May 16, 2025
Bäckström orders are a class of algebras over complete discrete valuation rings. Their Cohen-Macaulay representations are in correspondence with the representations of certain quivers/species by Ringel and Roggenkamp. In this paper, we give explicit descriptions of the singularity categories of Bäckström orders via certain von Neumann regular algebras and associated bimodules. We further provide singular equivalences between Bäckström orders and specific finite dimensional radical square zero algebras. We also classify weakly regular, Gorenstein, Iwanaga-Gorenstein and sg-Hom-finite Bäckström orders.
The study of Cohen-Macaulay modules is an active branch of representation theory of associative algebras, having an important connection with algebraic geometry and physics [@Yos90; @Sim92; @LW12; @IW14]. By definition, an \(R\)-order \(\Lambda\) is an \(R\)-algebra over a complete regular local ring which is free of finite rank over \(R\). This notion generalizes finite dimensional algebras to the case where the base rings have positive Krull dimension. Cohen-Macaulay modules are widely studied for such class of algebras, especially when \(\operatorname{dim}R=1\) [@CR90; @HN94; @Rei03].
A basic class of orders is called regular orders (see Figure (1 )), i.e. \(\operatorname{gl.dim}\Lambda=\operatorname{dim}R\). For instance, if \(R\) is a field, then \(\Lambda\) is regular if and only if \(\Lambda\) is semisimple; if \(R\) is a complete discrete valuation ring, then \(\Lambda\) is regular if and only if \(\Lambda\) is hereditary. In this case, \(\operatorname{CM}\Lambda = \operatorname{proj}\Lambda\). The well-known Artin-Wedderburn theorem characterizes regular orders over fields. A similar characterization holds for regular orders over complete discrete valuation rings (see Theorem 35).
Semisimple algebras are also characterized by the property that their Jacobson radicals are trivial. From this perspective, finite dimensional radical square zero algebras are close to semisimple algebras. We have \(\operatorname{ind}(\operatorname{mod}A) \xleftrightarrow{1:1} \operatorname{ind}(\operatorname{mod}H')\) for a radical square zero algebra \(A\), where \[H'= \begin{pmatrix} A / \operatorname{rad}A & 0 \\ \operatorname{rad}A & A / \operatorname{rad}A \end{pmatrix}.\] The representation theory of \(A\) is tractable since \(H\) is a finite dimensional hereditary algebra, whose representations correspond to the representations of a certain acyclic quiver/species. Moreover, a stable equivalence exists: \(\operatorname{\underline{mod}}A \simeq \operatorname{\underline{mod}}H'\). See [@Gab73] and [@ARS97].
For \(\operatorname{dim}R=1\), a class of orders which are ideal-theoretically close to hereditary orders is called Bäckström orders. By definition, a Bäckström order \(\Lambda=(\Lambda,\Gamma)\) contains a pair of orders \(\Lambda\subseteq \Gamma\) such that \(\Gamma\) is a hereditary order and \(\operatorname{rad}\Lambda =\operatorname{rad}\Gamma\); see Definition 36. They include several notable families of orders arising naturally in topology and geometry, such as ribbon graph orders [@KR01; @Gne19], gentle orders [@Gne25], nodal orders [@BD24a; @BD24b] and quadratic orders [@Iya05]. Furthermore, this framework has been fruitfully extended to the broader notion of Bäckström rings [@Dro23], which contain important classes of algebras such as gentle and skew-gentle algebras.
The representation theory of Bäckström orders was developed by Ringel and Roggenkamp [@RR79] (see also Green and Reiner [@GR78]). We have \(\operatorname{ind}(\operatorname{CM}\Lambda) \xleftrightarrow{\text{almost 1:1}} \operatorname{ind}(\operatorname{mod}H)\), where \[H= \begin{pmatrix} \Gamma / \operatorname{rad}\Gamma & 0 \\ \Gamma / \operatorname{rad}\Gamma & \Lambda / \operatorname{rad}\Lambda \end{pmatrix}\] is a finite dimensional hereditary algebra; see Theorem 40. Based on this correspondence, we provide a categorical equivalence.
Theorem 1 (=Theorem 44). Assume that \(\Lambda=(\Lambda,\Gamma)\) is a Bäckström order and keep the notations above. Then we have a categorical equivalence \(\operatorname{\underline{CM}}\Lambda \simeq \operatorname{\underline{mod}}_s H\), where \(\operatorname{\underline{mod}}_s H\) is the full subcategory of \(\operatorname{\underline{mod}}H\) whose objects do not contain any non-zero simple direct summand.
We have seen that the representation theory of radical square zero algebras and Bäckström orders behaves similarly: both correspond to hereditary algebras of the form of upper triangle matrices. Motivated by this phenomenon, we purpose the following slogan:
Slogan 2. Bäckström orders are one-dimensional counterparts of finite dimensional radical square zero algebras.
In this paper, we focus on the singularity category of Bäckström orders and provide further evidence to support this slogan.
The singularity category \(\operatorname{\rm{D}_{sg}}(\Lambda)\) of a noetherian ring \(\Lambda\) was introduced by Buchweitz [@Buc97]. It is defined as the Verdier quotient of the bounded derived category \(\rm{D}^{\rm{b}}(\Lambda)\) of \(\operatorname{mod}\Lambda\) by the bounded homotopy category \(\rm{K}^{\rm{b}}(\operatorname{proj}\Lambda)\). The singularity category captures the stable homological property of a noetherian ring. More precisely, as shown in [@KV87], the singularity category is equivalent to the stabilization \(\mathcal{S}(\operatorname{\underline{mod}}\Lambda, \Omega)\) of the stable module category \(\operatorname{\underline{mod}}\Lambda\) with the syzygy functor \(\Omega\) on it, where the morphism sets are direct limits of the syzygy functor; see Section 2.1. We will use this fact to study and give an explicit description of the singularity category of a Bäckström order. To illustrate our result, we introduce our settings.
Assume that \(\Lambda=(\Lambda,\Gamma)\) is a Bäckström order. Let \(\operatorname{Ker}\mu\) be the kernel of the multiplication map \(\mu:\Gamma \otimes_R \Lambda \rightarrow\mathrel{\mkern-14mu}\rightarrow\Gamma\). Let \(D:=\operatorname{\underline{End}}_\Lambda (\Gamma)\) and \(M:=\operatorname{\underline{Hom}}_\Lambda(\Gamma,\operatorname{Ker}\mu)\). Then \(M\) has a \((D,D)\)-bimodule structure which induces a functor \(?\otimes_D M: \operatorname{mod}D \to \operatorname{mod}D\); see Proposition 49. Let \(M^{\otimes 0}:=D\) and \(M^{\otimes i}:=M\otimes_D M^{\otimes i-1}\) for \(i\geq1\). Define \[V(\Lambda):=\underset{i\geq0}{\varinjlim} \operatorname{End}_D(M^{\otimes i}) \quad {\rm{and}} \quad K(\Lambda):=\underset{i\geq1}{\varinjlim} \operatorname{Hom}_D(M^{\otimes i}, M^{\otimes i-1}).\] The following is our first main result.
Theorem 3 (=Theorem 51). Assume that \(\Lambda=(\Lambda,\Gamma)\) is a Bäckström order. Then \(V(\Lambda)\) is a von Neumann regular algebra (see Theorem-Definition 11) and there are triangle equivalences \[(\operatorname{\rm{D}_{sg}}(\Lambda), [1]) \simeq \mathcal{S}(\operatorname{\underline{mod}}\Lambda, \Omega) \simeq (\operatorname{proj}V(\Lambda),? \otimes_{V(\Lambda)} K(\Lambda)),\] where the triangulated structures are trivial (see Section 2.1).
For a finite dimensional radical square zero algebra, a similar description was obtained by Xiao-Wu Chen using the theory of stabilization [@Che11]; see Theorem 14. We can see that our result and Chen’s result exhibit a close resemblance to each other, which provides another evidence for the guiding analogy in our slogan.
By a singular equivalence between two noetherian rings, we mean a triangle equivalence between their singularity categories. It is a weaker version of derived equivalence. Combining together Theorem 3 and Chen’s result, we construct a singular equivalence between a Bäckström order and a finite dimensional radical square zero algebra. This provides another connection between these two classes of algebras.
Corollary 4 (=Corollary 52). Let \(\Lambda=(\Lambda,\Gamma)\) be a Bäckström order and keep the notations as in Theorem 3. Define \(A(\Lambda)\) as the trivial extension \[A(\Lambda)=D\oplus M,\] which is a finite dimensional radical square zero algebra. Then we have a triangle equivalence \[\operatorname{\rm{D}_{sg}}(\Lambda)\simeq\operatorname{\rm{D}_{sg}}(A(\Lambda)).\]
Within the framework of orders, there are several notable classes whose hierarchy is presented below. Let \(\Lambda\) be an \(R\)-order. Then we have the following hierarchy: \[\label{eqa:hierarchy} \begin{tikzcd}[column sep={between origins, 14em}, row sep=large] \fbox{\begin{tabular}{c} Regular\\ (\operatorname{gl.dim}\Lambda=\operatorname{dim}R) \end{tabular}} \arrow[r,Rightarrow] \arrow[d,Rightarrow,to path={([yshift=-3.8em]\tikztostart.north) -- ([yshift=0em]\tikztotarget.north)}] & \fbox{\begin{tabular}{@{}c@{}} Gorenstein\\ (\operatorname{id}_{\Lambda}\Lambda=\operatorname{id}_{\Lambda^{\operatorname{op}}} \Lambda=\operatorname{dim}R) \end{tabular}} \arrow[d,Rightarrow] & \\ \fbox{\begin{tabular}{c} Weakly regular \\ (\operatorname{gl.dim}\Lambda < \infty)\end{tabular}} \arrow[r,Rightarrow, to path={([xshift=0.8em]\tikztostart.east) -- ([xshift=-0.35em]\tikztotarget.west)}] & \fbox{\begin{tabular}{c} Iwanaga-Gorenstein\\ (\operatorname{id}_{\Lambda}\Lambda=\operatorname{id}_{\Lambda^{\operatorname{op}}} \Lambda< \infty) \end{tabular}} \arrow[r,Rightarrow, to path={([xshift=0.2em]\tikztostart.east) -- ([xshift=-0.25em]\tikztotarget.west)}]& \fbox{\begin{tabular}{c} sg-Hom-finite \\ (\operatorname{\rm{D}_{sg}}(\Lambda) is Hom-finite) \end{tabular}} \end{tikzcd}\tag{1}\]
If \(R\) is a field, then \(\Lambda\) is Gorenstein if and only if \(\Lambda\) is self-injective. The singular equivalence in Corollary 4 additionally preserves some important homological properties. As application, we use it to classify some important classes of Bäckström orders. For the finite dimensional algebra \(A(\Lambda)\), we may assign a valued quiver \(Q_{A(\Lambda)}\) to it; see Section 4.2.
Theorem 5 (=Corollary 56, Corollary 57, Theorem 61, Theorem 62). Let \(\Lambda\) be a Bäckström order, \(A(\Lambda)\) the finite dimensional radical square zero algebra associated to it and \(Q_{A(\Lambda)}\) the valued quiver of \(A(\Lambda)\). We have the following criteria.
\(\Lambda\) is weakly regular if and only if \(A(\Lambda)\) is weakly regular if and only if \(Q_{A(\Lambda)}\) is acyclic.
\(\Lambda\) is Gorenstein if and only if \(A(\Lambda)\) is a product of some finite dimensional non-simple self-injective radical square zero algebras if and only if each connected component of \(Q_{A(\Lambda)}\) is a cycle with trivial valuations.
\(\Lambda\) is Iwanaga-Gorenstein if and only if \(A(\Lambda)\) is Iwanaga-Gorenstein if and only if each connected component of \(Q_{A(\Lambda)}\) is acyclic or is a cycle with trivial valuations.
\(\Lambda\) is sg-Hom-finite if and only if \(A(\Lambda)\) is sg-Hom-finite if and only if \(Q_{A(\Lambda)}\) is obtained from a disjoint union of oriented cycles with trivial valuations by adjoining sources and sinks with arbitrary values repeatedly.
This paper is organized as follows. In Section 2, we recall the theory of the stabilization of a left triangulated category and collect relevant facts about von Neumann regular rings and Gorenstein-projective modules. These will be used to study the singularity categories and to classify the specific classes of Bäckström orders. Sections 3.1 and 3.2 review the general theory of orders and the representation theory of Bäckström orders developed by Ringel and Roggenkamp. Section 4 is devoted to proving our main results, followed by the classification criteria. Finally in Section 5, we provide some examples.
::: {#conventions .conventions and notations} Conventions and Notations 1. Throughout this article, all modules are considered as finitely generated right modules unless otherwise specified. For a noetherian ring \(\Lambda\), we denote by \(\operatorname{mod}\Lambda\) the category of finitely generated \(\Lambda\)-modules. Let \(\mathcal{C}\) be an additive category. We denote by \(\operatorname{ind}\mathcal{C}\) the isomorphism classes of the objects in \(\mathcal{C}\). Let \(M\in\mathcal{C}\). We denote by \(\operatorname{add}M=\operatorname{add}_\mathcal{C} M\) the smallest subcategory of \(\mathcal{C}\) containing \(M\) which is closed under finite direct sums, direct summands and isomorphism. If \(\mathcal{C}=\operatorname{mod}\Lambda\), we abbreviate \(\operatorname{add}_{\operatorname{mod}\Lambda} M\) as \(\operatorname{add}_\Lambda M\) and denote by \(\operatorname{proj}\Lambda := \operatorname{add}_\Lambda \Lambda\) the category of finitely generated projective \(\Lambda\)-modules. Denote \(\operatorname{\underline{mod}}\Lambda\) as the stable module category of \(\Lambda\) and \(\Omega: \operatorname{\underline{mod}}\Lambda \to \operatorname{\underline{mod}}\Lambda\) as the syzygy functor. For \(M\in \operatorname{mod}\Lambda\), \(\operatorname{\underline{add}}_\Lambda M\) is defined as \(\operatorname{add}_{\operatorname{\underline{mod}}\Lambda} M\). We drop the subscripts if the base ring is clear. :::
Let \(\Lambda\) be a noetherian ring, \(\rm{D}^{\rm{b}}(\Lambda)\) the bounded derived category of \(\operatorname{mod}\Lambda\) and \(\rm{K}^{\rm{b}}(\operatorname{proj}\Lambda)\) the bounded homotopy category of \(\operatorname{proj}\Lambda\). The singularity category \(\operatorname{\rm{D}_{sg}}(\Lambda)\) of \(\Lambda\) is the Verdier quotient \(\operatorname{\rm{D}_{sg}}(\Lambda):=\rm{D}^{\rm{b}}(\Lambda)/\rm{K}^{\rm{b}}(\operatorname{proj}\Lambda)\). Denote by [1] the suspension functor of \(\operatorname{\rm{D}_{sg}}(\Lambda)\) and by \(q: \rm{D}^{\rm{b}}(\Lambda) \to \operatorname{\rm{D}_{sg}}(\Lambda)\) the quotient functor. The composition \(\operatorname{mod}A \hookrightarrow \mathbf{\rm{D}}^{\rm{b}}(\Lambda) \xrightarrow{q} \operatorname{\rm{D}_{sg}}(\Lambda)\) induces a functor \(\operatorname{\underline{mod}}A \to \operatorname{\rm{D}_{sg}}(\Lambda)\) since projectives vanish, which is also denoted by \(q\).
