:S\otimes_{S^{{\operatorname{en}}}}U^{\ell}\to S\otimes_{S^{{\operatorname{en}}}}U^{\ell}. \]

By (12), for $0\leq k\leq \ell$, we have a well-defined pairing

\[ \operatorname{ev}_{U^{k}}\otimes 1_{U^{\ell-k}}:(U^{k})^{{\vee}}\otimes_{\mathbb {F}} c(U^{\ell}) {\xrightarrow {\sim}} (U^{k})^{{\vee}}\otimes_{\mathbb {F}} c(U^{k}\otimes_{S}U^{\ell-k})\xrightarrow{\operatorname{ev}_{U^{k}}\otimes 1_{U^{\ell-k}}} U^{\ell-k}. \]

For $f\in (U^{k})^{\vee }$ and $x\in c(U^{\ell })$, we simply write $f\cdot x:=\operatorname {ev} _{U^{k}}\otimes 1_{U^{\ell -k}}(f\otimes x)$.

Definition 3.5 Let $S$ be a semisimple $\mathbb {F}$-algebra, $U$ an $S^{{\operatorname {en} }}$-module, and $T=\operatorname {T} _S(U)$. For a superpotential $W$ of degree $\ell$ and a non-negative integer $k\le \ell$, the $k$-Jacobi ideal of $T$ is the two-sided ideal

\[ J_S^{k}(U,W)=(f\cdot\varphi(W) \mid f\in\operatorname{Hom}_{S^{{\operatorname{en}}}}(U^{k},S^{{\operatorname{en}}})). \]

The $k$-Jacobi algebra is the quotient algebra

\[ P_S^{k}(U,W)=\operatorname{T}_S(U)/J_S^{k}(U,W). \]

We now explain a connection to notation used elsewhere.

Remark 3.6 Given a quiver $Q$, we have a semisimple algebra $S=\mathbb {F} Q_0$ with basis the vertices of $Q$ and an $S^{{\operatorname {en} }}$-module $U=\mathbb {F} Q_1$ with basis the arrows. For each $i\geq 0$, let $Q_i$ be the set of all paths of length $i$ on $Q$. Then $Q_i$ gives a basis of the $S^{{\operatorname {en} }}$-module $U^{i}$, and we denote by $\{p^{\vee }\mid p\in Q_i\}$ the dual basis of $(U^{i})^{\vee }$ in the obvious sense.

Let $W$ be a superpotential for $\mathbb {F} Q=\operatorname {T} _S(U)$. Define

\[ \partial_{p}W=p^{{\vee}}\cdot \varphi(W). \]

Then the $k$-Jacobi ideal is the ideal of $\operatorname {T} _S(U)$ generated by $\{\partial _{p}W\;|\; p\in Q_k\}$. Note that when $k=1$ and the superpotential $W$ is of odd degree we recover the usual notion of the Jacobi algebra of a quiver with potential $(Q,W)$.

Note also that some sources, such as [Reference Bocklandt, Schedler and WemyssBSW10], define the superpotential to be $\varphi (W)$ rather than $W$.

In the remainder of this section, we give general observations that are used later. Let $A$ be an $\mathbb {F}$-algebra.

Lemma 3.7 For $A^{{\operatorname {en} }}$-modules $X,Y,Z$, we have functorial morphisms

\[ \operatorname{Hom}_{A^{{\operatorname{en}}}}(X^{{\vee}},Y\otimes_AZ)\leftarrow c(X\otimes_AY\otimes_AZ)\to\operatorname{Hom}_{A^{{\operatorname{en}}}}(Y^{{\vee}},Z\otimes_AX). \]

The left (respectively, right) one is an isomorphism if $X$ (respectively, $Y$) is a projective $A^{{\operatorname {en} }}$-module.

As in (4) when $A$ is semisimple, for $A^{{\operatorname {en} }}$-modules $X,Y,Z$, we have functorial isomorphisms

(13)\begin{equation} Y^{{\vee}}\otimes_A\operatorname{Hom}_A(X,A)\cong(X\otimes_AY)^{{\vee}}\quad \mbox{and}\quad \operatorname{Hom}_{A^{{{\operatorname{op}}}}}(Y,A)\otimes_AX^{{\vee}}\cong(X\otimes_AY)^{{\vee}}. \end{equation}

The first map sends $f\otimes g$ to $(x\otimes y\mapsto \sum _ig(xs_i)\otimes s'_i)$ where $f(y)=\sum _is_i\otimes s'_i$, and the second one sends $f'\otimes g'$ to $(x\otimes y\mapsto \sum _jt_j\otimes f'(t'_jy))$ where $g'(x)=\sum _jt_j\otimes t'_j$. We have the following commutative diagram.

(14)

3.3 Higher preprojective algebras of Koszul algebras

Now we go back to the setting in § 3.1, that is, $\Lambda$ is a finite-dimensional $\mathbb {F}$-algebra with global dimension $d>0$. Moreover, we assume that $\Lambda$ is a Koszul algebra and

\[ \Lambda=\operatorname{T}_S(V)/I \]

for a separable $\mathbb {F}$-algebra $S$. Then the minimal $\mathbb {Z}$-graded projective resolution (6) of the $\Lambda ^{{\operatorname {en} }}$-module $\Lambda$ is given by the Koszul bimodule complex (5). Let $\bar {V}=V\oplus K_d^{\vee }$.

Definition 3.8 We define a superpotential $W$ of degree $d+1$ for $\operatorname {T} _S(\bar {V})$ as the image of $1_{\mathbb {F}}\in \mathbb {F}$ under the composition

\[ \mathbb{F}\xrightarrow{\operatorname{coev}_{K_d}} c\big(K_d\otimes_SK_d^{{\vee}}\big) \subset c\big(V^{d}\otimes_SK_d^{{\vee}}\big)\subset c(\bar{V}^{d}\otimes_S\bar{V})=c(\bar{V}^{d+1}), \]

where $\operatorname {coev} _{K_d}:\mathbb {F}\to \operatorname {End} _{S^{{\operatorname {en} }}}(K_d)\cong K_d\otimes _{S^{{\operatorname {en} }}}K_d^{\vee }\cong c\big (K_d\otimes _SK_d^{\vee }\big )$ is the coevaluation map. We call $W$ the superpotential associated to $\Lambda$, or the associated superpotential.

By Lemma 3.7, we have isomorphisms $\operatorname {Hom} _{S^{{\operatorname {en} }}}(K_i,V\otimes _SK_{i-1})\cong \operatorname {Hom} _{S^{{\operatorname {en} }}}(K_{i-1}^{\vee },K_i^{\vee }\otimes _SV)$ and $\operatorname {Hom} _{S^{{\operatorname {en} }}}(K_i,K_{i-1}\otimes _SV)\cong \operatorname {Hom} _{S^{{\operatorname {en} }}}(K_{i-1}^{\vee },V\otimes _SK_i^{\vee })$. Thus, the inclusions $\iota _i^{\ell }:K_i\to V\otimes _SK_{i-1}$ and $\iota _i^{r}:K_i\to K_{i-1}\otimes _SV$ give rise to

(15)\begin{equation} \theta_i^{\ell}:K_{i-1}^{{\vee}}\to K_i^{{\vee}}\otimes_SV\quad \mbox{and}\quad \theta_i^{r}:K_{i-1}^{{\vee}}\to V\otimes_SK_i^{{\vee}}. \end{equation}

We need the following observations.

