Singular Equivalences and ϕ-Dimensions

Abstract

The $\varphi$-dimension of an algebra is a homological invariant arising from the Igusa–Todorov function. In this paper, we establish a new bound on the $\varphi$-dimensions of two algebras related by singular equivalences of the Morita-type with level.

1. Introduction

In an attempt to prove the famous finitistic dimension conjecture, Igusa and Todorov defined in [1] two functions $\varphi$ and $\psi$, which map each finitely generated module to the natural numbers. These Igusa–Todorov functions determine new homological measures, generalizing the notion of projective dimension, and have become a powerful tool in the understanding of the finitistic dimension conjecture [1,2,3]. According to [4,5], the $\varphi$-dimension of an Artin algebra A is defined as

$$\varphi \dim \left(A\right):=\sup \left\{\varphi \right(M\left)\right|M\in \mathrm{mod}A\}.$$

The $\varphi$-dimension of an algebra A has a strong connection with its global dimension and finitistic dimension:

$$\mathrm{fin}.\dim \left(A\right)\le \varphi \dim \left(A\right)\le \mathrm{gl}.\dim \left(A\right),$$

and they all coincide in the case where $\mathrm{gl}.\dim \left(A\right)<\infty$. Moreover, the $\varphi$-dimension can be used to describe selfinjective algebras: an algebra A is selfinjective if and only if $\varphi \dim \left(A\right)=0$ [4]. Recently, various works have been dedicated to studying and generalizing the properties of Igusa–Todorov functions and the $\varphi$-dimension [3,5,6,7]. While many upper bounds for the $\varphi$-dimension of a given algebra have been established (see [7,8,9]), determining its precise value remains a challenging task. To gain a deeper understanding of the $\varphi$-dimensions for arbitrary algebras, it is reasonable to investigate the relationships between the $\varphi$-dimensions of different algebras that are connected in certain nice ways.

A useful type of natural linkage among algebras is through derived equivalences, and the main result in [6] shows that the difference between the $\varphi$-dimensions of two algebras, which are derived equivalent, is less than the term length of the tilting complex. In this paper, we will extend this result on derived equivalences to the broader framework of singular equivalences of Morita-type with level, a concept introduced by Wang [10]. This equivalence is analogous to the notion of stable equivalences of Morita-type [11], and it arises naturally in representation theory [12,13,14,15]. Indeed, derived equivalences are singular equivalences of Morita-type with level, and there are many other examples of singular equivalences that go beyond the derived equivalences. Nowadays, singular equivalences of Morita-type with level have been shown to preserve various homological conjectures, such as the finitistic dimension conjecture and Keller’s conjecture [10,16]. Moreover, many homological properties are invariant under these equivalences, such as the Hochschild (co)homology [17,18], the Fg condition [19,20], the properties of syzygy-finite and injectives generation [14], the extension dimensions [21] and the Igusa–Todorov distance [22]. In this paper, we will explore the behavior of the $\varphi$-dimensions under singular equivalences of Morita-type with level.

Theorem 1 (Refer to Theorem 3).

Let A and B be two finite dimensional algebras over a field. If A and B are singular equivalent of Morita-type with level n, then

$$\left|\varphi \dim \left(A\right)-\varphi \dim \left(B\right)\right|\le n.$$

Theorem 1 can be applied to homological ideals, bounded extensions, Morita rings, and triangular matrix algebras to give some estimations on $\varphi$-dimension; see Corollary 2, Corollary 3, Corollary 5, and Corollary 6.

The paper is structured as follows. In Section 2, we recall some relevant definitions and conventions. In Section 3, we investigate the behavior of the $\varphi$-dimensions under singular equivalences of Morita type with level, and we prove Theorem 1. In Section 4, we give some applications and concrete examples.

2. Definitions and Conventions

Throughout, k is a fixed field, all algebras are finite dimensional associative k-algebras with identity, and all modules are finitely generated left modules, unless stated otherwise.