The so-called stabilization is a useful tool to study singularity categories. It was first studied by Heller [@Hel68] in the context of topology and later studied in general settings by Keller and Vosseick [@KV87] and Beligiannis [@Bel00]. Before introducing stabilizations, we recall the notion of left triangulated categories, which are one-sided versions of triangulated categories. We refer to [@Che18] for an introduction to left triangulated categories and their stabilizations.
A left triangulated category \((\mathcal{C},\Omega,\Delta)\) consists of an additive category \(\mathcal{C}\), an additive endofunctor \(\Omega\) called a loop functor on \(\mathcal{C}\), and a class \(\Delta\) of sequences in \(\mathcal{C}\) called left triangles having the form \(\Omega Z \to X \to Y \to Z\) which satisfy the left-sided version of the axioms of a triangulated category. In particular, every triangulated category is left triangulated. The notions of a left triangle functor and a left triangulated subcategory are defined similarly to those in the theory of triangulated categories; see [@BM94].
Remark 6. It is clear that a left triangulated category is triangulated if and only if \(\Omega\) is an autoequivalence. We call a left triangulated subcategory \((\mathcal{B}, \Omega)\) of \((\mathcal{C}, \Omega)\) a triangulated subcategory of \((\mathcal{C}, \Omega)\) if \((\mathcal{B}, \Omega)\) is itself a triangulated category or equivalently \(\Omega|_\mathcal{B}\) gives an autoequivalence of \(\mathcal{B}\).
Recall that an abelian category is called semisimple if each short exact sequence splits. The (left) triangulated structure of a (left) triangulated category is called trivial if each (left) triangle is isomorphic to a direct sum of some trivial (left) triangles. The following properties are elementary.
Lemma 7. [@Che11]Let \(\mathcal{C}\) be a semisimple abelian category and \(\Omega\) be an endofunctor (resp. autoequivalence) on \(\mathcal{C}\). There exists exactly one left triangulated structure (resp. triangulated structure) on \(\mathcal{C}\), i.e. the trivial structure, with respect to \(\Omega\) as the loop functor (resp. cosuspension functor).
The stabilization \((\mathcal{S}(\mathcal{C}),S)\) of a left triangulated category \((\mathcal{C},\Omega, \Delta)\) is a triangulated category which formally inverts the loop functor \(\Omega\). It contains a triangulated category \(\mathcal{S}(\mathcal{C})=(\mathcal{S}(\mathcal{C}), \tilde{\Omega}, \tilde{\Delta})\) where \(\tilde{\Omega}\) is the cosuspension functor and a left triangle functor \(S: \mathcal{C} \to \mathcal{S}(\mathcal{C})\) called the stabilization functor. The construction of \((\mathcal{S}(\mathcal{C}),S)\) is as follows. The objects are \([M,m]\), where \(M \in \mathcal{C}\) and \(m\in \mathbb{Z}\). The morphism set of \([M,m],[N,n]\in\mathcal{S}(\mathcal{C})\) is \[\operatorname{Hom}_{\mathcal{S}(\mathcal{C})}([M,m],[N,n])=\underset{i\geq{m,n}}{\varinjlim} \operatorname{Hom}_\mathcal{C}(\Omega^{i-m}M,\Omega^{i-n}N).\] The cosuspension functor \(\tilde{\Omega}\) sends \([M,m]\) to \([M,m-1]\) acting in an obvious way on morphisms. Moreover, there is a natural isomorphism \([M,m-1]=\tilde{\Omega}[M,m]\simeq[\Omega(M),m]\). The stabilization functor \(S: \mathcal{C} \to \mathcal{S}(\mathcal{C})\) sends \(M\) to \([M,0]\) on objects and \(f:M \to N\) to the zero-representative on morphisms. We remark that the stabilization functor is neither full nor faithful in general.
The stabilization satisfies the following universal property: for any triangulated category \(\mathcal{D}\) and any left triangle functor \(F:\mathcal{C} \to \mathcal{D}\), there exists a unique triangle functor \(\tilde{F}: \mathcal{S}(\mathcal{C}) \to \mathcal{D}\) such that \(\tilde{F}S=F\). Consider a left triangle functor \(F: (\mathcal{C},\Omega) \to (\mathcal{C'}, \Omega')\) between two left triangulated categories. By the universal property, it induces a triangle functor \(\tilde{F}\) such that the following diagram commutes. \[\begin{CD} (\mathcal{C},\Omega) @>S>> S(\mathcal{C},\Omega) \\ @VFVV @V\tilde{F}VV \\ (\mathcal{C'},\Omega') @>S'>> S(\mathcal{C'},\Omega') \end{CD}\] The following lemma provides a criterion for when \(\tilde{F}\) is an equivalence.
Lemma 8. [@Che18]The following statements hold.
\(\tilde{F}\) is faithful if for any morphism \(g: FX \to FY\), there exists a morphism \(f: \Omega^i(X) \to \Omega^i(Y)\) for some \(i \geq 0\) such that \(\Omega'^i(g)=\delta_Y^i \circ F(f) \circ (\delta_X^i)^{-1}\). Here \(\delta: \Omega' \circ F \xrightarrow{\simeq} F \circ \Omega\) denotes the natural isomorphism of the left triangle functor \(F\).
\(\tilde{F}\) is full if and only if for any two morphisms \(f,f': X \to Y\) in \(\mathcal{C}\) such that \(F(f)=F(f')\), there exists \(i\geq 0\) such that \(\Omega^i(f)=\Omega^{i}(f')\).
\(\tilde{F}\) is dense if and only if for any \(C' \in \mathcal{C'}\), there exist \(i\geq 0\) and \(X\in \mathcal{C}\) such that \(\Omega'^{i}(C')\simeq F(X)\).
In particular, if \((\mathcal{C}, \Omega)\) is a left triangulated subcategory of \((\mathcal{C'}, \Omega)\), \(F\) is the inclusion functor and there exists \(i\geq 0\) such that \(\Omega^{i}(\mathcal{C'}) \subseteq \mathcal{C}\), then \(\tilde{F}: (\mathcal{C}, \Omega) \xrightarrow{\simeq} (\mathcal{C'}, \Omega)\).
A typical example of a left triangulated category is the stable module category \(\operatorname{\underline{mod}}\Lambda\) of a noetherian ring \(\Lambda\). The loop functor is given by the syzygy functor \(\Omega: \operatorname{\underline{mod}}\Lambda \to \operatorname{\underline{mod}}\Lambda\). An exact sequence \(0 \to L \to M \to N \to 0\) in \(\operatorname{mod}\Lambda\) gives a left triangle \(\Omega N \to L \to M \to N\) and any left triangle is isomorphic to a left triangle obtained in this way; see [@BM94]. For the stable module category \(\operatorname{\underline{mod}}\Lambda\), we denote its stabilization by \(\mathcal{S}(\Lambda):=\mathcal{S}(\operatorname{\underline{mod}}\Lambda, \Omega)\). The canonical functor \(q: \operatorname{\underline{mod}}\Lambda \to \operatorname{\rm{D}_{sg}}(\Lambda)\) is in fact a left triangle functor. By the universal property of stabilizations, we have a triangle functor \(\tilde{q}: (\mathcal{S}(\Lambda),\tilde{\Omega}^{-1}) \to (\operatorname{\rm{D}_{sg}}(\Lambda),[1])\).
Theorem 9. [@Bel00; @KV87] The functor \(\tilde{q}: (\mathcal{S}(\Lambda),\tilde{\Omega}^{-1}) \to (\operatorname{\rm{D}_{sg}}(\Lambda),[1])\) is a triangle equivalence. As a consequence, we have \[\operatorname{Hom}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(M,N)\simeq \underset{i\geq0}{\varinjlim} \operatorname{\underline{Hom}}_\Lambda(\Omega^{i}M,\Omega^{i}N)\] for any \(M,N\in \operatorname{mod}\Lambda\).
The following proposition will be useful in the proof of our main theorem. It is included in [@Che11]. For the convenience of readers, we provide a brief proof here. Recall that an additive category is called idempotent complete if each idempotent \(e: X\to X\) splits, i.e. it admits a factorization \(X\xrightarrow{u} Y \xrightarrow{v} X\) with \(uv=\operatorname{id}_Y\). For example, a Krull-Schmidt category is idempotent complete.
Proposition 10. Let \(\Lambda\) be a noetherian ring. Assume that \(\operatorname{mod}\Lambda\) is a Krull-Schmidt category and that there is a \(\Lambda\)-module \(X\) and \(n\geq0\) such that \(\Omega^n(\operatorname{\underline{mod}}\Lambda)\subseteq \operatorname{\underline{add}}X\), where \(\operatorname{\underline{add}}X\) is the full subcategory of \(\operatorname{\underline{mod}}\Lambda\) generated by \(X\). The following statements hold.
The singularity category \(\operatorname{\rm{D}_{sg}}(\Lambda)\) has an additive generator \(q(X)\), i.e. \(\operatorname{\rm{D}_{sg}}(\Lambda)= \operatorname{add}q(X)\).
The singularity category \(\operatorname{\rm{D}_{sg}}(\Lambda)\) is idempotent complete and we have \[\operatorname{Hom}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(q(X),?): \operatorname{\rm{D}_{sg}}(\Lambda) \xrightarrow{\sim} \operatorname{proj}\operatorname{End}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(q(X)).\]
**Proof.* Since \(\operatorname{mod}\Lambda\) is Krull-Schmidt, \(\operatorname{\underline{mod}}\Lambda\) is also Krull-Schmidt and in particular is idempotent complete. It is easy to see the stabilization \(\mathcal{S}(\Lambda)\) is idempotent complete by construction and so is \(\operatorname{\rm{D}_{sg}}(\Lambda) \simeq \mathcal{S}(\Lambda)\). Then the last statement of (2) follows from the projectivization of an idempotent complete category with an additive generator [@ARS97].*
For (1), since \(\Omega^n(\operatorname{\underline{mod}}\Lambda)\subseteq \operatorname{\underline{add}}E\), we have a sequence \[\operatorname{\underline{add}}X \supseteq \operatorname{\underline{add}}(\Omega^n(\operatorname{\underline{mod}}\Lambda))\supseteq \operatorname{\underline{add}}(\Omega^{n+1}(\operatorname{\underline{mod}}\Lambda)) \supseteq \cdots.\] This sequence must terminate, because \(\operatorname{\underline{add}}X\) has only finitely many indecomposable modules up to isomorphism. Thus, we may assume without loss of generality that \[\operatorname{\underline{add}}X \supseteq \operatorname{\underline{add}}(\Omega^n(\operatorname{\underline{mod}}\Lambda))=\operatorname{\underline{add}}(\Omega^{n+1}(\operatorname{\underline{mod}}\Lambda)).\]
Let \(M[m]\in\operatorname{\rm{D}_{sg}}(\Lambda)\). Since in the singularity category, the negative shift corresponds to the syzygy, we have \(M[m]\simeq\Omega^{n_1}(M)[m+n_1]\) for any \(n_1\geq-m,n\). Since \(\operatorname{\underline{add}}(\Omega^{n_1}(\operatorname{\underline{mod}}\Lambda))=\operatorname{\underline{add}}(\Omega^{n+m+n_1}(\operatorname{\underline{mod}}\Lambda))\), then \(M[m]\simeq\Omega^{n_1}(M)[m+n_1]\) is a direct summand of \(\Omega^{n+m+n_1}(Y)[m+n_1]\simeq\Omega^n(Y)\) for some \(Y \in \operatorname{\underline{mod}}\Lambda\). By assumption, \(\Omega^n(Y)\in \operatorname{\underline{add}}X\). Then \(M[m] \in \operatorname{add}q(X)\) as desired. ◻
The singularity category of a Bäckström order will be described as the category of finitely generated projective modules over a certain von Neumann regular algebra. In this subsection, we review some fundamental properties of a von Neumann regular ring. We provide brief proofs for the reader’s convenience.
Theorem-Definition 11. [@Fai73] Let \(V\) be any ring. The following statements are equivalent.
Each right \(V\)-module is flat.
For any \(a\in V\), there exists \(x\in V\) such that \(a=axa\).
Each finitely generated right ideal is generated by an idempotent.
A ring \(V\) satisfying one of the conditions above is called a von Neumann regular* ring.*
Proposition 12. Let \(V\) be any ring. Then the following statements hold.
If \(V\) is an artinian semisimple ring, then \(V\) is von Neumann regular.
A direct limit of von Neumann regular rings is also von Neumann regular. In particular, a direct limit of artinian semisimple rings is von Neumann regular.
If \(V\) is a von Neumann regular ring, then the category \(\operatorname{proj}V\) is semisimple abelian.
**Proof.* (1) and (2) follow immediately from Theorem-Definition 11 (1) and (2). For (3), we only need to show \(\operatorname{proj}V\) is abelian. To show this, it is enough to prove that \(\operatorname{proj}V\) is closed under kernels and cokernels. Let \(f:P \to Q\) be a homomorphism for some \(P,Q\in\operatorname{proj}V\). Then we have an exact sequence \(P \xrightarrow{f} Q \to \operatorname{Coker}f\to 0\), which implies \(\operatorname{Coker}f\) is finitely presented. By the condition Theorem-Definition 11 (1), we have \(\operatorname{Coker}f\in \operatorname{proj}V\). Here we use the well-known fact that a module over any ring is finitely generated and projective if and only if it is finitely presented and flat. So, \(\operatorname{Im}f \in \operatorname{proj}V\), which implies \(\operatorname{Ker}f\) is also in \(\operatorname{proj}V\). ◻*
Definition-Proposition 13. Assume that \(D\) is a semisimple ring and \(M\) is a \((D,D)\)-bimodule. Let \(M^{\otimes 0}:=D\) and \(M^{\otimes i}:=M \otimes_D M^{\otimes{i-1}}\) for \(i \geq 1\). Define \[V=V(D,M):=\underset{i\geq0}{\varinjlim} \operatorname{End}_D(M^{\otimes i}) \quad \text{and} \quad K=K(D,M):=\underset{i\geq1}{\varinjlim} \operatorname{Hom}_D(M^{\otimes i}, M^{\otimes i-1}).\] Then
\(V\) is a von Neumann regular algebra and \(K\) is a \((V,V)\)-bimodule.
The category \(\operatorname{proj}V\) is semisimple abelian and \((\operatorname{proj}V, ? \otimes_V K)\) is a trivial left triangulated category.
We review Xiao-wu Chen’s description of the singularity category of a finite dimensional radical square zero algebra.
Theorem 14. [@Che11]Let \(A\) be a finite dimensional radical square zero algebra and let \(J\) be the radical of \(A\). Then \(\Omega(\operatorname{\underline{mod}}A)\subseteq \operatorname{\underline{add}}(A/J)\) and the syzygy functor \(\Omega\) is isomorphic to \[\Omega \simeq ?\otimes_{A/J} J: \operatorname{\underline{add}}(A/J) \to \operatorname{\underline{add}}(A/J).\] Define \[V(A):=V(A/J,J)=\underset{i\geq0}{\varinjlim} \operatorname{End}_{A/J}(J^{\otimes i}) \quad {\rm{and}} \quad K(A):=K(A/J,J)=\underset{i\geq 1}{\varinjlim} \operatorname{Hom}_{A/J}(J^{\otimes i}, J^{\otimes i-1}).\] Then \(V(A)\) is a von Neumann regular algebra and there is a triangle equivalence \[(\operatorname{\rm{D}_{sg}}(A),[1]) \simeq (\operatorname{proj}V(A), ? \otimes_{V(A)}K(A)),\] where the triangulated structures are trivial.