Proof. (a) By definition, $W$ belongs to the subspace $c\big (K_{d-1}\otimes _SV\otimes _SK_d^{\vee }\big )$ of $c(\bar {V}^{d+1})$, and coincides with $\iota _d^{r}$ under the isomorphism $\operatorname {Hom} _{S^{{\operatorname {en} }}}(K_d,K_{d-1}\otimes _SV)\cong c\big (K_{d-1}\otimes _SV\otimes _SK_d^{\vee }\big )$ in Lemma 3.7. By definition, $\theta _d^{r}$ is the image of $W$ under the isomorphism $c\big (K_{d-1}\otimes _SV\otimes _SK_d^{\vee }\big )\cong \operatorname {Hom} _{S^{{\operatorname {en} }}}\big (K_{d-1}^{\vee },V\otimes _SK_d^{\vee }\big )$ in Lemma 3.7. Thus, $\theta _d^{r}$ coincides with

\[ K_{d-1}^{{\vee}}\xrightarrow{1\otimes W}K_{d-1}^{{\vee}}\otimes_{\mathbb {F}} c\big(K_{d-1}\otimes_SV\otimes_SK_d^{{\vee}}\big) \xrightarrow{\operatorname{ev}_{K_{d-1}}\otimes 1\otimes 1}V\otimes_SK_d^{{\vee}}. \]

On the other hand, because $W$ belongs to $c\big (K_{d-1}\otimes _SV\otimes _SK_d^{\vee }\big )$, the map $-\cdot W$ factors through the surjection $(\bar{V}^{d-1})^\vee\to K_{d-1}^\vee$. Thus, the assertion follows.

(b) Although the argument is mostly the same as (a), we record the details.

By definition, $W$ belongs to the subspace $c\big (V\otimes _SK_{d-1}\otimes _SK_d^{\vee }\big )$ of $c(\bar {V}^{d+1})$, and coincides with $\iota _d^{\ell }$ under the isomorphism $\operatorname {Hom} _{S^{{\operatorname {en} }}}(K_d,V\otimes _SK_{d-1})\cong c\big (V\otimes _SK_{d-1}\otimes _SK_d^{\vee }\big )$ in Lemma 3.7. By definition, $\theta _d^{\ell }$ is the image of $W$ under the isomorphism $c\big (V\otimes _SK_{d-1}\otimes _SK_d^{\vee }\big )\cong \operatorname {Hom} _{S^{{\operatorname {en} }}}\big (K_{d-1}^{\vee },K_d^{\vee }\otimes _SV\big )$ in Lemma 3.7. Thus, $\theta _d^{\ell }$ coincides with

\[ K_{d-1}^{{\vee}}\xrightarrow{1\otimes W}K_{d-1}^{{\vee}}\otimes_{\mathbb {F}} c\big(V\otimes_SK_{d-1} \otimes_SK_d^{{\vee}}\big)\xrightarrow{1\otimes\rho}K_{d-1}^{{\vee}}\otimes_{\mathbb {F}} c\big(K_{d-1} \otimes_SK_d^{{\vee}}\otimes_SV\big)\xrightarrow{\operatorname{ev}_{K_{d-1}}\otimes 1\otimes 1}V\otimes_SK_d^{{\vee}}, \]

which equals $K_{d-1}^{\vee }\xrightarrow {1\otimes \rho (W)}K_{d-1}^{\vee }\otimes _{\mathbb {F}} c\big (K_{d-1}\otimes _SK_d^{\vee }\otimes _SV\big )\xrightarrow {\operatorname {ev} _{K_{d-1}}\otimes 1\otimes 1}V\otimes _SK_d^{\vee }$.

On the other hand, because $\rho (W)$ belongs to $c\big (K_{d-1}\otimes _SK_d^{\vee }\otimes _SV\big )$, the map $-\cdot \rho (W)$ factors through the surjection $(\bar{V}^{d-1})^\vee\to K_{d-1}^\vee$. Thus, the assertion follows.

On the other hand, $\theta _i^{\ell }$ and $\theta _i^{r}$ induce morphisms $\hat {\theta }^{\ell }_i$ and ${\hat {\theta }^{r}_i:\Lambda \otimes _SK_{i-1}^{\vee }\otimes _S\Lambda \to \Lambda \otimes _SK_{i}^{\vee }\otimes _S\Lambda}$ of $\Lambda ^{{\operatorname {en} }}$-modules. This gives an explicit construction of $\delta _i'$ in (7) by the following observation.

Lemma 3.10 If $\Lambda$ is Koszul, then we have a commutative diagram

and therefore $\delta _i'=\hat {\theta }^{\ell }_i+(-1)^{i}\hat {\theta }^{r}_i$.

To prove this, we prepare the following observation.

Lemma 3.11 For $S^{{\operatorname {en} }}$-modules $X$ and $Y$, we have the following commutative diagram.

where we write $(-)^{\vee _{\Lambda} }=\operatorname {Hom} _{\Lambda ^{{\operatorname {en} }}}(-,\Lambda ^{{\operatorname {en} }})$.

Proof of Lemma 3.10 As $\delta _i=\hat {\iota }_i^{\ell }+(-1)^{i}\hat {\iota }_i^{r}$, it suffices to show that the following diagram commutes for $s\in \{\ell ,r\}$.

We just show the $s=r$ version; $s=\ell$ is the dual. We apply Lemma 3.11 to $X:=K_i$ and $Y:=K_{i-1}$. As $\iota _i^{r}\in \operatorname {Hom} _{S^{{\operatorname {en} }}}(K_i,K_{i-1}\otimes _SV)$ corresponds to $\theta _i^{r}\in \operatorname {Hom} _{S^{{\operatorname {en} }}}(K_{i-1}^{\vee },V\otimes _SK_i^{\vee })$, the map ${}_{\Lambda ^{{\operatorname {en} }}}(\hat {\iota }_i^{r},\Lambda ^{{\operatorname {en} }})$ coincides with $\hat {\theta }_i^{r}$ up to the isomorphisms in Lemma 2.3. This gives the commutativity of the diagram.

As $\Lambda$ is Koszul, we can regard $\delta _d'(K_{d-1}^{\vee })\subset V\otimes _SK_d^{\vee }\otimes _SS+S\otimes _SK_d^{\vee }\otimes _SV$ as a subspace of $T\otimes _SK_d^{\vee }\otimes _ST\subset \operatorname {T} _S(\bar {V})$ naturally. Now we show the following assertion.

Proposition 3.12 If $\Lambda =\operatorname {T} _S(V)/(R)$ is a finite-dimensional Koszul algebra of global dimension $d$ with $R\subset V^{2}$, then we have an isomorphism of algebras:

\[ \Pi\cong\operatorname{T}_S(\bar{V})/(R+\delta_d'(K_{d-1}^{{\vee}})). \]

In particular, $\Pi$ is quadratic.

Now we are ready to prove the following.

Theorem 3.13 If $\Lambda =\operatorname {T} _S(V)/(R)$ is a finite-dimensional Koszul algebra of global dimension $d$, then we have an isomorphism of algebras:

\[ \Pi\cong P_S^{d-1}(\bar{V},W)/(R). \]

Proof. The left-hand side is $\operatorname {T} _S(\bar {V})/(R+\delta _d'(K_{d-1}^{\vee }))$ by Proposition 3.12, and the right-hand side is $\operatorname {T} _S(\bar {V})/(R+(\bar {V}^{d-1})^{\vee }\cdot \varphi (W))$ by definition. It suffices to prove $R+\delta _d'(K_{d-1}^{\vee })=R+(\bar {V}^{d-1})^{\vee }\cdot \varphi (W)$.