Let A be a finite dimensional associative algebra over a field k. Denote by $\mathrm{mod}A$ the category of finitely generated left A-modules, and we view right A-modules as left $A^{op}$-modules, where $A^{op}$ is the opposite algebra of A. Let $\mathrm{proj}A$ be the full subcategory consisting of all finitely generated projective modules over A. We denote by $\underline{\mathrm{mod}}A$ the stable module category of $\mathrm{mod}A$ modulo morphisms factoring through projective modules. Let ${\mathcal{D}}^b\left(A\right)$ be the bounded derived category of complexes over $\mathrm{mod}A$, and let $K^b\left(\mathrm{proj}A\right)$ be the bounded homotopy category of complexes over $\mathrm{proj}A$.

Following [23,24], the singularity category of A is defined to be the Verdier quotient $D_{sg}\left(A\right)={\mathcal{D}}^b\left(A\right)/K^b\left(\mathrm{proj}A\right)$. Let $A^e=A{\otimes }_k{A}^{op}$ be the enveloping algebra of A. We identify A-A-bimodules with left $A^e$-modules, and denote by ${\Omega }_{A^e}(-)$ the syzygy functor on the stable category $\underline{\mathrm{mod}}{A}^e$ of A-A-bimodules. The following terminology is from Wang [10].

Definition 1

([10]). Let ${}_{A}M_B$ and ${}_{B}N_A$ be an A-B-bimodule and a B-A-bimodule, respectively, and let $n\ge 0$. We say $(M,N)$ defines a singular equivalence of Morita-type with level n, provided that the following conditions are satisfied:

(1) 

The four one-sided modules ${}_{A}M$, $M_B$, ${}_{B}N$, and $N_A$ are all finitely generated projective.

(2) 

There are isomorphisms $M{\otimes }_{B}N\cong {\Omega }_{A^e}^n\left(A\right)$ and $N{\otimes }_{A}M\cong {\Omega }_{B^e}^n\left(B\right)$ in $\underline{\mathrm{mod}}{A}^e$ and $\underline{\mathrm{mod}}{B}^e$, respectively.

For an algebra A, let $K\left(A\right)$ be the abelian group generated by all symbols $\left[M\right]$, with $M\in \mathrm{mod}A$, modulo the relations: (a) $\left[X\right]-\left[Y\right]-\left[Z\right]=0$ if $X\cong Y\oplus Z$ and (b) $\left[P\right]=0$ if P is projective. Then, $K\left(A\right)$ is the free abelian group generated by all isomorphism classes of finitely generated indecomposable non projective A-modules. The syzygy functor $\Omega$ gives rise to a group homomorphism $\Omega$ : $K\left(A\right)\to K\left(A\right)$. For any $M\in \mathrm{mod}A$, let $\langle \mathrm{add}M\rangle$ denote the subgroup of $K\left(A\right)$ generated by the classes of indecomposable summands of M. By the fitting Lemma, there is a non-negative integer n such that the group morphism $\Omega :{\Omega }^n\left(\langle \mathrm{add}M\rangle \right)\to {\Omega }^{n+1}\left(\langle \mathrm{add}M\rangle \right)$ is a monomorphism, where $\Omega$ is the syzygy functor. We define $\varphi \left(M\right)$ to be this n.

From [1,5], if $\mathrm{pd}\left(M\right)<\infty$, then $\varphi \left(M\right)=\mathrm{pd}\left(M\right)$, and $\varphi \left(M\right)<\infty$, in the case where $\mathrm{pd}\left(M\right)=\infty$. Therefore, $\varphi \left(M\right)$ is a generalization of $\mathrm{pd}\left(M\right)$.

Definition 2

([4,5]). For an algebra A, we define

$$\varphi \dim \left(A\right)=\sup \left\{\varphi \right(M\left)\right|M\in \mathrm{mod}A\}.$$

3. Singular Equivalence and ϕ-Dimension

In this section, we observe the behavior of the $\varphi$-dimensions under singular equivalences of Morita-type with level. In [6], the authors give another description of the $\varphi$-dimension in terms of the bi-functor ${\mathrm{Ext}}_A^i(-,-)$. Let us explain their work in detail.