In this subsection, we collect relevant facts about Iwanaga-Gorenstein rings and Gorenstein-projective modules, which will be used to classify Iwanaga-Gorenstein and Gorenstein Bäckström orders in Section 4.2. Throughout this subsection, we assume that \(\Lambda\) is a noetherian ring.
Definition 15. A noetherian ring \(\Lambda\) is called Iwanaga-Gorenstein if \(\operatorname{id}_\Lambda \Lambda< \infty\) and \(\operatorname{id}_{\Lambda^{\operatorname{op}}} \Lambda<\infty\). In this case, \(\operatorname{id}_\Lambda \Lambda=\operatorname{id}_{\Lambda^{\operatorname{op}}} \Lambda<\infty\); see [@EJ00].
Definition 16. A finitely generated \(\Lambda\)-module \(M\) is called Gorenstein-projective if there is an exact sequence \(P_\bullet=(\cdots \to P_{1} \xrightarrow{f_{_1}} P_{0} \xrightarrow{f_{0}} P_{-1} \xrightarrow{f_{-1}} P_{-2} \to \cdots)\), where \(P_i\in\operatorname{proj}\Lambda\) and \(M=\operatorname{Ker}f_{-1}\), such that \(\operatorname{Hom}_\Lambda(P_\bullet,Q)\) is also an exact sequence for any \(Q\in\operatorname{proj}\Lambda\).
We denote the full subcategory consisting of Gorenstein-projective modules by \(\operatorname{Gproj}\Lambda\).
It is clear by definition that \(\operatorname{proj}\Lambda \subseteq \operatorname{Gproj}\Lambda\), \(\Omega(\operatorname{Gproj}\Lambda)\subseteq\operatorname{Gproj}\Lambda\), \(\operatorname{Gproj}\Lambda\subseteq \bigcap_{i=1}^{\infty}\Omega^i(\operatorname{mod}\Lambda)\) and \(\operatorname{pd}M=\infty\) for any \(M\in \operatorname{Gproj}\Lambda \backslash \operatorname{proj}\Lambda\). Moreover, \(\operatorname{Gproj}\Lambda\) is a Frobenius category with projective-injective objects \(\operatorname{proj}\Lambda\) and thus the stable category \(\operatorname{\underline{Gproj}}\Lambda\) is a triangulated subcategory of \(s\operatorname{mod}\Lambda\) [@Hap88]. Moreover, Beligiannis proved that \(\operatorname{\underline{Gproj}}\Lambda\) is the largest triangulated subcategory of \(\operatorname{\underline{mod}}\Lambda\); see the following proposition.
Proposition 17. [@Bel00]If a full additive subcategory \(\mathcal{C}\) of \(\operatorname{\underline{mod}}\Lambda\) is a triangulated subcategory of \(\operatorname{\underline{mod}}\Lambda\), then \(\mathcal{C}\subseteq\operatorname{\underline{Gproj}}\Lambda\).
As for projective modules, we can define Gorenstein-projective dimension for Gorenstein-projective modules.
Definition 18. A finitely generated module \(M\) is said to have Gorenstein-projective dimension \(\operatorname{Gpd}M\) at most \(n\) if there exists a finite exact sequence \(0 \to G_n \to \dots \to G_0 \to M \to 0\) with \(G_i\in\operatorname{Gproj}\Lambda\). If no such a finite exact sequence exists, then \(\operatorname{Gpd}M\) is defined by \(\infty\).
Consider the canonical triangle functor \(i: \operatorname{\underline{Gproj}}\Lambda \hookrightarrow \operatorname{\underline{mod}}\Lambda \xrightarrow{S} \mathcal{S}(\Lambda)\simeq\operatorname{\rm{D}_{sg}}(\Lambda)\). The property that each module has finite Gorenstein-projective dimension can be characterized by the requirement that this functor is an equivalence.
Theorem 19. Let \(\Lambda\) be a noetherian ring. The canonical functor \(i:\operatorname{Gproj}\Lambda \to \operatorname{\rm{D}_{sg}}(\Lambda)\) is full and faithful. Moreover, this functor is dense if and only if \(\operatorname{Gpd}M <\infty\) for any \(M\in\operatorname{mod}\Lambda\).
**Proof.* The sufficiency is by Buchweitz and Happel [@Buc97; @Hap91]. The necessity is by Beligiannis [@Bel00]. ◻*
Any module over an Iwanaga-Gorenstein ring has finite Gorenstein-projective dimension; see the following proposition. The converse is not true in general, as there need not exist a uniform bound on Gorenstein-projective dimension. However, the converse does hold for an order; see Theorem 32.
Proposition 20. [@EJ00] Let \(\Lambda\) be an Iwanaga-Gorenstein ring. Then \(\operatorname{Gpd}M <\infty\) for any \(M\in\operatorname{mod}\Lambda\). In particular, the canonical functor \(i:\operatorname{Gproj}\Lambda \to \operatorname{\rm{D}_{sg}}(\Lambda)\) is an equivalence.
We end this subsection with a further property of Gorenstein-projective dimension. We remark that the notions Gorenstein-projective objects and Gorenstein-projective dimension can be defined similarly for an exact category with enough projectives.
Lemma 21. Let \(\mathcal{C}\) be an exact category with enough projectives and injectives. Then if \(M\in \mathcal{C}\) satisfies \(\operatorname{id}_\mathcal{C}M<\infty\), then \(\operatorname{Gpd}_\mathcal{C} M=\operatorname{pd}_{\mathcal{C}} M\).
**Proof.* This property was proved by Holm in the setting of module categories [@Hol04b]. The argument is still valid for arbitary exact categories. ◻*
In this subsection, we review the general theory for orders. Our main references are [@CR90], [@HN94] and [@Rei03]. For the reader’s convenience, we include short proofs.
Throughout this subsection we assume that \(R\) is a complete regular local ring of Krull dimension \(d\) and \(K\) is the fraction field of \(R\).
Definition 22. An \(R\)-order \(\Lambda\) is an \(R\)-algebra satisfying \(\Lambda\) is free of finite rank as an \(R\)-module.
For an order of positive Krull dimension, we mainly study the category of Cohen-Macaulay modules instead of the whole module category.
Definition 23. A finitely generated \(\Lambda\)-module \(M\) is called a Cohen-Macaulay \(\Lambda\)-module if \(M\) is free as an \(R\)-module. The full subcategory of \(\operatorname{mod}\Lambda\) consisting of Cohen-Macaulay \(\Lambda\)-modules is denoted by \(\operatorname{CM}\Lambda\).
Remark 24. The category \(\operatorname{CM}\Lambda\) is an extension-closed subcategory of \(\operatorname{mod}\Lambda\) and hence an exact category. We will see that it has enough projectives and injectives.
Remark 25. Since \(R\) is a complete local ring, the category \(\operatorname{mod}\Lambda\) is Krull-Schmidt and so is \(\operatorname{CM}\Lambda\); see [@LW12].
Remark 26. In some references, a Cohen-Macaulay \(\Lambda\)-module \(M\) is called a \(\Lambda\)-lattice since it is free over \(R\). The terminology of Cohen-Macaulay \(\Lambda\)-module arises from the fact that the defining condition is equivalent to \(M\) being a (maximal) Cohen-Macaulay \(R\)-module (i.e. \(\operatorname{depth}_R M=\operatorname{dim}R\)); see [@BH93]. Moreover, when \(\operatorname{dim}R=1\), the condition is further equivalent to \(M\) being \(R\)-torsion-free; see [@DF04].
Definition 27. Let \(\Lambda\) be an \(R\)-order.
Assume that \(M,N\in\operatorname{CM}\Lambda\). Then \(M\) is called an overmodule of \(N\) if \(N\subseteq M \subseteq N \otimes_R K\) and \(N\) is a \(\Lambda\)-submodule of \(M\).
An overorder \(\Gamma\) of \(\Lambda\) is an \(R\)-order such that \(\Lambda\) is an \(R\)-subalgebra of \(\Gamma\) and \(\Gamma\) is a \(\Lambda\)-overmodule of \(\Lambda\).
\(\Lambda\) is called a maximal order if \(\Lambda\) has no proper overorder.
\(\Lambda\) is called a regular order if \(\operatorname{gl.dim}\Lambda=\operatorname{dim}R\). If \(\operatorname{dim}R=1\), a regular order is also called a hereditary order.
The following properties are elementary for orders.
Proposition 28. Let \(\Lambda\) be an \(R\)-order.
There is an exact duality \((?)^{\vee}:=\operatorname{Hom}_R(?,R): \operatorname{CM}\Lambda \xrightarrow{\simeq} \operatorname{CM}\Lambda^{\operatorname{op}}\).
Define \(\operatorname{inj}\Lambda:=\operatorname{add}(_\Lambda\Lambda)^{\vee}\subseteq\operatorname{CM}\Lambda\). Then the duality in (1) gives dualities \((?)^{\vee}: \operatorname{proj}\Lambda \xrightarrow{\simeq} \operatorname{inj}\Lambda^{\operatorname{op}}\) and \((?)^{\vee}: \operatorname{inj}\Lambda \xrightarrow{\simeq} \operatorname{proj}\Lambda^{\operatorname{op}}\).
\(\operatorname{proj}\Lambda=\{\text{projective objects in } \operatorname{CM}\Lambda\}\) and \(\operatorname{inj}\Lambda=\{\text{injective objects in } \operatorname{CM}\Lambda\}\). Moreover, \(\operatorname{CM}\Lambda\) has enough projectives and injectives.
\(\Lambda\) is regular if and only if \(\operatorname{CM}\Lambda = \operatorname{proj}\Lambda\).
**Proof.* (1) follows by definition and \((?):\operatorname{proj}R \xrightarrow{\simeq} \operatorname{proj}R\) being an exact duality. (2)(3) follow easily by (1). For (4), see [@IW14]. ◻*
Lemma 29. Let \(\Lambda\) be an R-order and \(M,N\in \operatorname{CM}\Lambda\). Then
\((M)^{\vee}\simeq\{f\in \operatorname{Hom}_{K}(M\otimes_R K,K) \mid f(M)\subseteq R\}\)
\(\operatorname{Hom}_\Lambda(M,N)\simeq\{f\in \operatorname{Hom}_{\Lambda\otimes_R K}(M\otimes_R K,N\otimes_R K) \mid f(M)\subseteq N\}\).
Let \(M\) be an overmodule of \(N\). Then \(N\otimes_R K=M\otimes_R K\). Moreover \((N)^{\vee}\) is an overmodule of \((M)^{\vee}\) in \(\operatorname{CM}\Lambda^{\operatorname{op}}\).
**Proof.* (2): The left-hand side is an \(R\)-submodule of \(\operatorname{Hom}_{\Lambda\otimes_R K}(M\otimes_R K,N\otimes_R K)\) since \(N\) is \(R\)-free. Then the equality can be checked easily. The proof of (1) is similar. (3) follows immediately by (1). ◻*
Proposition 30. Let \(\Gamma\) be an overorder of an \(R\)-order \(\Lambda\). Then the restriction functor \[\operatorname{res}: \operatorname{CM}\Gamma \to \operatorname{CM}\Lambda\] is fully faithful.
**Proof.* This proposition follows easily from the definition and Lemma 29 ◻*
For \(R\)-orders, being Iwanaga-Gorenstein is equivalent to the property that each module has finite Gorenstein-projective dimension (c.f. Proposition 20). This is a natural generalization of the artinian case [@AB69].
Lemma 31. Let \(\Lambda\) be an \(R\)-order. If \(\operatorname{Gpd}M < \infty\) for any \(M\in\operatorname{mod}\Lambda\), then \(\Lambda\) is Iwanaga-Gorenstein.
**Proof.* Recall that our modules are right modules and \(\operatorname{inj}\Lambda\) is the subcategory of the injectives in \(\operatorname{CM}\Lambda\). Firstly, we show \(\operatorname{id}_{\Lambda^{\operatorname{op}}} \Lambda < \infty\). We have \(\operatorname{Gpd}_\Lambda (\Lambda^{\vee}) < \infty\). Since \(\Lambda^{\vee} \in \operatorname{inj}\Lambda\), by Lemma 21(2), we have \(\operatorname{pd}_\Lambda (\Lambda^{\vee}) = \operatorname{Gpd}_\Lambda (\Lambda^{\vee}) < \infty\). So, \(\operatorname{id}_{\operatorname{CM}\Lambda^{\operatorname{op}}} \Lambda < \infty\). By [@Iya05a], each injective object in \(\operatorname{CM}\Lambda\) has injective dimension \(d\) in \(\operatorname{mod}\Lambda\). Thus, \(\operatorname{id}_{\Lambda^{\operatorname{op}}} \Lambda =\operatorname{id}_{\operatorname{mod}\Lambda^{\operatorname{op}}} \Lambda < \infty\).*
Secondly, we show \(\operatorname{id}_\Lambda \Lambda < \infty\). Let \(S:=\Lambda/\operatorname{rad}\Lambda\). Since \(S':=\Omega^d(S) \in \operatorname{CM}\Lambda\), we have \(\operatorname{Gpd}_\Lambda S'=n<\infty\) for some \(n\geq 1\). Then by [@Hol04a], \(\Omega^n(S')\in \operatorname{Gproj}\Lambda\). Thus, \(\operatorname{Ext}_\Lambda^i(S,\Lambda)\simeq \operatorname{Ext}_\Lambda^{i-n-d}(\Omega^{n+d}(S),\Lambda)=\operatorname{Ext}_\Lambda^{i-n-d}(\Omega^{n}(S'),\Lambda)=0\) for \(i\geq n+d+1\), which implies \(\operatorname{id}_\Lambda\Lambda\leq n+d <\infty\) by the Bass-type lemma; see Corollary [@GN02]. ◻
Theorem 32. Let \(\Lambda\) be an \(R\)-order. The following statements are equivalent.
\(\Lambda\) is Iwanaga-Gorenstein .
\(\operatorname{Gpd}M < \infty\), for any \(M\in\operatorname{CM}\Lambda\).
The canonical functor \(\operatorname{\underline{Gproj}}\Lambda \to \operatorname{\rm{D}_{sg}}(\Lambda)\) is an equivalence.
In the rest of this subsection, we assume that \(R\) is a complete discrete valuation ring.
Definition-Proposition 33. Let \(\Gamma\) be an overorder of \(\Lambda\). For \(M\in\operatorname{CM}\Lambda\), define \[M\Gamma:=\{\sum_i (m_i\otimes 1) \gamma_i \in M\otimes_R K \mid m_i \in M, \gamma_i \in \Gamma\}\in \operatorname{CM}\Gamma.\] This induces a functor \((?)\Gamma: \operatorname{CM}\Lambda \to \operatorname{CM}\Gamma\). Moreover, there exist adjoint pairs \(((?)\Gamma, \operatorname{res})\) and \((\operatorname{res}, \operatorname{Hom}_\Lambda(\Gamma,?))\).
**Proof.* Since \(M\Gamma\) is \(R\)-torsion-free and \(\operatorname{dim}R=1\), \(M\Gamma\in\operatorname{CM}\Gamma\). The adjoint pairs can be checked easily. ◻*
There is a dual concept called coradical to the radical of a Cohen-Macaulay module.