As $K_d={\bigcap} _{i=2}^{d}V^{i-2}\otimes _SR\otimes _SV^{d-i}$, for each $2\le i\le d$ we have

\[ W\in c\big(K_d\otimes_SK_d^{{\vee}}\big)\subset c\big(V^{i-2}\otimes_SR\otimes_SV^{d-i}\otimes_SK_d^{{\vee}}\big) \]

and, hence, $\rho ^{i}(W)\in c\big (V^{d-i}\otimes _SK_d^{\vee }\otimes _SV^{i-2}\otimes _SR\big )$. Therefore, $(\bar {V}^{d-1})^{\vee }\cdot \rho ^{i}(W)\subset R$ holds. In particular,

\begin{align*} R+(\bar{V}^{d-1})^{{\vee}}\cdot\varphi(W) &= R+(\bar{V}^{d-1})^{{\vee}} \cdot(W+({-}1)^{d}\rho(W))\stackrel{\text{Lemma (3.9)}}{=}R+(\theta_d^{r}+({-}1)^{d}\theta_d^{\ell}) (K_{d-1}^{{\vee}})\\ &\hspace{-1.6pc}\stackrel{\text{Lemma (3.10)}}{=} R+\delta_d'(K_{d-1}^{{\vee}}) \end{align*}

holds as desired.

The extension condition in the following theorem is a special case of the following property of [Reference Iyama and OppermannIO13, Section 3]. Given a $d$-cluster tilting subcategory $\mathscr {U}$ of ${{D^{b}}}(\Lambda )$, we say that $\mathscr {U}$ has the vosnex property (‘vanishing of small negative extensions’) if $\operatorname {Hom} _{{{D^{b}}}(\Lambda )}(\mathscr {U}[j],\mathscr {U})=0$ for $j\in \{1,2,\ldots , d-2\}$. In this case, because $\Lambda ,\Lambda ^{*}[-d]\in \mathscr {U}$, we have $\operatorname {Ext} ^{d-j}_{\Lambda ^{{\operatorname {en} }}}(\Lambda ,\Lambda ^{{\operatorname {en} }})\cong \operatorname {Ext} ^{d-j}_{\Lambda }(\Lambda ^{*},\Lambda )\cong \operatorname {Hom} _{{{D^{b}}}(\Lambda )}(\Lambda ^{*}[j-d],\Lambda )=0$ for $j\in \{1,2,\ldots , d-2\}$.

Theorem 3.14 Suppose $\Lambda$ is a finite-dimensional Koszul algebra of global dimension $d$. If $\operatorname {Ext} ^{i}_{\Lambda ^{{\operatorname {en} }}}(\Lambda ,\Lambda ^{{\operatorname {en} }})_{-i}=0$ for $2\leq i\leq d-1$, then we have an isomorphism of algebras:

\[ \Pi\cong P_S^{d-1}(\bar{V},W). \]

Proof. By Theorem 3.13, it suffices to prove $(\bar {V}^{d-1})^{\vee }\cdot \varphi (W)\supseteq R$. In fact, for each $2\le i\le d$, we prove by downwards induction

(16)\begin{equation} (\bar{V}^{d-i+1})^{{\vee}}\cdot\varphi(W)\supseteq K_i. \end{equation}

First we prove (16) for $i=d$. Consider the decomposition $\bar {V}^{\vee }=V^{\vee }\oplus K_d$. Since $W=\operatorname {coev} _{K_d}(1_{\mathbb {F}})$, we have $K_d\cdot \rho ^{d}(W)=K_d$ and $K_d\cdot \rho ^{i}(W)=0$ for each $0\le i\le d-1$. Thus, $\bar {V}^{\vee }\cdot \varphi (W)\supseteq K_d\cdot \varphi (W)=K_d$ holds.

Next, for each $3\le i\le d$, we prove

(17)\begin{equation} \operatorname{Hom}_{S^{{\operatorname{op}}}}(V,S)\cdot K_{i}+K_{i}\cdot\operatorname{Hom}_S(V,S)=K_{i-1}, \end{equation}

where $\cdot$ are the maps $\operatorname {Hom} _{S^{{\operatorname {op} }}}(V,S)\otimes _SV^{i}\to V^{i-1}$ and $V^{i}\otimes _S\operatorname {Hom} _{S}(V,S)\to V^{i-1}$ given by the evaluations. We use the Koszul resolution together with Lemma 3.10. These tell us that $\operatorname {Ext} ^{i-1}_{\Lambda ^{{\operatorname {en} }}}(\Lambda ,\Lambda ^{{\operatorname {en} }})$ is the $(i-1)$th homology of the complex

\[ 0 \leftarrow \Lambda\otimes_SK_{d}^{{\vee}}\otimes_S\Lambda \leftarrow \Lambda\otimes_SK_{d-1}^{{\vee}}\otimes_S\Lambda \leftarrow \cdots \leftarrow \Lambda\otimes_SK_{1}^{{\vee}}\otimes_S\Lambda \leftarrow \Lambda\otimes_SK_{0}^{{\vee}} \otimes_S\Lambda \leftarrow 0, \]

where the differentials are induced by the maps

\[ \delta'_i=\theta_i^{\ell}+({-}1)^{i}\theta_i^{r}:K_{i-1}^{{\vee}} \to\big(K_{i}^{{\vee}}\otimes_SV\big)\oplus\big(V\otimes_SK_{i}^{{\vee}}\big)\subset\Lambda\otimes_SK_{i}^{{\vee}}\otimes_S\Lambda. \]

This is injective since its kernel is $\operatorname {Ext} ^{i-1}_{\Lambda ^{{\operatorname {en} }}}(\Lambda ,\Lambda ^{{\operatorname {en} }})_{1-i}=0$ by our assumption. Applying $(-)^{\vee }$, we have a surjective map

\[ (\operatorname{Hom}_{S^{{\operatorname{op}}}}(V,S)\otimes_SK_{i})\oplus(K_{i}\otimes_S\operatorname{Hom}_S(V,S)) \stackrel{(13)}{\cong}\big(K_{i}^{{\vee}}\otimes_SV\big)^{{\vee}} \oplus\big(V\otimes_SK_{i}^{{\vee}}\big)^{{\vee}}\xrightarrow{(\delta'_i)^{{\vee}}} K_{i-1}. \]

This is a restriction of the map $(\operatorname {Hom} _{S^{{\operatorname {op} }}}(V,S)\otimes _SV^{i})\oplus (V^{i}\otimes _S\operatorname {Hom} _{S}(V,S))\to V^{i-1}$ given by the evaluations. Thus, (17) holds.