Definition 3

([6]). Let d be a positive integer, $M\in \mathrm{mod}A$, and $X,Y\in \mathrm{add}\left(M\right)$ with $\mathrm{add}\left(X\right)\cap \mathrm{add}\left(Y\right)=0$. The pair $(X,Y)$ is called a d-Division of M, if ${\mathrm{Ext}}_A^d(X,-)\ncong {\mathrm{Ext}}_A^d(Y,-)$, and ${\mathrm{Ext}}_A^{d+1}(X,-)\cong {\mathrm{Ext}}_A^{d+1}(Y,-)$.

Theorem 2

([6] Theorem 3.6). Let A be a finite dimensional algebra, and $M\in \mathrm{mod}A$. Then, $\varphi \left(M\right)=\max \left(\right\{d\in \mathbb{N}:there is a d\text{-}Division of M\}\cup \{0\left\}\right)$.

With this homological description, Fernandes, Lanzilotta, and Mendoza showed that the difference between the $\varphi$-dimensions of two derived equivalence algebras is less than the length of the tilting complex [6]. Now, we will extend this result to singular equivalence.

Theorem 3.

Let A and B be two finite dimensional algebras, and let ${}_{A}M_B$ be an A-B-bimodule and ${}_{B}N_A$ be a B-A-bimodule. If $(M,N)$ defines a singular equivalence of Morita-type with level n, then

$$\left|\varphi \dim \left(A\right)-\varphi \dim \left(B\right)\right|\le n.$$

Proof. 

We first claim $\varphi \dim \left(B\right)\le \varphi \dim \left(A\right)+n$. If $\varphi \dim \left(B\right)\le n$, the claim holds clearly. Now, assume $\varphi \dim \left(B\right)>n$. Then, there exists $L\in \mathrm{mod}B$ such that ${\varphi }_B\left(L\right)=d>n$. It follows from Theorem 2 that there exist $U,V\in \mathrm{add}L$ such that

$$\left\{\begin{array}{cc}{\mathrm{Ext}}_B^d(U,-)\ncong {\mathrm{Ext}}_B^d(V,-),\hfill & \left(1\right)\hfill \\ {\mathrm{Ext}}_B^{d+1}(U,-)\cong {\mathrm{Ext}}_B^{d+1}(V,-).\hfill & \left(2\right)\hfill \end{array}\right.$$

Since $(M{\otimes }_B-,{\mathrm{Hom}}_A(M,-))$ is an adjoint pair, and $M{\otimes }_B-$ is an exact functor which preserves the projective modules, there is an induced isomorphism ${\mathrm{Ext}}_A^{d+1}(M{\otimes }_{B}U,-)\cong {\mathrm{Ext}}_B^{d+1}(U,{\mathrm{Hom}}_A(M,-))$. Then, it follows from (2) that ${\mathrm{Ext}}_A^{d+1}(M{\otimes }_{B}U,-)\cong {\mathrm{Ext}}_B^{d+1}(V,{\mathrm{Hom}}_A(M,-)),$ which is isomorphic to ${\mathrm{Ext}}_A^{d+1}(M{\otimes }_{B}V,-)$ by adjointness.

On the other hand, assume ${\mathrm{Ext}}_A^{d-n}(M{\otimes }_{B}U,-)\cong {\mathrm{Ext}}_A^{d-n}(M{\otimes }_{B}V,-)$. Then, we get ${\mathrm{Ext}}_B^{d-n}(N{\otimes }_{A}M{\otimes }_{B}U,-)\cong {\mathrm{Ext}}_A^{d-n}(N{\otimes }_{A}M{\otimes }_{B}V,-)$ by the same argument. That is, ${\mathrm{Ext}}_B^{d-n}({\Omega }_{B^e}^n\left(B\right){\otimes }_{B}U,-)\cong {\mathrm{Ext}}_B^{d-n}({\Omega }_{B^e}^n\left(B\right){\otimes }_{B}V,-)$, and thus, ${\mathrm{Ext}}_B^{d-n}({\Omega }_B^n\left(U\right),-)\cong {\mathrm{Ext}}_B^{d-n}({\Omega }_B^n\left(V\right),-).$ Therefore, ${\mathrm{Ext}}_B^d(U,-)\cong {\mathrm{Ext}}_B^d(V,-)$, which is a contradiction with $\left(1\right)$.