Definition-Proposition 34. For \(M\in\operatorname{CM}\Lambda\), the coradical* \(\operatorname{corad}M\) of \(M\) is defined as \[M\subseteq\operatorname{corad}M:=(\operatorname{rad}_{\Lambda^{\operatorname{op}}} (M^{\vee}))^\vee = \sum \{M\subsetneq N\mid \text{N is a minimal overmodule of M}\}\in\operatorname{CM}\Lambda.\] In particular, \(\operatorname{corad}I\) is the unique minimal overmodule of \(I\) if \(I\in\operatorname{inj}\Lambda\) is indecomposable.*
**Proof.* Note that \(\operatorname{CM}\Lambda\) is closed under submodules since \(\operatorname{dim}R=1\). Then the statements follow from the fact that \(\operatorname{rad}M\) is the intersection of all maximal submodule of \(M\) and from Propositions [prop:duality] and 30. ◻*
The following is the structure theorem for hereditary orders which is analogous to Artin-Wedderburn theorem for finite dimensional semisimple algebras.
Theorem 35. [@Rei03] [@HN94] The following statements hold.
Let \(D\) be a finite dimensional division \(K\)-algebra. Then there exists a unique maximal \(R\)-order \(\Delta\) in \(D\). Moreover, \(\Delta\) is local and a (non-commutative) PID.
Let \(D\) and \(\Delta\) be as in (1). Then \(M_n(\Delta)\subseteq M_n(D)\) is a maximal order in \(M_n(D)\). If \(\Lambda\) is another maximal order, then \(\Lambda\) is conjugate to \(M_n(D)\) by an invertible element in \(M_n(D)\).
Assume that \(\Lambda\) is a ring-indecomposable hereditary order, then \(B:=\Lambda \otimes_R K\) is a simple \(K\)-algebra. Moreover, there exists an identification \(B=M_n(D)\) for some finite dimensional division \(K\)-algebra \(D\) such that \[\Lambda= \begin{pmatrix} \begin{matrix} M_{n_1}(\Delta) & \\ & M_{n_2}(\Delta) \end{matrix} & & \Delta \\ & \ddots & \\ \operatorname{rad}\Delta & & M_{n_r}(\Delta) \end{pmatrix}_{n\times n}\] where \(n_1,n_2,\dots,n_r\geq 1\) and \(\Delta\) is the unique maximal order in \(D\).
In this subsection, we introduce the definition of Bäckström orders and their representation theory developed by Ringel and Roggenkamp.
Assumption 1. In the rest of this article, we assume that \((R,\mathfrak{m}=\pi R, k)\) is a complete discrete valuation ring and that \(K=\operatorname{Frac}R\) is the fraction field of \(R\).
Definition 36. A Bäckström order \(\Lambda=(\Lambda,\Gamma)\) is an \(R\)-order \(\Lambda\) together with an overorder \(\Gamma\) of \(\Lambda\) satisfying both of the following conditions
\(\Gamma\) is a hereditary order;
\(\operatorname{rad}\Lambda =\operatorname{rad}\Gamma\).
Remark 37. By definition and Propositions 28 and [prop:overorder], we have \(\operatorname{CM}\Gamma = \operatorname{proj}\Gamma \subseteq \operatorname{CM}\Lambda\).
Remark 38. Since \(\Gamma\) is hereditary, \(\Lambda \otimes_R K\) is semisimple by Theorem 35 (3). It is equivalent to say that \(\Lambda\) is an isolated singularity, which is also equivalent to \(\operatorname{CM}\Lambda\) admitting Auslander-Reiten sequences; see [@Aus86].
Lemma 39. Let \(\Lambda=(\Lambda,\Gamma)\) be a Bäckström order. The following statements hold.
The class of Bäckström orders is invariant under Morita equivalence. Therefore, we may always assume that Bäckström orders are basic.
The hereditary order \(\Gamma\) is uniquely determined by \(\Lambda\). Therefore, for a Bäckström order \(\Lambda=(\Lambda,\Gamma)\), we simply write \(\Lambda\) when no ambiguity arises.
For any \(M\in\operatorname{CM}\Lambda\), the radical of \(M\) coincides with the radical of \(M\Gamma\) as a \(\Lambda\)-module, and also with the radical of \(M\Gamma\) as a \(\Gamma\)-module. That is, we have \[\operatorname{rad}_\Lambda M=\operatorname{rad}_\Lambda (M\Gamma)=\operatorname{rad}_\Gamma(M\Gamma)\in\operatorname{CM}\Gamma.\] Therefore, we simply denote the radicals above by \(\operatorname{rad}M\).
**Proof.* For (1), see [@RR79]. For (2), there is a fact that, for any \(R\)-order \(\Gamma\), \(\Gamma\) is hereditary if and only if \[\Gamma=\{a \in \Gamma \otimes_R K \mid (\operatorname{rad}\Gamma)a\subseteq\operatorname{rad}\Gamma\};\] see [@HN94]. In our case, since \(\operatorname{rad}\Lambda=\operatorname{rad}\Gamma\), we have \[\Gamma=\{a\in \Lambda \otimes_R K \mid (\operatorname{rad}\Lambda)a\subseteq\operatorname{rad}\Lambda\},\] which is uniquely determined by \(\Lambda\). For (3), since \(\operatorname{rad}\Lambda =\operatorname{rad}\Gamma\) is a two-sided ideal both in \(\Lambda\) and \(\Gamma\), we have \(\operatorname{rad}_\Gamma(M\Gamma)=(M\Gamma)(\operatorname{rad}\Gamma)=(M\Gamma)(\operatorname{rad}\Lambda)=M(\operatorname{rad}\Lambda)=\operatorname{rad}_\Lambda M\) and the middle term is \(\operatorname{rad}_\Lambda (M\Gamma)\). ◻*
From now on, we assume that \(\Lambda=(\Lambda,\Gamma)\) is a Bäckström order. Define a finite dimensional \(k\)-algebra \(H\) associated to \(\Lambda\) as follows \[\label{eqa:32H} H=H(\Lambda,\Gamma):= \begin{pmatrix} \Gamma / \operatorname{rad}\Gamma & 0 \\ \Gamma / \operatorname{rad}\Gamma & \Lambda / \operatorname{rad}\Lambda \end{pmatrix}\tag{2}\] This is a hereditary radical square zero algebra.
Define a functor \[\begin{tikzcd}[row sep=0.1em, column sep=3em] \mathbb{F}: \operatorname{CM}\Lambda \arrow[r] & \operatorname{mod}H \\ \quad M \arrow[r,|->,to path={([xshift=1em]\tikztostart.east) -- ([xshift=-0.25em]\tikztotarget.west)}] & ( M \Gamma /\operatorname{rad}M, M/\operatorname{rad}M) \end{tikzcd}\] The \(H\)-action on \((M \Gamma /\operatorname{rad}M, M/\operatorname{rad}M)\) is given by the usual matrix multiplication and the canonical map \(M/\operatorname{rad}M \otimes_{\Lambda/\operatorname{rad}\Lambda} \Gamma/\operatorname{rad}\Gamma \to M\Gamma/\operatorname{rad}M\). For a homomorphism \(f:M \to N\) in \(\operatorname{CM}\Lambda\), it induces a \(\Gamma\)-module homomorphism \(f_1=(f)\Gamma: M\Gamma \to N\Gamma\) such that the following diagram commutes, in which the horizontal maps are the counits of the adjoint pair \(((?)\Gamma, \operatorname{res})\); see Definition-Proposition 33 \[\begin{tikzcd} M \arrow[r,hook] \arrow[d,"f"] & M\Gamma \arrow[d,"f_1"]\\ N \arrow[r,hook] & N\Gamma \end{tikzcd}\] Passing to the quotient, we have another commutative diagram: \[\begin{tikzcd} M/\operatorname{rad}M \arrow[r,hook] \arrow[d,"\bar{f}"] & M\Gamma /\operatorname{rad}M \arrow[d,"\bar{f_1}"]\\ N/\operatorname{rad}N \arrow[r,hook] & N\Gamma /\operatorname{rad}N \end{tikzcd}\] This defines an \(H\)-module homomorphism \[\mathbb{F}(f):=(\bar{f_1},\bar{f}):(M\Gamma /\operatorname{rad}M, M/\operatorname{rad}M) \to (N\Gamma /\operatorname{rad}N, N/\operatorname{rad}N).\]
Denote by \(\operatorname{mod}_s H\) the full subcategory of \(\operatorname{mod}H\) whose objects contain no non-zero simple direct summand and \(\operatorname{sim}H:=\operatorname{ind}(\operatorname{mod}H/\operatorname{rad}H)\). The next theorem by Ringel and Roggenkamp establishes the representation theory of Bäckström orders.
Theorem 40. [@RR79]Let \(\Lambda=(\Lambda,\Gamma)\) be a Bäckström order and keep the notations above. The functor \[\begin{tikzcd}[row sep=0.1em, column sep=2em] \mathbb{F}: \operatorname{CM}\Lambda \arrow[r] & \operatorname{mod}H \\ \end{tikzcd}\] is full with essential image \(\operatorname{mod}_s H\). The kernel \(\operatorname{Ker}\mathbb{F}_{M,N}\) for \(M,N\in \operatorname{CM}\Lambda\) is \(\operatorname{Hom}_\Lambda(M,\operatorname{rad}N)\). Moreover, this functor gives one-to-one correspondences:
\(\operatorname{ind}(\operatorname{CM}\Lambda) \xleftrightarrow{1:1} \operatorname{ind}(\operatorname{mod}H)\backslash \operatorname{sim}H\).
\(\operatorname{ind}(\operatorname{proj}\Lambda) \xleftrightarrow{1:1} \operatorname{ind}(\operatorname{proj}H)\backslash\operatorname{sim}H\).
\(\operatorname{ind}(\operatorname{proj}\Gamma) \xleftrightarrow{1:1} \operatorname{ind}(\operatorname{inj}H)\backslash\operatorname{sim}H\).
Remark 41. For the finite dimensional \(k\)-algebra \(H\), we may associate a species to it. The Bäckström orders of finite Cohen-Macaulay type, i.e. those with finitely many indecomposable Cohen-Macaulay modules up to isomorphism, are classified by the shape of the species [@RR79]. More precisely, a Bäckström order is of finite Cohen-Macaulay type if and only if its species is a finite union of Dynkin diagrams. Moreover, Roggenkamp also studied the Auslander-Reiten quiver of a Bäckström order [@Rog83]. It is obtained by the Auslander-Reiten quiver of \(\operatorname{mod}H\) under some combinatorial constructions.
In this subsection, we investigate further properties of Bäckström orders. In this subsection, we assume that \(\Lambda=(\Lambda,\Gamma)\) is a Bäckström order.
Let \(N\in\operatorname{CM}\Lambda\). For any surjective homomorphism \(f: P \rightarrow\mathrel{\mkern-14mu}\rightarrow N\) with \(P\in \operatorname{proj}\Lambda\), it induces a commutative diagram of exact sequences: \[\label{cartesian95diagram} \begin{CD} @. @. 0 @. 0 @. \\ @. @. @VVV @VVV @. \\ 0 @>>> \Omega(N) @>>> \operatorname{rad}P @>>> \operatorname{rad}N @>>> 0 \\ @. @VV\rotatebox{270}{\simeq}V @VVV @VVV @. \\ 0 @>>> \Omega(N) @>>> P @>f>> N @>>> 0 \\ @. @. @VVV @VVV @. \\ @. @. P/\operatorname{rad}P @>\bar{f}>\simeq> N/\operatorname{rad}N @. \\ @. @. @VVV @VVV @. \\ @. @. 0 @. 0 @. \end{CD}\tag{3}\]
Proposition 42. For a Bäckström order \(\Lambda\), we have \(\Omega(\operatorname{\underline{CM}}\Lambda)\subseteq \operatorname{\underline{add}}\Gamma\).
**Proof.* Let \(N\in\operatorname{CM}\Lambda\). Since \(\operatorname{rad}P, \operatorname{rad}N \in \operatorname{CM}\Gamma\) in the commutative diagram (3 ), then \(\Omega(N) \in \operatorname{CM}\Gamma=\operatorname{proj}\Gamma\). ◻*
Lemma 43. The following statements hold.
For any \(N \in \operatorname{CM}\Lambda\), the inclusion map \(\operatorname{rad}N \hookrightarrow N\) factors through a projective \(\Lambda\)-module.
Let \(M,N \in \operatorname{CM}\Lambda\) be indecomposable and non-projective. Then we have \[\mathcal{P}(M,N)=\operatorname{Hom}_\Lambda(M,\operatorname{rad}N),\] where \(\mathcal{P}(M,N)\) is the set of homomorphisms from \(M\) to \(N\) factoring through a projective module.
**Proof.* For (1), the exact sequence \(0\to\Omega(N)\to\operatorname{rad}P \to \operatorname{rad}N \to 0\) splits since it is in \(\operatorname{CM}\Gamma=\operatorname{proj}\Gamma\). So, \(\operatorname{rad}N \hookrightarrow N\) factors through \(P\in \operatorname{proj}\Lambda\).*
For (2), assume that \(f: M \to N\) factors through \(g: M \to P\) for some \(P \in \operatorname{proj}\Lambda\). Since \(M\) is not projective, then \(\operatorname{Im}g \subseteq \operatorname{rad}P\). So, \(\operatorname{Im}f \subseteq \operatorname{rad}N\). Conversely, assume that the image of \(f: M \to N\) is in \(\operatorname{rad}N\). This means \(f\) factors through the inclusion map \(\operatorname{rad}N \hookrightarrow N\). By (1), this inclusion factors through a projective \(\Lambda\)-module and so does \(f\). ◻
The following result shows that the functor \(\mathbb{F}:\operatorname{CM}\Lambda \to \operatorname{mod}H\) in Theorem 40 induces a stable equivalence. This is a 1-dimensional analogue of a result for finite dimensional radical square zero algebras [@ARS97], providing evidence for our Slogan 2.
Theorem 44. The functor \(\mathbb{F}: \operatorname{CM}\Lambda \to \operatorname{mod}H\) induces an equivalence \[\underline{\mathbb{F}}: \operatorname{\underline{CM}}\Lambda \xlongrightarrow{\sim} \operatorname{\underline{mod}}_s H,\] where \(\operatorname{\underline{mod}}_s H\) is the full subcategory of \(\operatorname{\underline{mod}}H\) whose objects do not contain any non-zero simple direct summand.
**Proof.* By construction, \(\mathbb{F}\) sends \(\operatorname{proj}\Lambda\) to \(\operatorname{proj}H\). So it induces a functor \(\underline{\mathbb{F}}: \operatorname{\underline{CM}}\Lambda \to \operatorname{\underline{mod}}H\). It is full and dense in \(\operatorname{\underline{mod}}_s H\) by Theorem 40. By Lemma 43, if \(M,N\) have no projective direct summands, then \(\operatorname{Ker}\mathbb{F}_{M,N}=\operatorname{Hom}_\Lambda(M,\operatorname{rad}N)=\mathcal{P}(M,N)\). Therefore, \(\underline{\mathbb{F}}\) is also faithful. ◻*
Next, we analyze the connection between the syzygies of \(N\in \operatorname{proj}\Lambda\) and of \(\mathbb{F}(N) \in \operatorname{mod}_s H\) under the functor \(\mathbb{F}: \operatorname{CM}\Lambda \to \operatorname{mod}H\).
Proposition 45. Consider \(N\in\operatorname{CM}\Lambda\) and \(\mathbb{F}(N) \in \operatorname{mod}_s H\). The following statements hold.
Assume that \(f:P \rightarrow\mathrel{\mkern-14mu}\rightarrow N\) is a projective cover of \(N\) with \(P\in\operatorname{proj}\Lambda\). It gives rise to a commutative diagram (?? ) of exact sequences.