Now assume (16) holds. Applying the upper part of (14) to $(X,Y,Z)=(\bar {V}^{d-i+1},\bar {V},\bar {V}^{i-1})$ and the lower part to $(X,Y,Z)=(\bar {V},\bar {V}^{d-i+1},\bar {V}^{i-1})$, we obtain

\begin{align*} & (\bar{V}^{d-i+2})^{{\vee}}\cdot\varphi(W)\stackrel{(14)}{=} \operatorname{Hom}_{S^{{\operatorname{op}}}}(\bar{V},S)\cdot((\bar{V}^{d-i+1})^{{\vee}}\cdot\varphi(W)) \stackrel{(16)}{\supseteq} \operatorname{Hom}_{S^{{\operatorname{op}}}}(\bar{V},S)\cdot K_i,\\ & (\bar{V}^{d-i+2})^{{\vee}}\cdot\varphi(W)\stackrel{(14)}{=} ((\bar{V}^{d-i+1})^{{\vee}}\cdot\varphi(W))\cdot\operatorname{Hom}_S(\bar{V},S) \stackrel{(16)}{\supseteq} K_i\cdot\operatorname{Hom}_S(\bar{V},S). \end{align*}

Thus, $(\bar {V}^{d-i+2})^{\vee }\cdot \varphi (W)\supseteq \operatorname {Hom} _{S^{{\operatorname {op} }}}(V,S)\cdot K_i+K_i\cdot \operatorname {Hom} _S(V,S)\stackrel {(17)}{=}K_{i-1}$ holds, which completes the induction.

Note that the condition of Theorem 3.14 is vacuous when $d=2$, so this result agrees with Keller's description of $3$-preprojective algebras (see [Reference KellerKel11, Theorem 6.10, Reference Herschend and IyamaHI11, Section 2.2]).

We will see in Corollary 4.3 that this theorem is particularly applicable to $d$-hereditary algebras.

Example 3.15 (a) Consider the quiver

and the algebra $\Lambda =\mathbb {F} Q/(\alpha \beta ,\beta \gamma )$. One can check that it satisfies the conditions of Theorem 3.14 for $d=3$. (In fact, $\Lambda$ is Koszul and $3$-representation finite, see Definition 4.4.) We have $K_3=\langle {\alpha \beta \gamma }\rangle$ so the quiver $\bar {Q}$ of $\Pi =\Pi (\Lambda )$ is

where $\eta =(\alpha \beta \gamma )^{*}$. The superpotential $W$ is represented by $\alpha \beta \gamma {\eta }$ and the space of relations of $\Pi$ is given by $\bar {V}^{-2}\cdot \varphi (W) =\langle {\alpha \beta ,\beta \gamma ,\gamma {\eta },{\eta }\alpha }\rangle$.

(b) Next consider the quiver

and the algebra $\Lambda =\mathbb {F} Q/(R)$ with $R=\langle \alpha \beta ,\beta \gamma ,\delta \varepsilon \rangle$. One can check that $\Lambda$ has global dimension $3$ and is Koszul, but it does not satisfy the condition of Theorem 3.14 as $\operatorname {Ext} ^{2}_{\Lambda }(\Lambda ^{*} e_6,\Lambda e_4)_{-2}\neq 0$. Again, we have $K_3=\langle {\alpha \beta \gamma }\rangle$ so the quiver $\bar {Q}$ of $\Pi (\Lambda )$ is

where ${\eta =}(\alpha \beta \gamma )^{*}$. The superpotential $W$ is represented by $\alpha \beta \gamma {\eta }$ and the $2$-Jacobi ideal is generated by $\bar {V}^{-2}\cdot \varphi (W) =\langle {\alpha \beta ,\beta \gamma ,\gamma {\eta },{\eta }\alpha }\rangle$. We see that this does not include $\delta \varepsilon$, and so to obtain the whole space of relations of $\Pi$ we need to consider $R+\bar {V}^{-2}\cdot \varphi (W)$.

Remark 3.16 It is worth pointing out that higher preprojective algebras are sometimes higher Jacobi algebras even in the non-Koszul case. For example, consider the following example, due to Vaso [Reference VasoVas19, Example 5.3], of an algebra of global dimension 4 that satisfies the Ext-vanishing condition of Theorem 3.14. (In fact, $\Lambda$ is $4$-representation finite, see Definition 4.4.) We take the quiver

and the algebra $\Lambda =\mathbb {F} Q/(\operatorname {rad} \mathbb {F} Q)^{4}=\mathbb {F} Q/(\alpha \beta \gamma \delta , \beta \gamma \delta \varepsilon , \gamma \delta \varepsilon \zeta , \delta \varepsilon \zeta \eta , \varepsilon \zeta \eta \theta )$. We know from Proposition 3.2 that the quiver for $\Pi$ is

where $\iota =(\alpha \beta \gamma \delta \varepsilon \zeta \eta \theta )^{*}$, and one can check that $\Pi$ is, in fact, a $5$-Jacobi algebra: we obtain its relations by differentiating the superpotential represented by $W=\alpha \beta \gamma \delta \varepsilon \zeta \eta \theta \iota$ with respect to paths of length $5$.

We do not know an example of a non-Koszul algebra that satisfies the Ext-vanishing condition of Theorem 3.14 but is not a higher Jacobi algebra.

4.1 Preliminaries on $d$-hereditary algebras

Let $\Lambda$ be a finite-dimensional $\mathbb {F}$-algebra with $\mathop {\textrm {gl.dim}} \Lambda \le d$, and ${{D^{b}}}(\Lambda )$ the derived category of finitely generated left $\Lambda$-modules with bounded homology. Then we have the following result on formality.

Let $\nu$ denote the Nakayama functor

\[ \nu:=\Lambda^{*}\otimes^{\mathbb{L}}_{\Lambda}-:{{D^{b}}}(\Lambda)\overset{\sim}{\rightarrow}{{D^{b}}}(\Lambda) \]

of $\Lambda$ and let $\nu ^{-1}$ denote its quasi-inverse, defined using the internal hom,

\[ \nu^{{-}1}:=\mathbb{R}\strut\kern-.2em\operatorname{Hom}_{\Lambda}(\Lambda^{*},-):{{D^{b}}}(\Lambda)\overset{\sim}{\rightarrow}{{D^{b}}}(\Lambda). \]

Let $\nu _d$ denote the shifted Nakayama functor $\nu _d=\nu \circ [-d]$ and $\nu _d^{-1}=\nu ^{-1}\circ [d]$. Then we have

(18)\begin{equation} \tau_d=H^{0}(\nu_d-):\operatorname{mod}\Lambda\to\operatorname{mod}\Lambda\quad \mbox{and}\quad \tau_d^{-}=H^{0}\big(\nu_d^{{-}1}-\big):\operatorname{mod}\Lambda\to\operatorname{mod}\Lambda. \end{equation}

One of the important properties of $d$-hereditary algebras $\Lambda$ follows from Lemma 4.1: for any $j\in \mathbb {Z}$ and an indecomposable projective $\Lambda$-module $P$, there exists $i\in \mathbb {Z}$ such that

(19)\begin{equation} \nu_d^{j}(P)\cong H^{di}(\nu_d^{j}(P))[{-}di]. \end{equation}

Note that in [Reference Herschend, Iyama and OppermannHIO14], the weaker condition $\operatorname {gldim}\Lambda \leq d$ instead of $\operatorname {gldim}\Lambda =d$ was imposed. The only difference between the two definitions is whether we allow $\Lambda$ to be semisimple, which is a case we are not interested in. Therefore, we always assume $\operatorname {gldim}\Lambda =d$.

The following result is an immediate consequence of Theorem 3.14.