Under the assumptions in (1), the homomorphism \(\mathbb{F}(f):\mathbb{F}(P) \rightarrow\mathrel{\mkern-14mu}\rightarrow\mathbb{F}(N)\) is also a projective cover in \(\operatorname{mod}H\).
Assume that \(\mathbb{F}(f):\mathbb{F}(P) \rightarrow\mathrel{\mkern-14mu}\rightarrow\mathbb{F}(N)\) is a projective cover of \(\mathbb{F}(N)\) induced by \(P\in\operatorname{proj}\Lambda\) and a homomorphism \(f:P\rightarrow\mathrel{\mkern-14mu}\rightarrow N\). It also gives rise to the commutative diagram (?? ).
Under the assumptions in (3), the homomorphism \(f:P \rightarrow\mathrel{\mkern-14mu}\rightarrow N\) is also a projective cover in \(\operatorname{CM}\Lambda\).
\[\label{CD:big95comm95diagram} \begin{tikzcd}[row sep=1.25em, column sep={between origins, 4em}] & & & & &0 \arrow[dd] & &0 \arrow[dd] & & \\ & &0 \arrow[dd] & &0 \arrow[dd] & &0 \arrow[dd] & & & \\ &0 \arrow[rr] & &X \arrow[rr] \arrow[dl,equal] \arrow[dd] & &\operatorname{rad}P \arrow[rr] \arrow[dl,equal] \arrow[dd] & &\operatorname{rad}N \arrow[rr] \arrow[dl,equal] \arrow[dd] & &0\\ 0 \arrow[rr] & &X \arrow[rr,crossing over] \arrow{dd} & &\operatorname{rad}P \arrow[rr,crossing over] \arrow[dd] & &\operatorname{rad}N \arrow[rr, crossing over] \arrow[dd] & &0 & \\ &0 \arrow[rr] & &X \arrow[rr] \arrow[dl] & &P \arrow[rr,"f"near start] \arrow[dl] \arrow[dd] & &N \arrow[rr] \arrow[dl] \arrow[dd] \arrow[dd] & &0\\ 0 \arrow[rr] & &Y \arrow[rr] \arrow[dd] & & P \Gamma \arrow[rr,"f_1"near end,crossing over] \arrow[dd] & & N \Gamma \arrow[rr,crossing over] \arrow[dd] & &0 & \\ & & & & &P/\operatorname{rad}P \arrow[rr,"\bar{f}"near start,"\simeq"'near start] \arrow[dl] \arrow[dd] & &N/\operatorname{rad}N \arrow[dl] \arrow[dd] & &\\ 0 \arrow[rr] & &Z \arrow[rr] \arrow[dd] & &P\Gamma/\operatorname{rad}P \arrow[rr,"\bar{f_1}"near start,crossing over] \arrow[dd] & &N\Gamma /\operatorname{rad}N \arrow[rr,crossing over] \arrow[dd] & &0 & \\ & & & & &0 & &0 & & \\ & &0 & &0 & &0 & & & \arrow[from=2-3, to=4-3, crossing over] \arrow[from=2-5, to=4-5, crossing over] \arrow[from=3-4, to=5-4, "\rotatebox{270}{\simeq}" near start] \arrow[from=4-3, to=6-3, crossing over] \arrow[from=4-3, to=4-5, crossing over] \arrow[from=4-5, to=6-5, crossing over] \arrow[from=4-7, to=6-7, crossing over] \arrow[from=6-7, to=8-7, crossing over] \end{tikzcd}\qquad{(1)}\]
**Proof.* (1): Assume that \(f: P \rightarrow\mathrel{\mkern-14mu}\rightarrow N\) is a projective cover in \(\operatorname{CM}\Lambda\). Let \(X\) be the kernel of \(f\) and \(Y\) be the kernel of the induced map \(f_1: P \Gamma\rightarrow\mathrel{\mkern-14mu}\rightarrow N\Gamma\). There exists a commutative diagram of exact sequences: \[\label{CD:bigcommute1} \begin{tikzcd} 0 \arrow[r] &X \arrow[r] \arrow[d,dotted] &P \arrow[r,"f"] \arrow[d] &N \arrow[r] \arrow[d] &0\\ 0 \arrow[r] &Y \arrow[r] &P\Gamma \arrow[r,"f_1"] &N\Gamma \arrow[r] &0 \end{tikzcd}\tag{4}\] Let \(Z\) be the kernel of \(\bar{f_1}:P\Gamma/\operatorname{rad}P \to N\Gamma/\operatorname{rad}N\). By the 3 \(\times\) 3 lemma, there exists a commutative diagram with exact columns and rows: \[\label{CD:bigcommute2} \begin{tikzcd} & 0\arrow[d,dotted] & 0\arrow[d] &0 \arrow[d] & \\ 0 \arrow[r] & X \arrow[r] \arrow[d,dotted] &\operatorname{rad}P \arrow[r] \arrow[d] & \operatorname{rad}N \arrow[r] \arrow[d] &0 \\ 0 \arrow[r] & Y \arrow[r] \arrow[d,dotted] &P\Gamma \arrow[r,"f_1"] \arrow[d] & N\Gamma \arrow[r] \arrow[d] &0 \\ 0 \arrow[r] & Z \arrow[r] \arrow[d,dotted] &P\Gamma/\operatorname{rad}P \arrow[r,"\bar{f_1}"] \arrow[d] & N\Gamma/\operatorname{rad}N \arrow[r] \arrow[d] &0 \\ & 0 &0 &0 & \end{tikzcd}\tag{5}\] Combining the diagrams (3 ), (4 ) and (5 ), we obtain the desired commutative diagram (?? ).*
(2): Note that as an \(H\)-module, we have \(\operatorname{rad}( P\Gamma / \operatorname{rad}P, P / \operatorname{rad}P) = (P\Gamma / \operatorname{rad}P, 0)\). So, the exact sequence in \(\operatorname{mod}H\) \[\begin{tikzcd}[column sep=2em] 0 \arrow[r] & (Z,0) \arrow[r] & ( P \Gamma / \operatorname{rad}P,P / \operatorname{rad}P) \arrow[r,"\mathbb{F}(f)"] & ( N \Gamma / \operatorname{rad}N , N / \operatorname{rad}N) \arrow[r] & 0 \end{tikzcd}\] is a minimal projective resolution of \(\mathbb{F}(M)\).
(3)(4): We just need to note that each step above is invertible. In detail, assume that \[\mathbb{F}(f): \mathbb{F}(P)=(P\Gamma /\operatorname{rad}P, P/\operatorname{rad}P) \rightarrow\mathrel{\mkern-14mu}\rightarrow(N\Gamma /\operatorname{rad}N, N/\operatorname{rad}N)=\mathbb{F}(N)\] is a projective cover. Then the kernel of \(\mathbb{F}(f)\) is of the form \((Z,0)\) and \(P/\operatorname{rad}{P} \simeq N/\operatorname{rad}N\). So, again let \(X\) be the kernel of \(f\) and \(Y\) be the kernel of the induced map \(P\Gamma \rightarrow\mathrel{\mkern-14mu}\rightarrow N\Gamma\) by \(f\), we obtain the desired big commutative diagram (?? ). Finally, since \(X=\operatorname{Ker}f \subseteq \operatorname{rad}P\), the homomorphism \(f:P \rightarrow\mathrel{\mkern-14mu}\rightarrow N\) is a projective cover in \(\operatorname{CM}\Lambda\). ◻
Corollary 46. Let \(0\to X \to P\xrightarrow{f}M\to0\) be an exact sequence where \(f:P\rightarrow\mathrel{\mkern-14mu}\rightarrow M\) is a projective cover for \(M\in\operatorname{CM}\Lambda\). Then \(\operatorname{Ker}\mathbb{F}(f)\)=\(((\operatorname{corad}X)/X,0)\) (recall Definition 34 for coradicals).
**Proof.* Consider the commutative diagram (?? ). The top and middle outer exact sequences split, since they are in \(\operatorname{CM}\Gamma\). So, \(X\simeq \operatorname{rad}Y\). Thus, \(Y \simeq \operatorname{corad}X\) and \(Z \simeq Y/X \simeq (\operatorname{corad}X)/X\). ◻*
The following example is to explain Proposition 45
Example 47. Let \((R,\pi R,k)\) be a complete discrete valuation ring. \[\Lambda=\begin{pNiceMatrix}[last-col] R & R & \Block{2-1}{P_1}\\ \pi R & R & \CodeAfter \tikz \draw ([xshift=0.3em,yshift=0.2em] 1-1.south east) -- ([xshift=-0.1em,yshift=-0.1em] 2-2.north west); \tikz \draw ([xshift=0.2em,yshift=0.05em] 1-1.south east) -- ([xshift=-0.2em, yshift=-0.25em] 2-2.north west);\end{pNiceMatrix} \quad \text{and} \quad \Gamma=\begin{pNiceMatrix}[last-col] R & R & Q_2\\ \pi R & R & Q_3 \end{pNiceMatrix}\] where \(\left(R=R\right):=\{(r_1,r_2)\in R \times R \mid r_1-r_2 \in \pi R\}\). The labels on the right indicate the row vectors, namely, \(P_1\) denotes the indecomposable projective \(\Lambda\)-module, and \(Q_2,Q_3\) denote the indecomposable projective \(\Gamma\)-modules. These constitute all the indecomposable Cohen-Macaulay \(\Lambda\)-modules up to isomorphism. We have \(\operatorname{corad}Q_2 \simeq Q_3\) and \(\operatorname{corad}Q_3 \simeq Q_2\). The finite dimensional \(k\)-algebra \(H(\Lambda,\Gamma)\) is isomorphic to the path algebra of the quiver \(\begin{tikzcd}[column sep=1.5em] 2 & 1 \arrow[l] \arrow[r] & 3. \end{tikzcd}\)
Under the functor \(\mathbb{F}\), the indecomposable non-simple projective \(H\)-module \(P(1)\) at 1 and the indecomposable non-simple injective \(H\)-module \(I(2)\), \(I(3)\) at 2 and 3 correspond to \(P_1\), \(Q_2\) and \(Q_3\) respectively. Then the minimal projective resolutions in \(\operatorname{mod}H\) \[0 \to P(2) \to P(1) \to I(3) \to 0, \quad 0 \to P(3) \to P(1) \to I(2) \to 0,\] give rise to the projective covers in \(\operatorname{CM}\Lambda\) \[0 \to Q_2 \to P_1 \to Q_2 \to 0, \quad 0 \to Q_3 \to P_1 \to Q_3 \to 0.\]
We prove our main theorems in this section. Throughout this section, let \(\Lambda=(\Lambda,\Gamma)\) be a Bäckström order and keep the notations in Section 3.2.
We first give a concrete description of the syzygy functor. Recall that we have \(\Omega^2(\operatorname{\underline{mod}}\Lambda) \subseteq \Omega(\operatorname{\underline{CM}}\Lambda)\subseteq \operatorname{\underline{add}}\Gamma\); see Proposition 42. So, the syzygy functor \(\Omega: \operatorname{\underline{mod}}\Lambda \to \operatorname{\underline{mod}}\Lambda\) restricts to a functor \(\Omega: \operatorname{\underline{add}}\Gamma \to \operatorname{\underline{add}}\Gamma\). Consider the multiplication map \(\mu: \Gamma \otimes_R \Lambda \rightarrow\mathrel{\mkern-14mu}\rightarrow\Gamma\). Since \(\Gamma\) is free as an \(R\)-module, \(\Gamma \otimes_R \Lambda\) is projective as a right \(\Lambda\)-module. Hence, it gives a projective presentation of \(\Gamma\) in \(\operatorname{CM}\Lambda\): \[\label{eqa:32proj4632resol46of32Gamma} 0\to\operatorname{Ker}\mu \to \Gamma \otimes_R \Lambda \xrightarrow{\mu} \Gamma \to 0,\tag{6}\] Then \(\operatorname{Ker}\mu\) is a \((\Gamma,\Lambda)\)-bimodule and \(\Omega(\Gamma)=\operatorname{Ker}\mu\). We consider the stable endomorphism ring \(\operatorname{\underline{End}}_\Lambda(\Gamma)\) and the stable Hom-space \(\operatorname{\underline{Hom}}_\Lambda(\Gamma,\operatorname{Ker}\mu)\).
Lemma 48. The stable endomorphism ring \(\operatorname{\underline{End}}_\Lambda(\Gamma)\) is a finite dimensional semisimple \(k\)-algebra.
**Proof.* First, recall that the restriction functor \(\operatorname{res}: \operatorname{CM}\Gamma \to \operatorname{CM}\Lambda\) is fully faithful. Since \(\operatorname{rad}\Gamma \simeq \operatorname{Hom}_\Gamma(\Gamma,\operatorname{rad}\Gamma)\subseteq \operatorname{Hom}_\Gamma(\Gamma,\Gamma)\), by Lemma 43, we have \(\operatorname{rad}\Gamma \subseteq \mathcal{P}_\Lambda(\Gamma,\Gamma)\). Thus, the proposition follows from the projection \(\operatorname{End}_\Gamma (\Gamma)/\operatorname{rad}\operatorname{End}_\Gamma(\Gamma) \rightarrow\mathrel{\mkern-14mu}\rightarrow\operatorname{\underline{End}}_\Lambda(\Gamma)\). ◻*
Henceforth, we denote \[D:=\operatorname{\underline{End}}_\Lambda(\Gamma) \quad \text{and} \quad M:=\operatorname{\underline{Hom}}_\Lambda(\Gamma,\operatorname{Ker}\mu),\] The following proposition realizes the syzygy functor as a tensor functor.
Proposition 49. The following statements hold.
M is an \((D,D)\)-bimodule.
The following diagram commutes up to a natural isomorphism of functors: \[\begin{CD} \operatorname{\underline{add}}\Gamma @>\operatorname{\underline{Hom}}_\Lambda(\Gamma,?)>\simeq> \operatorname{mod}D \\ @VV\Omega V @VV? \otimes_D MV \\ \operatorname{\underline{add}}\Gamma @>\operatorname{\underline{Hom}}_\Lambda(\Gamma,?)>\simeq> \operatorname{mod}D \end{CD}\]
**Proof.* The Hom-space \(M=\operatorname{\underline{Hom}}_\Lambda(\Gamma, \operatorname{Ker}\mu)\) is a right \(\operatorname{\underline{End}}_\Lambda (\Gamma)\)-module via composition of maps. In addition, the syzygy functor gives a map \(\Omega: \operatorname{\underline{End}}_\Lambda (\Gamma) \to \operatorname{\underline{End}}_\Lambda(\Omega(\Gamma))= \operatorname{\underline{End}}_\Lambda (\operatorname{Ker}\mu)\), which induces a left \(\operatorname{\underline{End}}_\Lambda (\Gamma)\)-module structure on \(M\). These indeed define a \((D,D)\)-bimodule structure on \(M\).*
For (2), the vertical equivalence is by projectivization and \(\operatorname{proj}D=\operatorname{mod}D\), since \(D\) is semisimple by Lemma 48. For the commutative diagram, we first construct a natural transformation between these two functors. Let \(Q\in\operatorname{CM}\Gamma\). Tensoring \(Q\) with the exact sequence (6 ), we obtain an exact sequence \[0 \to Q\otimes_\Gamma \operatorname{Ker}\mu \to Q\otimes_R \Lambda \to Q \to 0.\] Thus, \(\Omega(Q)=Q\otimes_\Gamma \operatorname{Ker}\mu\). We then have a natural isomorphism \[\operatorname{Hom}_\Lambda(\Gamma,Q)\otimes_\Gamma M \simeq Q\otimes_\Gamma M \xrightarrow{\simeq} \operatorname{\underline{Hom}}_\Lambda(\Gamma,Q\otimes_\Gamma\operatorname{Ker}\mu),\] where the first isomorphism is by Proposition [prop:overorder] and the second isomorphism is because \(Q\in\operatorname{proj}\Gamma\). On the other hand, we have a natural projection \[\operatorname{Hom}_\Lambda(\Gamma,Q)\otimes_\Gamma M \rightarrow\mathrel{\mkern-14mu}\rightarrow\operatorname{\underline{Hom}}_\Lambda(\Gamma,Q)\otimes_D M,\] which is induced by \(\Gamma \xrightarrow{\simeq} \operatorname{End}_\Gamma(\Gamma) \rightarrow\mathrel{\mkern-14mu}\rightarrow D\). Combining together with the natural isomorphism above, we obtain the desired natural transformation. The natural transformation is an isomorphism when evaluated at \(Q=\Gamma\). Therefore, we obtain (2). ◻
Proposition 50. The categories \((\operatorname{\underline{add}}\Gamma, \Omega)\) and \((\operatorname{mod}D, ?\otimes_D M)\) are trivial left triangulated categories. The functor \(\operatorname{\underline{Hom}}_\Lambda(\Gamma,?):(\operatorname{\underline{add}}\Gamma, \Omega)\to (\operatorname{mod}D, ?\otimes_D M)\) gives a left triangle equivalence.
**Proof.* Since \(\Omega^2(\operatorname{\underline{mod}}\Lambda)\subseteq \operatorname{\underline{add}}\Gamma\), it is easy to see that \((\operatorname{\underline{add}}\Gamma, \Omega)\) forms a left triangulated subcategory of \((\operatorname{\underline{mod}}\Lambda,\Omega)\). By Lemmas 48 and 7, the left triangulated structure of \((\operatorname{\underline{add}}\Gamma, \Omega)\) is trivial. By the same lemmas, \((\operatorname{mod}D, ?\otimes_D M)\) is also a trivial left triangulated category. Therefore, Proposition 49 implies that \(\operatorname{\underline{Hom}}_\Lambda(\Gamma,?)\) is a left triangle equivalence. ◻*
We are now prepared to prove the main theorem in this article. Recall Definition-Theorem 13, define \[V(\Lambda):=V(D,M)=\underset{i\geq0}{\varinjlim} \operatorname{End}_D(M^{\otimes i}) \quad {\rm{and}} \quad K(\Lambda):=K(D,M)=\underset{i\geq1}{\varinjlim} \operatorname{Hom}_D(M^{\otimes i}, M^{\otimes i-1}).\] Then \(V(\Lambda)\) is a von Neumann regular algebra. The category \(\operatorname{proj}V(\Lambda)\) of finitely generated projective \(V(\Lambda)\)-modules is semisimple abelian, and \((\operatorname{proj}V(\Lambda),? \otimes_{V(\Lambda)} K(\Lambda))\) forms a trivial left triangulated category.
Theorem 51. For a Bäckström order \((\Lambda,\Gamma)\), there exists a triangle equivalence \[(\operatorname{\rm{D}_{sg}}(\Lambda),[1]) \simeq (\operatorname{proj}V(\Lambda), ? \otimes_{V(\Lambda)} K(\Lambda)),\] where the triangulated structures are trivial.
**Proof.* By Propositions 42 and 10, we have \(\operatorname{\rm{D}_{sg}}(\Lambda) \simeq \operatorname{proj}\operatorname{End}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(q(\Gamma))\) as additive categories. On the other hand, Theorem 9 and Proposition 49 imply \[\operatorname{End}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(q(\Gamma)) \simeq \underset{i\geq0}{\varinjlim} \operatorname{\underline{End}}_\Lambda(\Omega^i (\Gamma)) \simeq \underset{i\geq0}{\varinjlim} \operatorname{End}_D(M^{\otimes i})=V(\Lambda).\] Therefore, \(\operatorname{\rm{D}_{sg}}(\Lambda) \simeq \operatorname{proj}V(\Lambda)\) as additive categories. Let \(\Sigma\) be the corresponding automorphism to \([1]\). Then \((\operatorname{proj}V(\Lambda),\Sigma)\) becomes a triangulated category inherited by the categorical equivalence. In addition, by Lemma 7, there is a unique triangulated structure with respect to \(\Sigma\), which is the trivial triangulated structure. So, these two triangulated structures coincide. Therefore, we obtain a triangle equivalence \[\operatorname{Hom}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(q(\Gamma),?):(\operatorname{\rm{D}_{sg}}(\Lambda),[1]) \xlongrightarrow{\sim} (\operatorname{proj}V(\Lambda), \Sigma).\]*
Next, we determine the suspension functor \(\Sigma\). Note that \(\Sigma(V(\Lambda))\) is a \((V(\Lambda),V(\Lambda))\)-bimodule whose left module structure is given by \(V(\Lambda) \simeq \operatorname{End}_{V(\Lambda)}(V(\Lambda)) \xrightarrow{\Sigma} \operatorname{End}_{V(\Lambda)}(\Sigma(V(\Lambda)))\). Moreover, we have a natural isomorphism \(\Sigma \simeq ? \otimes_{V(\Lambda)} \Sigma (V(\Lambda)): \operatorname{proj}V(\Lambda) \to \operatorname{proj}V(\Lambda)\).
A similar calculation shows that there exists a \((V(\Lambda),V(\Lambda))\)-bimodule isomorphism \[\Sigma(V(\Lambda))\simeq\operatorname{Hom}_{\operatorname{\rm{D}_{sg}}(\Lambda)}(q(\Gamma),q(\Gamma) [1]) \simeq \underset{i\geq1}{\varinjlim} \operatorname{\underline{Hom}}_\Lambda(\Omega^i(\Gamma), \Omega^{i-1} (\Gamma)) \simeq \underset{i\geq1}{\varinjlim} \operatorname{Hom}_D(M^{\otimes i}, M^{\otimes i-1})=K(\Lambda).\] Therefore, \(\Sigma\) is isomorphic to the functor \(? \otimes_{V(\Lambda)} K (\Lambda): \operatorname{proj}V(\Lambda) \to \operatorname{proj}V(\Lambda)\). ◻
As noted in the introduction, our result on the singularity category of a Bäckström order closely parallels Chen’s result; see Theorem 14. Using the notations above, we associate to a Bäckström order \(\Lambda=(\Lambda,\Gamma)\) a finite dimensional \(k\)-algebra \(A(\Lambda)\) defined as the trivial extension \[\label{eqa:A40Lambda41} A(\Lambda):=D\oplus M,\tag{7}\] where the multiplication is given by \((a,m)(a',m'):=(aa',am'+ma')\). Since \(D\) is semisimple and \(M^2=0\) in \(A(\Lambda)\), we have \(\operatorname{rad}A(\Lambda)\simeq M\) and \(A(\Lambda)/\operatorname{rad}A(\Lambda)\simeq D\). Therefore, \(A(\Lambda)\) is a finite dimensional radical square zero \(k\)-algebra.
Corollary 52. Let \(\Lambda\) be a Bäckström order and \(A(\Lambda)=D\oplus M\) be the finite dimensional radical square zero algebra associated to it. We have a triangle equivalence \[\operatorname{\rm{D}_{sg}}(\Lambda) \simeq \operatorname{\rm{D}_{sg}}(A(\Lambda)).\]
We also construct the equivalence functor explicitly using the theory of stabilizations. Recall that \(\operatorname{\underline{add}}D=\operatorname{\underline{add}}_{A(\Lambda)}D\) denotes the full subcategory of \(\operatorname{\underline{mod}}A(\Lambda)\) generated by \(D\).
Proposition 53. Let \(\Lambda\) be a Bäckström order and \(A:=A(\Lambda)=D\oplus M\) be the finite dimensional radical square zero algebra associated to it. Then
There is a natural full and dense left triangle functor between two trvial left triangulated categories \[\pi: (\operatorname{mod}D, ? \otimes_D M) \longrightarrow (\operatorname{\underline{add}}D, ? \otimes_D M).\]
The functor \(\pi\) induces a triangle equivalence \(\tilde{\pi}: \mathcal{S}(\operatorname{mod}D, ? \otimes_D M) \xlongrightarrow{\simeq} \mathcal{S}(\operatorname{\underline{add}}D, ? \otimes_D M)\) such that the following diagram commutes: \[\label{CD:32explicit32equivalence32functor} \begin{CD} (\operatorname{mod}D, ? \otimes_D M) @>S>> \mathcal{S}(\operatorname{mod}D,? \otimes_D M)\\ @VV\pi V @V\rotatebox{90}{\tiny \simeq} V\tilde{\pi}V \\ (\operatorname{\underline{add}}D, ? \otimes_D M) @>S>> \mathcal{S}(\operatorname{\underline{add}}D,? \otimes_D M) \end{CD}\qquad{(2)}\]
There are triangle equivalences \(\operatorname{\rm{D}_{sg}}(\Lambda)\simeq\mathcal{S}(\operatorname{mod}D, ?\otimes_D M)\) and \(\operatorname{\rm{D}_{sg}}(A)\simeq\mathcal{S}(\operatorname{\underline{add}}D, ?\otimes_D M)\). Hence, the functor \(\tilde{\pi}\) gives the equivalence \(\operatorname{\rm{D}_{sg}}(\Lambda)\xlongrightarrow{\simeq} \operatorname{\rm{D}_{sg}}(A)\).
**Proof.* (3): By Theorem 9, we have \(\operatorname{\rm{D}_{sg}}(\Lambda) \simeq \mathcal{S}(\operatorname{\underline{mod}}\Lambda, \Omega_\Lambda)\) and \(\operatorname{\rm{D}_{sg}}(A) \simeq \mathcal{S}(\operatorname{\underline{mod}}A, \Omega_A)\). Then, by Lemma 8, we have \(\mathcal{S}(\operatorname{\underline{mod}}\Lambda, \Omega_\Lambda)\simeq \mathcal{S}(\operatorname{\underline{add}}\Gamma, \Omega_\Lambda)\) and \(\mathcal{S}(\operatorname{\underline{mod}}A, \Omega_A) \simeq \mathcal{S}(\operatorname{\underline{mod}}A, \Omega_A)\). Finally, by Proposition 50 and Theorem 14, we have \(\mathcal{S}(\operatorname{\underline{add}}\Gamma, \Omega_\Lambda) \simeq \mathcal{S}(\operatorname{mod}D, ?\otimes_D M)\) and \(\mathcal{S}(\operatorname{\underline{mod}}A, \Omega_A)\simeq \mathcal{S}(\operatorname{\underline{add}}D, ?\otimes_D M)\).*
(1): Since \(D \simeq A/\operatorname{rad}A\), we have \(\operatorname{mod}D=\operatorname{add}_D D=\operatorname{add}_A D\). Thus there is a canonical projection \(\pi: \operatorname{mod}D \to \operatorname{\underline{add}}_A D\) which is full and dense. By Theorems 51 and 14, the left triangle structures of \((\operatorname{mod}D, ?\otimes_D M)\) and \((\operatorname{\underline{add}}_A D, ? \otimes_D M)\) are trivial. Therefore, \(\pi\) is a left triangle functor.
(2): By the universal property of stabilization, \(\pi\) induces a triangle functor \(\tilde{\pi}\) between their stabilizations such that the commutative diagram (?? ) holds. Now, we show that \(\tilde{\pi}\) is an equivalence. Since \(\pi\) is full and dense, by Lemma 8 (1)(3), \(\tilde{\pi}\) is also full and dense. For the faithfulness, we consider two morphisms \(f,f': S \to T\) between two simple \(A\)-modules \(S,T\) such that \(\pi(f)=\pi(f')\). If \(S \not\simeq T\), since \(\operatorname{Hom}_A(S,T)=0\), trivially \(f=f'=0\). If \(S\simeq T \not\in \operatorname{proj}A\), then \(\mathcal{P}(S,S)=0\) and hence \(\operatorname{Hom}_A(S,S)=\operatorname{\underline{Hom}}_A(S,S)\). Thus, \(f=\pi(f)=\pi(f')=f'\). If \(S\simeq T \in \operatorname{proj}A\), then \(S\otimes_D M \simeq \Omega(S) \simeq \operatorname{rad}S\)=0. Therefore, \(f \otimes_D M = f' \otimes_D M=0\). By Lemma 8 (2), \(\tilde{{\pi}}\) is faithful. ◻
We apply Corollary 52 to give criteria for weakly regular, Gorenstein, Iwanaga-Gorenstein and sg-Hom-finite Bäckström orders via the associated finite dimensional radical square zero algebras. The relationships between these classes of Bäckström orders are as presented in the introduction; see the figure (1 ).
Definition 54. [@ARS97]. Let \(A\) be a basic finite dimensional algebra. Assume that \(A/\operatorname{rad}A \simeq \prod_{i\in I}D_i\), where \(I\) is a finite index set and \(D_i\) are finite dimensional division algebras. Let \(\{e_i\}_{i\in I}\) be a complete set of primitive orthogonal idempotents of \(A\) in which \(e_i\) corresponds to \(D_i\). Let \(M_{ij}:=e_i(\operatorname{rad}A/\operatorname{rad}^2 A)e_j\) be a \((D_i,D_j)\)-bimodule for any \(i,j\in I\). The valued quiver \(Q_A\) is defined as follows.
The vertex set is \(I\), corresponding to the division algebras \(\{D_i\}_{i\in I}\). For any \(i,j\in I\), there is an arrow from \(i\) to \(j\) endowed with a valuation \((\operatorname{dim}(M_{ij})_{D_j}, \operatorname{dim}_{D_i}(M_{ij}))\). For a general finite dimensional algebra \(A\), its valued quiver \(Q_A\) is defined as the valued quiver of its corresponding basic algebra.
A valuation is called trivial if it is (1,1). The valued quiver \(Q_A\) is called acyclic if it contains no oriented cycle.
We first recall the results by Xiao-wu Chen on the classifications of the similar classes of a finite dimensional radical square zero algebras in terms of their valued quivers.
Proposition 55. [@Che11] [@Che12] Let \(A\) be a finite dimensional radical square zero algebra and \(Q_A\) be its valued quiver. The following statements hold.
\(A\) is weakly regular if and only if \(Q_A\) is acyclic.
\(A\) is self-injective if and only each connected component of \(Q_A\) is a single vertex or a cycle with trivial valuations.
\(A\) is Iwanaga-Gorenstein if and only if each connected component of \(Q_A\) is acyclic or is a cycle with trivial valuations.
\(A\) is sg-Hom-finite if and only if \(V(A)\) is semisimple if and only if \(Q_A\) is obtained from a disjoint union of oriented cycles with trivial valuations by adjoining sources and sinks with arbitrary values repeatedly.
Let \(\Lambda\) be a Bäckström order and \(A(\Lambda)\) be the finite dimensional radical square zero \(k\)-algebra associated to it; recall (7 ).
Corollary 56. The following statements are equivalent.
\(\Lambda\) is sg-Hom-finite;
\(A(\Lambda)\) is sg-Hom-finite;
\(V(\Lambda)=V(A(\Lambda))\) is semisimple.
\(Q_{A(\Lambda)}\) is obtained from a disjoint union of oriented cycles with trivial valuations by adjoining sources and sinks with arbitrary values repeatedly.