Proof. The assertion is immediate from Theorem 3.14 because

\[ \operatorname{Ext}^{d-i}_{\Lambda^{{\operatorname{en}}}}(\Lambda,\Lambda^{{\operatorname{en}}})\cong\operatorname{Hom}_{{{D^{b}}}(\Lambda)}(\Lambda^{*},\Lambda[d-i]) \cong\operatorname{Hom}_{{{D^{b}}}(\Lambda)}\big(\Lambda[i],\nu_d^{{-}1}(\Lambda)\big)\cong H^{i}\big(\nu_d^{{-}1}(\Lambda)\big)=0 \]

holds for any $0 < i < d$.

Then we have a dichotomy theorem.

In the study of $d$-hereditary algebras, the subcategory

\[ \mathscr{U}:=\operatorname{add}\big\{\nu_d^{i}(\Lambda)\mid i\in\mathbb{Z}\big\} \]

of ${{D^{b}}}(\Lambda )$ plays an important role.

We give a few properties of $\mathscr {U}$ and the categories $\mathscr {P}$ and $\mathscr {I}$ of $d$-preprojective $\Lambda$-modules and $d$-preinjective $\Lambda$-modules (Definition 2.8). By the following result, any $d$-RF algebra has a unique $d$-cluster tilting module up to additive equivalence, which is given by $\Pi$. For a full subcategory $\mathscr {X}$ and $\mathscr {Y}$ of an additive category $\mathscr {C}$, we denote by $\mathscr {X}\vee \mathscr {Y}$ the full subcategory $\operatorname {add} (\mathscr {X}\cup \mathscr {Y})$ of $\mathscr {C}$.

In the final part of our preparations for this section, we recall the generalization of almost split sequences, or Auslander–Reiten sequences, to $d$-hereditary algebras.

Definition 4.7 [Iya07] Let $\mathscr {C}$ be a Krull–Schmidt $\mathbb {F}$-linear category with Jacobson radical $\operatorname {rad} _{\mathscr{C}}$ and let

(20)\begin{equation} Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X \end{equation}

be a complex in $\mathscr {C}$ where $X$ and $Y$ are indecomposable and each $f_i$ belongs to $\operatorname {rad} _{\mathscr{C}}$. Then we say the sequence (20) is $d$-almost split in $\mathscr {C}$ if both of the following sequences are exact for all objects $M$ in $\mathscr {C}$:

\begin{align*} & 0\to\operatorname{Hom}_{\mathscr{C}}(M,Y)\xrightarrow{{f_{d}}_*} \operatorname{Hom}_{\mathscr{C}}(M,C_{d-1})\xrightarrow{{f_{d-1}}_*} \cdots\xrightarrow{{f_{1}}_*} \operatorname{Hom}_{\mathscr{C}}(M,C_0)\xrightarrow{{f_{0}}_*} \operatorname{rad}_{\mathscr{C}}(M,X)\to 0;\\ & 0\to\operatorname{Hom}_{\mathscr{C}}(X,M)\xrightarrow{{f_{0}}^{*}} \operatorname{Hom}_{\mathscr{C}}(C_0,M)\xrightarrow{{f_{1}}^{*}} \cdots\xrightarrow{{f_{d-1}}^{*}} \operatorname{Hom}_{\mathscr{C}}(C_{d-1},M)\xrightarrow{{f_{d}}^{*}} \operatorname{rad}_{\mathscr{C}}(Y,M) \to 0. \end{align*}

More generally, we say the sequence (20) is weak $d$-almost split in $\mathscr {C}$ if these sequences are exact except at $\operatorname{Hom}_{\mathscr{C}}(M,Y)$ and $\operatorname{Hom}_{\mathscr{C}}(X,M)$, respectively.

It was shown in [Reference Herschend, Iyama and OppermannHIO14] (respectively, [Reference IyamaIya07]) that the category $\mathscr {P}\vee \mathscr {I}$ has $d$-almost split sequences when $\Lambda$ is $d$-RI (respectively, $d$-RF). In addition, it was shown in [Reference Iyama and YoshinoIY08, Reference Iyama and OppermannIO13] that $d$-cluster tilting subcategories of triangulated categories have certain analogue of $d$-almost split sequences called AR $(d+2)$-angles. From these results, one can deduce the following results on $d$-almost split sequences in the category $\mathscr {U}$, which play a key role in this section.

Theorem 4.9 Let $\Lambda$ be a $d$-hereditary algebra.

(a) If $\Lambda$ is $d$-RI, then any indecomposable object $X$ (respectively, $Y$) in $\mathscr {U}$ has a $d$-almost split sequence in $\mathscr {U}$

\[ Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X. \]

Moreover, we have $Y\cong \nu _d(X)$ (respectively, $X\cong \nu _d^{-1}(Y)$).

(b) If $\Lambda$ is $d$-RF, then any indecomposable object $X$ (respectively, $Y$) in $\mathscr {U}$ has a weak $d$-almost split sequence in $\mathscr {U}$

\[ Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X. \]

Moreover, we have $Y\cong \nu _d(X)$ (respectively, $X\cong \nu _d^{-1}(Y)$), $\operatorname {Ker} (f_d{}_*)=\operatorname {soc} \operatorname {Hom} _{\mathscr {U}}(-,Y)$ and $\operatorname {Ker} (f_0{}^{*})=\operatorname {soc} \operatorname {Hom} _{\mathscr {U}}(X,-)$.

Proof. In both cases, we only show the assertion for $X$ because the assertion for $Y$ is the dual.

(a) Let $X\in \mathscr {U}$ be an indecomposable object. If $X$ is a projective $\Lambda$-module, then $\nu _d^{-1}(X)$ is not projective, as otherwise $X\cong \nu _d\nu _d^{-1}(X)$ would be concentrated in degree $d$ which contradicts our assumption that $\Lambda$ is $d$-RI. As $\nu _d:\mathscr {U}\to \mathscr {U}$ is an equivalence, it preserves $d$-almost split sequences in $\mathscr {U}$. Thus, we can assume that $X$ is a non-projective object in $\mathscr {P}$.

It was shown in [Reference Herschend, Iyama and OppermannHIO14, Theorem 4.25] that there exists an exact sequence

(21)\begin{equation} 0\to Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X\to 0 \end{equation}

in $\operatorname {mod} \Lambda$ that has terms in $\mathscr {P}$, $Y=\nu _d(X)$, and gives a $d$-almost split sequence in $\mathscr {P}\vee \mathscr {I}$. Thus, because Proposition 4.6(b) implies $Y\notin \mathscr {I}$, which implies $\operatorname{rad}_{\Lambda} (Y,\mathscr {I})=\operatorname {Hom}_{\Lambda} (Y,\mathscr {I})$, the following sequences are exact:

\begin{align*} & 0\to\operatorname{Hom}_{\Lambda}(\mathscr{P},Y)\xrightarrow{{f_{d}}_*} \operatorname{Hom}_{\Lambda}(\mathscr{P},C_{d-1})\xrightarrow{{f_{d-1}}_*} \cdots\xrightarrow{{f_{1}}_*} \operatorname{Hom}_{\Lambda}(\mathscr{P},C_0)\xrightarrow{{f_{0}}_*} \operatorname{rad}_{\Lambda}(\mathscr{P},X)\to 0; \\ & 0\to\operatorname{Hom}_{\Lambda}(X,\mathscr{I})\xrightarrow{{f_{0}}^{*}} \operatorname{Hom}_{\Lambda}(C_0,\mathscr{I})\xrightarrow{{f_{1}}^{*}} \cdots\xrightarrow{{f_{d-1}}^{*}} \operatorname{Hom}_{\Lambda}(C_{d-1},\mathscr{I})\xrightarrow{{f_{d}}^{*}} \operatorname{Hom}_{\Lambda}(Y,\mathscr{I}) \to 0. \end{align*}