Corollary 57. The following statements are equivalent.
\(\Lambda\) is weakly regular;
\(\operatorname{\rm{D}_{sg}}(\Lambda)=0\);
\(A(\Lambda)\) is weakly regular.
\(\operatorname{\rm{D}_{sg}}(A(\Lambda))=0\);
\(Q_{A(\Lambda)}\) is acyclic.
Next, we classify Iwanaga-Gorenstein Bäckström orders. The following property is basic.
Lemma 58. [@Che12] Let \(\Lambda\) be any \(R\)-orders. Let \(Q\) be a indecomposable non-projective Gorenstein-projective \(\Lambda\)-module. Then the kernel of projective cover of \(Q\) is also indecomposable non-projective Gorenstein-projective.
Proposition 59. Assume that \(\Lambda\) is a Bäckström order and \(A:=A(\Lambda)=D\oplus M\) is the finite dimensional radical square algebra associated to it. Let \(\pi,\tilde{\pi}\) be the functors in the commutative diagram (?? ) of Proposition 53. Then \(\pi\) restricts to a triangle equivalence \(\pi':\operatorname{\underline{Gproj}}\Lambda \to \operatorname{\underline{Gproj}}A\). More precisely, we have the following commutative diagram: \[\begin{tikzcd} (\operatorname{\underline{Gproj}}\Lambda, \Omega_\Lambda) \arrow[d,"\pi'","\rotatebox{90}{\tiny \simeq}"'] \arrow[r,hookrightarrow, "\subseteq"] & (\operatorname{\underline{add}}_\Lambda \Gamma, \Omega_\Lambda) \arrow[r, "\simeq"] & (\operatorname{mod}D, ? \otimes_D M) \arrow[r, "S"] \arrow[d,"\pi"] & \mathcal{S}(\operatorname{mod}D,? \otimes_D M) \arrow[d,"\tilde{\pi}","\rotatebox{90}{\tiny \simeq}"']\\ (\operatorname{\underline{Gproj}}A,\Omega_A) \arrow[r, hookrightarrow, "\subseteq"] & (\operatorname{\underline{add}}_A D, \Omega_A) \arrow[r,"\simeq"]& (\operatorname{\underline{add}}D, ? \otimes_D M) \arrow[r,"S"] & \mathcal{S}(\operatorname{\underline{add}}D,? \otimes_D M) \end{tikzcd}\]
**Proof.* Recall that by Proposition 50 and Theorem 14, we have left triangle equivalences \(a: (\operatorname{\underline{add}}_\Lambda \Gamma, \Omega_\Lambda) \xrightarrow{\simeq}(\operatorname{mod}D, ? \otimes_D M)\) and \(b: (\operatorname{\underline{add}}_A D, \Omega_A) \xrightarrow{\simeq} (\operatorname{\underline{add}}D, ?\otimes_D M)\). Recall that \(\operatorname{\underline{Gproj}}\Lambda\) and \(\operatorname{\underline{Gproj}}A\) are triangulated subcategories of \((\operatorname{\underline{add}}_\Lambda \Gamma, \Omega_\Lambda)\) and \((\operatorname{\underline{add}}_A D, \Omega_A)\) respectively; see Section 2.3.*
Firstly, we show that the essential image \(b^{-1} \pi a (\operatorname{\underline{Gproj}}\Lambda)\) is contained in \(\operatorname{\underline{Gproj}}A\), and thus the triangle functor \(\pi':=b^{-1}\pi a : \operatorname{\underline{Gproj}}\Lambda \to \operatorname{\underline{Gproj}}A\) is well-defined. Since \(\pi\) is a full left triangle functor, \(b^{-1} \pi a (\operatorname{\underline{Gproj}}\Lambda)\) is a triangle subcategory of \((\operatorname{\underline{add}}_A D, \Omega_A)\) and hence of \((\operatorname{\underline{mod}}A, \Omega_A)\). By Proposition 17, we have \(b^{-1} \pi a (\operatorname{\underline{Gproj}}\Lambda) \subseteq \operatorname{\underline{Gproj}}A\). Thus, we have the following commutative diagram. \[\begin{tikzcd} (\operatorname{\underline{Gproj}}\Lambda, \Omega_\Lambda) \arrow[r,hookrightarrow,"\subseteq"] \arrow[d,dotted, "\pi'"]& (\operatorname{\underline{add}}_\Lambda \Gamma, \Omega_\Lambda) \arrow[r, "a", "\simeq"'] & (\operatorname{mod}D, ? \otimes_D M) \arrow[r, "S"] \arrow[d, "\pi"] & \mathcal{S}(\operatorname{mod}D,? \otimes_D M) \arrow[d,"\tilde{\pi}","\rotatebox{90}{\tiny \simeq}"']\\ (\operatorname{\underline{Gproj}}A,\Omega_A) \arrow[r, hookrightarrow, "\subseteq"] & (\operatorname{\underline{add}}_A D, \Omega_A) \arrow[r, "b", "\simeq"']& (\operatorname{\underline{add}}D, ? \otimes_D M) \arrow[r,"S"] & \mathcal{S}(\operatorname{\underline{add}}D,? \otimes_D M) \end{tikzcd}\]
Secondly, we show that \(\pi'\) is full and faithful. Trivially, \(\pi'\) is full since \(\pi\) is full. To prove the faithfulness, we consider a morphism \(f: X \to Y\) between two indecomposable non-projective Gorenstein-projective \(\Lambda\)-modules \(X\) and \(Y\) such that \(b^{-1}\pi a (f)=0\). We will show that \(a(f)=0\). Note that \(a(X)\) and \(a(Y)\) are simple \(A\)-modules. If \(a(X) \not\simeq a(Y)\), then clearly \(a(f)=0\). If \(a(X)\simeq a(Y) \not\in \operatorname{proj}A\), then \(\operatorname{Hom}_A(a(X),a(Y))=\operatorname{\underline{Hom}}_A(a(X),a(Y))\), and thus \(a(f)=\pi a(f)=0\). If \(a(X)\simeq a(Y) \in \operatorname{proj}A\), then \(a(\Omega_\Lambda(X)) \simeq a(X)\otimes_D M =\operatorname{rad}a(X)=0\), which is a contradiction to \(\operatorname{pd}X=\infty\).
Thirdly, we show that \(\pi'\) is dense. Consider the decomposition of \(D\) as an \(A\)-module \(D=D_0 \oplus D_1\) such that \(D_1 \in \operatorname{proj}A\) and each direct summand of \(D_0\) is not \(A\)-projective. Then there is an equivalence \(c: \operatorname{add}D_0 \xrightarrow{\simeq} \operatorname{\underline{add}}D\) as additive categories. Moreover, the inclusion functor \(i: \operatorname{add}D_0 \hookrightarrow \operatorname{mod}D\) is left adjoint to the projection \(c^{-1}\pi: \operatorname{mod}D \to \operatorname{add}D_0\). By Lemma 58, the essential image \(ic^{-1} b (\operatorname{\underline{Gproj}}A)\) is a triangle subcategory of \((\operatorname{mod}D, \otimes_D M)\). So, \(a^{-1}ic^{-1} b (\operatorname{\underline{Gproj}}A)\) is a triangle subcategory of \((\operatorname{\underline{add}}_\Lambda \Gamma, \Omega_\Lambda)\) and hence of \((\operatorname{\underline{mod}}\Lambda, \Omega_\Lambda)\). By Proposition 17, we have \(a^{-1}ic^{-1} b (\operatorname{\underline{Gproj}}A) \subseteq \operatorname{\underline{Gproj}}\Lambda\). Then for any \(X\in \operatorname{\underline{Gproj}}A\), we have \[\pi'a^{-1}ic^{-1}ib(X)=(b^{-1}\pi a)(a^{-1}ic^{-1}ib)(X)\simeq X,\] which implies that \(\pi'\) is dense. ◻
Remark 60. In the proof, \(\operatorname{add}D_0 \subseteq \operatorname{mod}D\) is not necessarily a left triangulated subcategory. Because even for a non-projective simple \(A\)-module \(S\), \(\Omega(S)\) may have an \(A\)-projective direct summand. However, by Lemma 58, this is true for \(ic^{-1} b (\operatorname{Gproj}A) \subseteq \operatorname{mod}D\).
Theorem 61. Assume that \(\Lambda\) is a Bäckström order and \(A:=A(\Lambda)=D\oplus M\) is the finite dimensional radical square algebra associated to it. The following statements are equivalent.
\(\Lambda\) is Iwanaga-Gorenstein;
\(A(\Lambda)\) is Iwanaga-Gorenstein;
Each connected component in the species of \(Q_{A(\Lambda)}\) is acyclic or is a cycle with trivial valuations.
**Proof.* By Proposition 59, the canonical functor \(\operatorname{\underline{Gproj}}\Lambda \to \operatorname{\rm{D}_{sg}}(\Lambda)\) is a triangle equivalence if and only if the canonical functor \(\operatorname{\underline{Gproj}}A \to \operatorname{\rm{D}_{sg}}(A)\) is a triangle equivalence. Then (1) \(\Leftrightarrow\) (2) follows by Theorem 32. (2) \(\Leftrightarrow\) (3) is by Proposition 55 (3). ◻*
We finally classify Gorenstein Bäckström orders.
Theorem 62. Assume that \(\Lambda\) is a Bäckström order and \(A(\Lambda)=D\oplus M\) is the finite dimensional radical square zero algebra associated to it. The following statements are equivalent.
\(\Lambda\) is Gorenstein;
\(A(\Lambda)\) is a product of non-simple self-injective algebras;
Each connected component of \(Q_{A(\Lambda)}\) is a cycle with trivial valuations.
**Proof.* (1) \(\Rightarrow\) (2): Let \(\Lambda\) be Gorenstein. Then we have \(\operatorname{CM}\Lambda=\operatorname{Gproj}\Lambda\); see [@EJ00]. So, each non-projective Cohen-Macaulay module has an infinite projective dimension, which implies that each component of \(Q_{A(\Lambda)}\) cannot be acyclic. Note that \(\Lambda\) is in particular Iwanaga-Gorenstein. Then, by Theorem 61, each connected component of \(Q_{A(\Lambda)}\) is a cycle with trivial valuations. So, \(A(\Lambda)\) is a product of some non-simple self-injective algebras.*
(2) \(\Rightarrow\) (1): Assume conversely that \(A(\Lambda)\) is a product of some non-simple self-injective (i.e. non-simple Gorenstein) algebras. Then \(\operatorname{\underline{Gproj}}A(\Lambda) \subseteq \operatorname{\underline{add}}D \subseteq \operatorname{\underline{mod}}A(\Lambda)\) is an equality. To show that \(\Lambda\) is Gorenstein, it is equivalently to show \(\operatorname{proj}\Lambda = \operatorname{inj}\Lambda\); see [@IW; @14 Lemma 2.15]. Without loss of generality, we may assume \(\Lambda\) to be ring-indecomposable and non-hereditary. Since each projective \(A(\Lambda)\)-module is not simple, we have \(\operatorname{\underline{add}}D = \operatorname{mod}D\). Therefore, \(\operatorname{\underline{Gproj}}\Lambda = \operatorname{\underline{add}}\Gamma\) by Proposition 59.
We claim that \(\operatorname{proj}\Lambda \cap \operatorname{proj}\Gamma=\{0\}\). In fact, if there exists \(0\neq Q\in\operatorname{ind}(\operatorname{proj}\Lambda \cap \operatorname{proj}\Gamma)\), then by the assumptions that \(\Lambda\) is ring-indecomposable and \(\Lambda \neq \Gamma\), we may assume that \(\operatorname{corad}Q\in\operatorname{ind}(\operatorname{proj}\Gamma)\backslash\operatorname{proj}\Lambda\). Otherwise, using the notions in [@HN94], the unique \(\Gamma\)-composition series \(Q\subsetneq \operatorname{corad}Q \subsetneq \operatorname{corad}^2(Q) \subsetneq \dots \subsetneq \operatorname{corad}^i(Q)\simeq Q\) with \(\operatorname{corad}^j (Q)\in\operatorname{ind}(\operatorname{proj}\Lambda \cap \operatorname{proj}\Gamma)\), is also the unique \(\Lambda\)-composition series of \(Q\). Then \(\Lambda=\Gamma\), which is a contradiction.
Since \(\operatorname{corad}Q \not \in \operatorname{proj}\Lambda\), \(\mathbb{F}(\operatorname{corad}Q)\) is a non-projective injective \(H\)-module (recall Theorem 40). Since \(H\) is a finite dimensional hereditary radical square zero \(k\)-algebra and \(\Lambda/\operatorname{rad}\Lambda \neq \Gamma/\operatorname{rad}\Gamma\), there exists a projective cover \(p: P \to I\) in \(\operatorname{mod}H\) of some indecomposable non-projective injective \(H\)-module \(I\) such that \(\operatorname{soc}\mathbb{F}(\operatorname{corad}Q)=(\operatorname{corad}Q/Q,0)\) is a direct summand of the kernel of \(p: P\rightarrow\mathrel{\mkern-14mu}\rightarrow I\), namely, we have an exact sequence in \(\operatorname{mod}H\): \[0 \to (\operatorname{corad}Q/Q,0) \oplus S \to P \xrightarrow{p} I \to 0,\] for some semisimple projective \(H\)-module \(S\). Then Corollary 46 implies that there exists an exact sequence of \(\Lambda\)-modules: \[0 \to Q \oplus \tilde{S} \to \tilde{P} \xrightarrow{\tilde{p}} \tilde{I} \to 0,\] where \(\mathbb{F}(\tilde{p})=p\) is a projective cover of some \(\tilde{I} \in \operatorname{ind}(\operatorname{proj}\Gamma)\). Since \(\operatorname{\underline{Gproj}}\Lambda = \operatorname{\underline{add}}\Gamma\), \(\tilde{I}\in \operatorname{ind}(\operatorname{Gproj}\Lambda)\). Since \(Q\in\operatorname{proj}\Lambda\), then \(Q=0\) by Lemma 58, which is a contradiction. This finishes the proof of our claim.
Now, Proposition 42 and our claim imply that every non-projective \(\Lambda\)-module has infinite projective dimension. Let \(N \in \operatorname{inj}\Lambda\). By Lemma 21, we have \(\operatorname{Gpd}N=\operatorname{pd}N\). If \(N\) is not projective, then \(\operatorname{Gpd}N=\infty\), which leads to a contradiction to \(\operatorname{\underline{Gproj}}\Lambda = \operatorname{\underline{add}}\Gamma\). Therefore, \(\operatorname{inj}\Lambda\subseteq\operatorname{proj}\Lambda\). Finally, since we have the same number of the isomorphism classes of indecomposable projective and injective objects, we have \(\operatorname{inj}\Lambda=\operatorname{proj}\Lambda\). ◻
Throughout this section, we assume that \((R,\mathfrak{m},k)\) is a complete discrete valuation ring and \(\mathfrak{m}=\pi R\). Let \(\Lambda=(\Lambda,\Gamma)\) be a Bäckström order and keep the notations in Sections 3.2 and 4. In this section, we present several examples to illustrate our results.
To compute the associated finite dimensional radical square zero \(k\)-algebra \(A(\Lambda)\), we need to compute the semisimple algebra \(D=\operatorname{\underline{End}}_\Lambda(\Gamma)\) and the \((D,D)\)-bimodule \(M=\operatorname{\underline{Hom}}_\Lambda(\Gamma, \operatorname{Ker}\mu)\), where \(\mu: \Gamma \otimes_R \Lambda \rightarrow\mathrel{\mkern-14mu}\rightarrow\Gamma\) is the multiplication map.
Let \(\Omega(\Gamma)\) be the kernel of the projective cover of \(\Gamma\) in \(\operatorname{CM}\Lambda\). Then, as a right \(D\)-module, we have \[M\simeq \operatorname{\underline{Hom}}_\Lambda(\Gamma,\Omega(\Gamma)),\] which can be computed via Proposition 48. The left \(D\)-module structure is given by the \(k\)-algebra homomorphism on the endomorphism rings induced by the syzygy functor \[\Omega: D=\operatorname{\underline{End}}_\Lambda(\Gamma) \to \operatorname{\underline{End}}_\Lambda(\Omega(\Gamma)).\]
We start with an easy example, Example 63. An example involving a field extension is given in Example 64. Example 65 illustrates the hierarchy discussed in the introduction.
Example 63. Let \(\Gamma=k[\![x_1]\!] \times k[\![x_2]\!] \times \dots \times k[\![x_{n}]\!]\) and \(x=(x_1, x_2, \dots, x_{n})\) for \(n\geq 2\). Let \(\Lambda=k+x\Gamma\). Denote \(P_{n+1}=\Lambda\), \(Q_{j}=k[\![x_j]\!]\) and \(k_j=Q_j/\operatorname{rad}Q_j\simeq k\) for \(1\leq j\leq n\). The projective cover of \(Q_{j}\) is \[0 \to \bigoplus_{j'\neq j} Q_{j'} \to P_{n+1} \to Q_j \to 0.\] Thus, as a right \(D\)-module, \(M=\bigoplus^n_{j=1}(k^{\oplus n-1}_j)\). The syzygy functor is the diagonal map \[\begin{CD} \Omega: \quad @. \operatorname{\underline{End}}_\Lambda (Q_j) @>>> \operatorname{\underline{End}}_\Lambda (\Omega(Q_j))\\ @. @| @| \\ \Delta: \quad @. k_j @>>> \prod_{j'\neq j}k_{j'} \end{CD}\] Represented as matrices, the semisimple algebra \(D\) and the \((D,D)\)-bimodule \(M\) are given by \[D= \begin{pmatrix} k & 0 & \cdots & 0& 0 \\ 0 & k & \cdots & 0& 0 \\ \vdots& \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & k& 0 \\ 0 & 0 & \cdots & 0& k \end{pmatrix}, \quad M = \begin{pmatrix} 0 & k & \cdots &k &k \\ k & 0 & \cdots &k &k\\ \vdots&\vdots & \ddots & \vdots & \vdots\\ k & k & \cdots &0 &k\\ k & k & \cdots &k &0 \end{pmatrix}.\]
Therefore, \(Q_{A(\Lambda)}\) is a quiver with vertices \(1\leq j \leq n\) and exactly one arrow in each direction between any two distinct vertices. The finite dimensional \(k\)-algebra with radical square zero \(A(\Lambda)\) is the path algebra \(kQ_{A(\Lambda)}\) with relations that all paths with length at least two vanish. For instance, when \(n=3\), then \[Q_{A(\Lambda)}: \begin{tikzcd}[column sep=1em] & 1 \arrow[ld, shift left] \arrow[rd,shift left] & \\ 2 \arrow[ru] \arrow[rr,shift left]& & 3 \arrow[lu] \arrow[ll] \end{tikzcd}\]
In this example, \(\Lambda\) is sg-Hom-finite if and only if \(\Lambda\) is Iwanaga-Gorenstein if and only it is Gorenstein if and only if \(n=2\).
Example 64. Assume \(l\) to be a field extension of \(k\) with degree \(n\geq 2\). Put \[\Lambda=k+xl[\![x]\!],\qquad \Gamma=l[\![x]\!],\] According to Proposition 48, the semisimple \(k\)-algebra \(D=\operatorname{\underline{End}}_\Lambda(\Gamma)=\Gamma/\operatorname{rad}\Gamma=l\). Consider the exact sequence (6 ) \[0 \to \operatorname{Ker}\mu \to \Gamma \otimes_R \Lambda \xrightarrow{\mu} \Gamma \to 0.\]
Claim 1. Consider an exact sequence \[0 \to Z \to l \otimes_k l \xrightarrow{\mu'} l \to 0,\] where \(\mu'\) is the multiplication map. Then the \((D,D)\)-bimodule \(M=\operatorname{\underline{Hom}}_\Lambda(\Gamma,K)\) is isomorphic to \(Z\), where \(K=\operatorname{Ker}\mu\).
Therefore, their \(k\)-algebra with radical square zero \(A(\Lambda)\) and its valued quiver are \[A(\Lambda)=l\oplus Z = l \oplus (\operatorname{Ker}(l \otimes_k l \xrightarrow[]{\mu'}\mathrel{\mkern-14mu}\rightarrow l)), \quad Q_{A(\Lambda)}: \begin{tikzcd} \bullet \arrow[out=-30,in=30,loop,swap,"{(n-1,n-1)}"] \end{tikzcd}\]
In this example, \(\Lambda\) is sg-Hom-finite if and only if \(\Lambda\) is Iwanaga-Gorenstein if and only it is Gorenstein if and only if \(n=2\).
\[\label{eqa:ex2} \begin{tikzcd}[row sep=1.25em, column sep={between origins, 3.5em}] & & & & &0 \arrow[dd] & &0 \arrow[dd] & & \\ & &0 \arrow[dd]
& &0 \arrow[dd] & &0 \arrow[dd] & & & \\ &0 \arrow[rr] & &K \arrow[rr] \arrow[dl,equal] \arrow[dd] & & l \otimes_k xl[\![x]\!]\arrow[rr] \arrow[dl,equal] \arrow[dd] & &xl[\![x]\!] \arrow[rr]
\arrow[dl,equal] \arrow[dd] & &0\\
0 \arrow[rr] & &K \arrow[rr,crossing over] \arrow{dd} & &l\otimes_k xl[\![x]\!] \arrow[rr,crossing over] \arrow[dd] & &xl[\![x]\!] \arrow[rr, crossing over] \arrow[dd] & &0 & \\ &0 \arrow[rr] & &K \arrow[rr]
\arrow[dl] & &l\otimes_k \Lambda \arrow[rr,"\mu"near start] \arrow[dl] \arrow[dd] & &l[\![x]\!] \arrow[rr] \arrow[dl,equal] \arrow[dd] \arrow[dd] & &0\\
0 \arrow[rr] & &Y \arrow[rr] \arrow[dd] & & l\otimes_k l[\![x]\!] \arrow[rr,crossing over] \arrow[dd] & & l[\![x]\!] \arrow[rr,crossing over] \arrow[dd] & &0 & \\ & & & & &l
\arrow[rr,"\simeq"near start] \arrow[dl] \arrow[dd] & &l \arrow[dl,equal] \arrow[dd] & &\\
0 \arrow[rr] & &Z \arrow[rr] \arrow[dd] & &l \otimes_k l \arrow[rr,"\mu'",crossing over] \arrow[dd] & &l \arrow[rr,crossing over] \arrow[dd] & &0 & \\ & & & & &0 & &0 & &
\\ & &0 & &0 & &0 & & & \arrow[from=2-3, to=4-3, crossing over] \arrow[from=2-5, to=4-5, crossing over] \arrow[from=3-4, to=5-4, "\rotatebox{270}{\simeq}" near start] \arrow[from=4-3, to=6-3, crossing over]
\arrow[from=4-3, to=4-5, crossing over] \arrow[from=4-5, to=6-5, crossing over] \arrow[from=4-7, to=6-7, crossing over] \arrow[from=6-7, to=8-7, crossing over] \end{tikzcd}\tag{8}\]
Note that \(\Gamma \otimes_R \Lambda \simeq l \otimes_k \Lambda\) and \(\mu: \Gamma \otimes_R \Lambda \to \Gamma\) induces an isomorphism \[\mu: \Gamma \otimes_R
\Lambda \otimes_\Lambda (\Lambda/\operatorname{rad}\Lambda) \xrightarrow{\simeq} \Gamma \otimes_\Lambda (\Lambda/\operatorname{rad}\Lambda),\] where both sides are isomorphic to \(l\). So, in this case, \(\mu\) is a projective cover and \(K\) is a \((\Gamma,\Gamma)\)-bimodule. Then by Proposition 48, we have \[M=\operatorname{\underline{Hom}}_\Lambda(\Gamma,K) = K/(K\operatorname{rad}\Gamma)\] as a \((D,D)\)-bimodule. Moreover, by
Proposition 45 and Corollary 46, we have the big
commutative diagram (8 ) and \(K=Y\operatorname{rad}\Gamma, Z \simeq Y/(Y\operatorname{rad}\Gamma)\). On the other hand, multiplying \(x\) induces an isomorphism
\(Y\operatorname{rad}\Gamma\simeq Y\). Therefore, \[M = K/(K\operatorname{rad}\Gamma)= (Y \operatorname{rad}\Gamma / Y (\operatorname{rad}\Gamma)^2)\simeq Y/(Y\operatorname{rad}\Gamma)\simeq
Z.\]
Example 65. Define an \(n\times n\) matrix \[\Gamma= \begin{pNiceMatrix}[last-col] R & R & \cdots & R &Q_1 \\ \pi R & R & \cdots & R &Q_2 \\ \vdots & \vdots & \ddots & \vdots & \;\, \vdots \\ \pi R & \pi R & \cdots & R & Q_n \end{pNiceMatrix}\] The labels on the right denote the row vectors which are the indecomposable projective \(\Gamma\)-modules.
Consider a partition \[J=\mathbb{Z}/n\mathbb{Z}=\{1,2,\dots,n\}=J_1 \sqcup J_2 \sqcup \dots \sqcup J_u.\]
Define \[\Lambda=\Lambda[J_1,J_2,\dots,J_u]:=\{(\gamma_{st})\in \Gamma \mid \gamma_{ss}-\gamma_{tt} \in \pi R, \text{ for any s,t \in J_v and 1 \leq v \leq u}\}.\] Then \((\Lambda,\Gamma)\) is a Bäckström order. Define a subset \(J'\subseteq J\) by removing all parts of the partition which consist of a single element. This is equivalent to say that \(J'=\{j \in J \mid Q_j \not \in \operatorname{proj}\Lambda\}\).
Let \(k_j=Q_j/\operatorname{rad}Q_j\simeq k\) for \(j\in J\). For \(1\leq v \leq u\), let \(P_v\) be the indecomposable projective \(\Lambda\)-module corresponding to the partition \(J_v\). The projective cover of a non-\(\Lambda\)-projective \(Q_j\) for \(j \in J_v\) is \[0 \to \bigoplus_{j'\in J_v \backslash \{j\}} Q_{j'+1} \to P_v \to Q_j \to 0.\] So, we have the semisimple \(k\)-algebra \[D=\operatorname{\underline{End}}_\Lambda (\Gamma)=\prod_{j\in J'}k_j,\] and as a right \(D\)-module \[\operatorname{\underline{Hom}}_\Lambda(\Gamma, \Omega(Q_j))\simeq \bigoplus_{j'} k_{j'+1},\] where the direct sum is over all \(j'\in J_v \backslash \{j\}\) such that \(j'+1 \in J'\), or equivalently such that \(Q_{j'+1}\) is not a projective \(\Lambda\)-module. The syzygy functor is given by the diagonal map \[\begin{CD} \Omega: \quad @. \operatorname{\underline{End}}_\Lambda (Q_j) @>>> \operatorname{\underline{End}}_\Lambda (\Omega(Q_j))\\ @. @| @| \\ \Delta: \quad @. k_j @>>> \prod_{j'}k_{j'+1} \end{CD}\] where the product is over all \(j'\in J_v \backslash \{j\}\) satisfying the condition as the direct sum above.
Therefore, the valued quiver \(Q_{A(\Lambda)}\) can be obtained combinatorially as follows. For \(1 \leq v \leq u\), let \(G_v\) be the quiver with vertices \(J_v\) and exactly one arrow in each direction between any two distinct vertices. The valued quiver \(Q_{A(\Lambda)}\) is obtained from the disjoint union of all \(G_v\) for \(1 \leq v \leq u\), by
For each arrow from \(j \to j'\), change the target \(j'\) to \(j'+1\).
Delete the vertices \(j\) such that \(Q_{j+1}\in \operatorname{proj}\Lambda\) and the arrows adjoining to them.
The finite dimensional \(k\)-algebra with radical square zero \(A(\Lambda)\) is the path algebra \(kQ_{A(\Lambda)}\) with relations that all paths with length at least two vanish.
For instance,
For \(n=2s,s\geq 1\) and \(\Lambda=\Lambda[\{1,2\},\{2,3\},\dots,\{2s-1,2s\}]\). Then \[Q_{A(\Lambda)}: \begin{tikzcd}[column sep=0.8em] 1 \arrow[r] & 3 \arrow [r] & \cdots \arrow[r] & 2s-1 & \times & 2 \arrow[out=65,in=110,loop,swap] & \times & 4 \arrow[out=65,in=110,loop,swap] & \times & \cdots & \times & 2s \arrow[out=65,in=110,loop,swap] \arrow[from=1-4, to=1-1, bend right=45] \end{tikzcd}\] and \(\Lambda\) is a Gorenstein Bäckström order.
For \(n=2s+1, s \geq 1\) and \(\Lambda=\Lambda[\{1,2\},\{2,3\},\dots,\{2s-1,2s\},\{2s+1\}]\). Then \[Q_{A(\Lambda)}: \begin{tikzcd}[column sep=0.8em] 1 \arrow[r] & 3 \arrow [r] & \cdots \arrow[r] & 2s-1 & \times & 2 \arrow[out=65,in=110,loop,swap] & \times & 4 \arrow[out=65,in=110,loop,swap] & \times & \cdots & \times & 2s \arrow[out=65,in=110,loop,swap] \end{tikzcd}\] and \(\Lambda\) is an Iwanaga-Gorenstein but not a Gorenstein Bäckström order.
For \(n=2s+1,s\geq 1\) and \(\Lambda=\Lambda[\{1,3,\dots,2s+1\},\{2\},\{4\} \cdots,\{2s\}]\). Then \[Q_{A(\Lambda)}: \begin{tikzcd}[row sep=1em, column sep=1em] & & 1 \arrow[out=65,in=110,loop,swap] & & \\ 3 \arrow[urr] & 5\arrow[ur] & \cdots & 2s-3 \arrow[ul] & 2s-1 \arrow[ull] \end{tikzcd} \times 2s+1\] and \(\Lambda\) is a sg-Hom-finite but not an Iwanaga-Gorenstein Bäckström order.
For \(n=2s+2,s\geq 1\) and \(\Lambda=\Lambda[\{1,3,\dots,2s+1\},\{2\},\{4\} \cdots,\{2s+2\}]\). Then \[Q_{A(\Lambda)}: \begin{tikzcd}[column sep=0.5em] 1 & \times & 3 & \times & \cdots & \times & 2s+1 \end{tikzcd}\] and \(\Lambda\) is of global dimension 2.
Acknowledgement 1. The author would like to thank his supervisor Osamu Iyama for suggesting this problem and for his constant encouragement, insightful discussions, and valuable comments. The author also thanks to Xiao-wu Chen for helpful comments on the paper. The author is also grateful to his labmates, especially Ryu Tomonaga, for their helpful discussions and support throughout his studies.
*labels=alphabetic