Using Serre duality, we have $\operatorname {Hom} _{\Lambda} (\mathscr {P},\mathscr {I})=\operatorname {Hom} _{\mathscr {U}}(\nu ^{-1}_d(\mathscr {I})[-d],\mathscr {P})^{*}$. As $\Lambda$ is $d$-RI, we have $\mathscr {I}\subseteq \nu ^{-1}_d(\mathscr {I})$ by Proposition 4.6(b). Therefore, the latter exact sequence gives an exact sequence

\begin{align*} 0\to\operatorname{Hom}_{\mathscr{U}}(\mathscr{I}[{-}d],Y)\xrightarrow{{f_{d}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{I}[{-}d],C_{d-1})\xrightarrow{{f_{d-1}}_*}\cdots &\xrightarrow{{f_{1}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{I}[{-}d],C_0)\\ &\xrightarrow{{f_{0}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{I}[{-}d],X)\to 0. \end{align*}

As $\mathscr {U}=\mathscr {I}[-d]\vee \mathscr {P}$ by [Reference Herschend, Iyama and OppermannHIO14, Proposition 4.10(c)], the above exact sequences give an exact sequence

\[ 0\to\operatorname{Hom}_{\mathscr{U}}(\mathscr{U},Y)\xrightarrow{{f_{d}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{U},C_{d-1})\xrightarrow{{f_{d-1}}_*} \cdots \xrightarrow{{f_{1}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{U},C_0)\xrightarrow{{f_{0}}_*} \operatorname{rad}_{\mathscr{U}}(\mathscr{U},X)\to 0. \]

Dually, the following sequence is exact:

\[ 0\to\operatorname{Hom}_{\mathscr{U}}(X,\mathscr{U})\xrightarrow{{f_{0}}^{*}} \operatorname{Hom}_{\mathscr{U}}(C_0,\mathscr{U})\xrightarrow{{f_{1}}^{*}} \cdots\xrightarrow{{f_{d-1}}^{*}} \operatorname{Hom}_{\mathscr{U}}(C_{d-1},\mathscr{U})\xrightarrow{{f_{d}}^{*}} \operatorname{rad}_{\mathscr{U}}(Y,\mathscr{U}) \to 0. \]

Thus, the sequence (21) is a $d$-almost split sequence in $\mathscr {U}$.

(b) By [Reference IyamaIya11, Theorem 1.23], $\mathscr {U}$ is a $d$-cluster tilting subcategory of ${{D^{b}}}(\Lambda )$. By [Reference Iyama and YoshinoIY08, Theorem 3.10], there exist triangles

\[ X_{i+1}\xrightarrow{h_{i+1}} C_i\xrightarrow{{g_i}} X_i\to X_{i+1}[1] \]

in ${{D^{b}}}(\Lambda )$ for $0\le i\le d-1$ satisfying the following conditions:

• • $X_0=X$, $X_{d}=\nu _d(X)$, and $C_i\in \mathscr {U}$ for any $0\le i\le d-1$;

• • $\operatorname {Hom} _{\mathscr{U}}(\mathscr {U},C_0)\xrightarrow {{g_{0}}_*} \operatorname {rad} _{\mathscr {U}}(\mathscr {U},X)\to 0$ and $\operatorname {Hom} _{\mathscr{U}}(C_{d-1},\mathscr {U})\xrightarrow {{h_{d}}^{*}} \operatorname {rad} _{\mathscr {U}}(\nu _d(X),\mathscr {U})\to 0$ are exact.

Let $f_d:=h_d$, $f_i:=h_ig_i$, and $f_0:=g_0$. Then we have a complex

\[ \nu_d(X)\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2} C_1\xrightarrow{f_1} C_0\xrightarrow{f_0} X. \]

Moreover, as $\Lambda$ is $d$-RF, $\nu (\mathscr {U})=\mathscr {U}$ by [Reference Iyama and OppermannIO13, Theorem 3.1(1)$\Rightarrow$(3)] and hence $\mathscr {U}[d]=\mathscr {U}$. Thus, by [Reference Iyama and OppermannIO13, Lemma 4.3], we have an exact sequence

\begin{align*} &\cdots\to\operatorname{Hom}_{\mathscr{U}}(\mathscr{U},C_0[{-}d])\xrightarrow{{f_{0}[{-}d]}_*}\operatorname{Hom}_{\mathscr{U}}(\mathscr{U},X[{-}d])\to\\ &\operatorname{Hom}_{\mathscr{U}}(\mathscr{U},\nu_d(X))\xrightarrow{{f_{d}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{U},C_{d-1})\xrightarrow{{f_{d-1}}_*} \cdots\xrightarrow{{f_{1}}_*} \operatorname{Hom}_{\mathscr{U}}(\mathscr{U},C_0)\xrightarrow{{f_{0}}_*} \operatorname{rad}_{\mathscr{U}}(\mathscr{U},X)\to 0. \end{align*}

Thus, $\operatorname {Cok} (f_{0}{}_*:\operatorname {Hom} _{\mathscr{U}}(-,C_0)\to \operatorname {Hom} _{\mathscr {U}}(-,X))$ is a simple $\mathscr {U}$-module, and hence $\operatorname {Ker} (f_d{}_*)=\operatorname {Cok} (f_{0}[-d]{}_*)$ is a simple $\mathscr {U}$-module because $[d]:\mathscr {U}\to \mathscr {U}$ is an autoequivalence. As $X[-d]\in \mathscr {U}$ is indecomposable, $\operatorname {Hom} _{\mathscr {U}}(X[-d],-)$ is an indecomposable projective functor and, thus, it has a simple top. Hence, the $\mathscr {U}$-module $\operatorname {Hom} _{\mathscr {U}}(-,\nu _d(X))\cong \operatorname {Hom} _{\mathscr {U}}(X[-d],-)^{*}$ has a simple socle. Therefore, $\operatorname {Ker} (f_d{}_*)=\operatorname {soc} \operatorname {Hom} _{\mathscr {U}}(\mathscr {U},\nu _d(X))$.

Dually, we have an exact sequence

\[ \operatorname{Hom}_{\mathscr{U}}(X,\mathscr{U})\xrightarrow{{f_{0}}^{*}} \operatorname{Hom}_{\mathscr{U}}(C_0,\mathscr{U})\xrightarrow{{f_{1}}^{*}} \cdots\xrightarrow{{f_{d-1}}^{*}} \operatorname{Hom}_{\mathscr{U}}(C_{d-1},\mathscr{U})\xrightarrow{{f_{d}}^{*}} \operatorname{rad}_{\mathscr{U}}(Y,\mathscr{U}) \to 0 \]

such that $\operatorname {Ker} ({f_{0}}^{*})=\operatorname {soc} \operatorname {Hom} _{\mathscr{U}}(X,\mathscr {U})$. Thus, the assertions hold.

4.2 Resolutions of simple modules over higher preprojective algebras

For the rest of this section, $\Lambda$ is a $d$-hereditary algebra and $\Pi$ is its higher preprojective algebra. We assume that $\Lambda$ is basic and ring-indecomposable. We regard $\Pi$ as a $\mathbb {Z}$-graded algebra with the tensor grading. Then we have an isomorphism

\[ \Pi\cong\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{Hom}_{{{D^{b}}}(\Lambda)}\big(\Lambda,\nu_d^{{-}i}(\Lambda)\big) \]

of $\mathbb {Z}$-graded algebras.

For a group $\Psi$ and a $\Psi$-graded ring $\Gamma$, we denote by $\Gamma \operatorname {-mod^{\Psi }}$ (respectively, $\Gamma \operatorname {-proj^{\Psi }}$) the category of finitely generated (respectively, finitely generated projective) $\Psi$-graded $\Gamma$-modules. We start with the following easy observation.

Lemma 4.10 Let $\mathscr {C}$ be an additive category and $\Psi$ a group acting on $\mathscr {C}$. Assume that $M\in \mathscr {C}$ is an object satisfying $\mathscr {C}=\operatorname {add} \{\psi M\mid \psi \in \Psi \}$. Define a $\Psi$-graded ring by $\Gamma :={\bigoplus} _{\psi \in \Psi }\operatorname {Hom} _{\mathscr {C}}(M,\psi M)$. Then there are equivalences of additive categories

\[ \mathop{\bigoplus}\limits_{\psi\in \Psi}\operatorname{Hom}_{\mathscr{C}}(M,\psi -):\mathscr{C}\to\operatorname{proj^{\Psi}-}\Gamma\quad {\rm and}\quad \mathop{\bigoplus}\limits_{\psi\in \Psi}\operatorname{Hom}_{\mathscr{C}}(-,\psi M):\mathscr{C}\to\Gamma\operatorname{-proj^{\Psi}}. \]

Applying Lemma 4.10 to the category $\mathscr {U}$ and the group $\big \{\nu _d^{-i}\mid i\in \mathbb {Z}\big \}\cong \mathbb {Z}$, we have the following description of the category $\mathscr {U}$.

Proposition 4.11 (a) There are equivalences of additive categories

\begin{align*} & G:=\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{Hom}_{{{D^{b}}}(\Lambda)}\big(\Lambda,\nu_d^{{-}i}(-)\big):\mathscr{U}\to\operatorname{proj^{\mathbb{Z}}-}\Pi,\\ & H:=\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{Hom}_{{{D^{b}}}(\Lambda)}\big(-,\nu_d^{{-}i}(\Lambda)\big):\mathscr{U}\to\Pi\operatorname{-proj^{\mathbb{Z}}}. \end{align*}

In particular, there are equivalences of additive categories

\[ G_*:\operatorname{mod^{\mathbb{Z}}-}\Pi\to\operatorname{mod-}\mathscr{U}\quad {\rm and}\quad H_*:\Pi\operatorname{-mod^{\mathbb{Z}}}\to\mathscr{U}\operatorname{-mod}. \]

(b) The following diagram commutes up to natural isomorphism.

Now we are ready to state the main result of this subsection. It asserts that minimal projective resolutions of $\mathbb {Z}$-graded simple modules over the higher preprojective algebra $\Pi$ of a $d$-hereditary algebra $\Lambda$ are induced from $d$-almost split sequences in $\mathscr {U}$.

Theorem 4.12 Let $X$ be an indecomposable object in $\mathscr {U}$, and

\[ Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X \]

a weak $d$-almost split sequence in $\mathscr {U}$.

1. (a) There exist exact sequences

   \begin{gather*} GY\xrightarrow{Gf_d} GC_{d-1}\xrightarrow{Gf_{d-1}}\cdots\xrightarrow{Gf_2} GC_1\xrightarrow{Gf_1} GC_0\xrightarrow{Gf_0} GX\to T\to 0,\\ HX\xrightarrow{Hf_0} HC_0\xrightarrow{Hf_1}HC_1\xrightarrow{Hf_2}\cdots\xrightarrow{Hf_{d-1}} HC_{d-1}\xrightarrow{Hf_d} HY\to U\to 0 \end{gather*}
   in $\operatorname {mod^{\mathbb {Z}}-}\Pi$ and $\Pi \operatorname {-mod^{\mathbb {Z}}}$, where $T$ and $U$ are simple.

2. (b) If $\Lambda$ is $d$-RI, then $Gf_d$ and $Hf_0$ are monomorphisms.

3. (c) If $\Lambda$ is $d$-RF, then $\operatorname {Ker} Gf_d=\operatorname {soc} GY$ and $\operatorname {Ker} Hf_0=\operatorname {soc} HX$. Moreover, these are simple.

Proof. (a) By Theorem 4.9(a) and (b), we have an exact sequence

\begin{align*} \mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{Hom}_{\mathscr{U}}\big(\nu_d^{i}(\Lambda),Y\big)\xrightarrow{f_d{}_*} \mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{Hom}_{\mathscr{U}}\big(\nu_d^{i}(\Lambda),C_{d-1}\big)\xrightarrow{f_{d-1}{}_*}\cdots &\xrightarrow{f_1{}_*}\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{Hom}_{\mathscr{U}}\big(\nu_d^{i}(\Lambda),C_0\big)\\ &\xrightarrow{f_0{}_*}\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\operatorname{rad}_{\mathscr{U}}\big(\nu_d^{i}(\Lambda),X\big)\to 0. \end{align*}

This gives the first sequence. Dually, we obtain the second sequence. It follows from Proposition 4.11(a) that $T$ and $U$ are simple.

(b),(c) These follow from Theorem 4.9(a) and (b). It follows from Proposition 4.11(a) that $\operatorname {Ker} Gf_d$ and $\operatorname {Ker} Hf_0$ are simple if $\Lambda$ is $d$-RF.

We say that an algebra $A$ is twisted-periodic if, for some $i\geq 1$, $\Omega _{A^{{\operatorname {en} }}}^{i}(A)\cong A_{\sigma}$ as $A^{{\operatorname {en} }}$-modules for some $\sigma \in \operatorname {Aut} (A)$, i.e., the projective resolution of the identity bimodule is periodic up to a twist by some algebra automorphism.

As an application of our results, we have the following result for $d$-RF case. The self-injectivity was first proved in [Reference Iyama and OppermannIO13], and the twisted-periodicity was first proved by Dugas [Reference DugasDug12].

We note that the twisted-periodicity is closely related to the stably Calabi–Yau property (e.g. [Reference Ivanov and VolkovIV14, Theorem 1.8]). In fact, $\Pi$ is known to be stably $(d+1)$-Calabi–Yau [Reference Iyama and OppermannIO13, Theorem 1.1(a)].

As another application our results, we have the following result for $d$-RI case.

Corollary 4.14 says that $\Pi$, with the tensor grading, is a generalized Artin–Schelter regular algebra of dimension $d+1$ and Gorenstein parameter $1$ in the sense of [Reference Martínez-VillaMV98, Reference Martínez-Villa and SolbergMS11, Reference Minamoto and MoriMM11, Reference Reyes and RogalskiRR18] (see also [Reference Artin and SchelterAS87]). This is equivalent to a result [Reference Minamoto and MoriMM11, Theorem 4.2] of Minamoto and Mori up to [Reference Reyes and RogalskiRR18, Theorem 5.2], and also can be deduced from results of Keller [Reference KellerKel11].

4.3 $\mathbb {Z}^{2}$-graded higher preprojective algebras

Here, we consider gradings on higher preprojective algebras, which are used in § 4.4. Let $\Lambda$ be a $\mathbb {Z}$-graded algebra

\[ \Lambda =\mathop{\bigoplus}\limits_{i\in\mathbb{Z}}\Lambda_i. \]

The enveloping algebra $\Lambda ^{{\operatorname {en} }}$ of $\Lambda$ has a $\mathbb {Z}$-grading given by

\[ (\Lambda^{{\operatorname{en}}})_i=\mathop{\bigoplus}\limits_{i=j+k}\Lambda_j\otimes_{\mathbb {F}}\Lambda_k. \]

Using the $\mathbb {Z}$-grading on $\Lambda$, we define a new $\mathbb {Z}$-grading on the higher preprojective algebra $\Pi$.

For $i>0$ and finitely generated $\mathbb {Z}$-graded $\Lambda$-modules $M$ and $N$, let $\operatorname {ext} ^{i}_{\Lambda} (M,N)$ denote the $\mathbb {Z}$-graded $i$th ext space (our notation follows [Reference Beilinson, Ginzburg and SoergelBGS96, Section 2.1]). Then we have an equality

\[ \operatorname{Ext}^{i}_{\Lambda}(M,N)=\mathop{\bigoplus}\limits_{j\in\mathbb{Z}}\operatorname{ext}^{i}_{\Lambda}(M,N({j})). \]

Hence, $\operatorname {Ext} ^{i}_{\Lambda} (M,N)$ has a $\mathbb {Z}$-grading whose degree $j$ part is $\operatorname {ext} ^{i}_{\Lambda} (M,N({j}))$.

Now we define the $\mathbb {Z}$-grading on the $\Lambda ^{{\operatorname {en} }}$-module $E=\operatorname {Ext} ^{d}_{\Lambda} (\Lambda ^{*},\Lambda )$ by

(22)\begin{equation} E=\mathop{\bigoplus}\limits_{j\in \mathbb{Z}}\operatorname{ext}^{d}_{\Lambda}(\Lambda^{*},\Lambda({j})). \end{equation}

Then, as in Lemma 2.9, we can show that there are isomorphisms

\[ E\cong\mathop{\bigoplus}\limits_{j\in \mathbb{Z}}\operatorname{ext}^{d}_{\Lambda^{{\operatorname{en}}}}(\Lambda,\Lambda^{{\operatorname{en}}}({j}))\cong \mathop{\bigoplus}\limits_{j\in \mathbb{Z}}\operatorname{ext}^{d}_{\Lambda^{{\operatorname{op}}}}(\Lambda^{*},\Lambda({j})) \]

of $\mathbb {Z}$-graded $\Lambda ^{{\operatorname {en} }}$-modules. Let $\Lambda \operatorname {-mod^{\mathbb {Z}}}$ denote the category of finitely generated $\mathbb {Z}$-graded left $\Lambda$-modules. We lift the functors $\tau _d$ and $\tau _d^{-}$ to $\mathbb {Z}$-graded $\Lambda$-modules as follows:

\[ \tau_d:=\operatorname{Hom}_{\Lambda}(E,-):\Lambda\operatorname{-mod^{\mathbb{Z}}}\to\Lambda\operatorname{-mod^{\mathbb{Z}}}\quad \mbox{and}\quad \tau_d^{-}:=E\otimes_{\Lambda}-:\Lambda\operatorname{-mod^{\mathbb{Z}}}\to\Lambda\operatorname{-mod^{\mathbb{Z}}}. \]

Definition 4.15 (a) The $\mathbb {Z}^{2}$-graded $(d+1)$-preprojective algebra of a $\mathbb {Z}$-graded algebra $\Lambda ={\bigoplus} _{i\in \mathbb {Z}}\Lambda _i$ is the tensor algebra of the $\mathbb {Z}$-graded $\Lambda ^{{\operatorname {en} }}$-module $E$:

\[ \Pi(\Lambda)=\operatorname{T}_{\Lambda}(E). \]

The first part of the $\mathbb {Z}^{2}$-grading is the tensor grading (Definition 2.11). The second part of the $\mathbb {Z}^{2}$-grading is called the $\Lambda$-grading, which is a natural grading on $E^{i}$ for any $i\ge 0$ given by the $\mathbb {Z}$-grading on $E$ in (22).

(b) We consider a single $\mathbb {Z}$-grading on $\Pi$, called the $(d+1)$-total grading, by defining

\[ \Pi_{\ell}:=\mathop{\bigoplus}\limits_{(d+1)i+j=\ell}\Pi_{i,j}, \]

where $\Pi _{i,j}=(E^{i})_j$ denotes the $j$th graded component of $E^{i}$.

Later we use the following observation.

4.4 Koszul properties of higher preprojective algebras

Let $\Lambda$ be a $d$-hereditary $\mathbb {F}$-algebra. In this section, we further assume that $\Lambda$ is a $\mathbb {Z}$-graded algebra $\Lambda ={\bigoplus} _{i\in \mathbb {Z}}\Lambda _i$. We denote by ${{D^{b}}}(\Lambda \operatorname {-mod^{\mathbb{Z}}})$ the bounded derived category of $\Lambda \operatorname {-mod^{\mathbb{Z}}}$. As in the ungraded case, we define an autoequivalence

\[ \nu_d=\Lambda^{*}[{-}d]\otimes^{\mathbb{L}}_{\Lambda}-:{{D^{b}}}(\Lambda\operatorname{-mod^{\mathbb{Z}}})\to{{D^{b}}}(\Lambda\operatorname{-mod^{\mathbb{Z}}}) \]

and a full subcategory

\[ \mathscr{U}^{\mathbb{Z}}:=\operatorname{add}\big\{\nu_d^{{-}i}(\Lambda)(j)\mid i,j\in\mathbb{Z}\big\}\subseteq{{D^{b}}}(\Lambda\operatorname{-mod^{\mathbb{Z}}}). \]

We have the following graded version of Theorem 4.9.

Theorem 4.17 Let $\Lambda$ be a $\mathbb {Z}$-graded $d$-hereditary algebra.

(a) If $\Lambda$ is $d$-RI, then any indecomposable object $X$ (respectively, $Y$) in $\mathscr {U}^{\mathbb {Z}}$ has a $d$-almost split sequence in $\mathscr {U}^{\mathbb {Z}}$

\[ Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X. \]

Moreover, we have $Y\cong \nu _d(X)$ (respectively, $X\cong \nu _d^{-1}(Y)$).

(b) If $\Lambda$ is $d$-RF, then any indecomposable object $X$ (respectively, $Y$) in $\mathscr {U}^{\mathbb {Z}}$ has a weak $d$-almost split sequence in $\mathscr {U}^{\mathbb {Z}}$

\[ Y\xrightarrow{f_d} C_{d-1}\xrightarrow{f_{d-1}} C_{d-2}\xrightarrow{f_{d-2}}\cdots\xrightarrow{f_2}C_1\xrightarrow{f_1}C_0\xrightarrow{f_0} X. \]

Moreover, we have $Y\cong \nu _d(X)$ (respectively, $X\cong \nu _d^{-1}(Y)$), $\operatorname {Ker} (f_d{}_*)=\operatorname {soc} \operatorname {Hom} _{\mathscr {U}}(-,Y)$ and $\operatorname {Ker} (f_0{}^{*})=\operatorname {soc} \operatorname {Hom} _{\mathscr {U}}(X,-)$.