Above all, we obtain

$$\left\{\begin{array}{cc}{\mathrm{Ext}}_A^{d-n}(M{\otimes }_{B}U,-)\ncong {\mathrm{Ext}}_A^{d-n}(M{\otimes }_{B}V,-),\hfill & \left(3\right)\hfill \\ {\mathrm{Ext}}_A^{d+1}(M{\otimes }_{B}U,-)\cong {\mathrm{Ext}}_A^{d+1}(M{\otimes }_{B}V,-).\hfill & \left(4\right)\hfill \end{array}\right.$$

Then, it follows from Theorem 2 that ${\varphi }_A\left(M{\otimes }_B(U\oplus V)\right)\ge d-n$. Hence, we get $d-n\le \varphi \dim \left(A\right)$; that is, $d\le \varphi \dim \left(A\right)+n$, and hence, $\varphi \dim \left(B\right)\le \varphi \dim \left(A\right)+n$. Analogously, it can be shown that $\varphi \dim \left(A\right)\le \varphi \dim \left(B\right)+n$. □

By [10] (Theorem 2.3), a derived equivalence always induces a singular equivalence of Morita-type with level, but the converse is not true in general. Therefore, Theorem 3 extends the main result in [6] from the derived equivalence to a singular equivalence.

Corollary 1

([6] Theorem 4.10). Let A and B be two algebras, which are derived equivalent. If the length of the tilting complex is n, then

$$\left|\varphi \dim \left(A\right)-\varphi \dim \left(B\right)\right|\le n.$$

Proof. 

By [25] (Lemma 2.1) and [10] (Theorem 2.3), there exist an A-B-bimodule M and a B-A-bimodule N such that $(M,N)$ defines a singular equivalence of Morita-type with level n between A and B. Then, the statement follows from Theorem 3. □

4. Applications and Examples

In this section, we apply Theorem 3 to homological ideals, bounded extensions, Morita rings, and triangular matrix algebras to give some estimations on the $\varphi$-dimension.

Let A be an algebra and $I\subseteq A$ be a two-sided ideal. Following [26], I is a homological ideal if the canonical map $A\to A/I$ is a homological epimorphism; that is, the naturally functor ${\mathcal{D}}^b(A/I)\to {\mathcal{D}}^b\left(A\right)$ is fully faithful. Applying Theorem 3 to the homological ideal, we get the following corollary.

Corollary 2.

Let $I\subseteq A$ be a homological ideal, which has a finite projective dimension as an A-A-bimodule. Then,

$$\left|\varphi \dim \left(A\right)-\varphi \dim (A/I)\right|\le 2·\mathrm{pd}{(}_{A^e}A/I).$$

Proof. 

By [14] (Theorem 3.9), A and $A/I$ are singularly equivalent of Morita-type with level $2·\mathrm{pd}{(}_{A^e}A/I)$. Then, the corollary follows from Theorem 3. □

Recall that an extension $B\subseteq A$ of finite dimensional algebras is bounded if there exists $p\ge 1$ such that the tensor power ${(A/B)}^{{\otimes }_{B}p}$ vanishes, $A/B$ has finite projective dimension as a B-B-bimodule, and ${\mathrm{Tor}}_i^B(A/B,{(A/B)}^{{\otimes }_{B}j})=0$ for all $i,j\ge 1$. Applying Theorem 3 to the bounded extension, we get the following corollary.

Corollary 3.

Let $B\subseteq A$ be a bounded extension with ${(A/B)}^{{\otimes }_{B}p}=0$ for some $p\ge 1$. Then,

$$\left|\varphi \dim \left(A\right)-\varphi \dim \left(B\right)\right|\le 2·\mathrm{pd}{(}_{B^e}A/B)+p-1.$$

Proof. 

By [15] (Theorem 0.3) and [14] (Proposition 3.3), A and B are singularly equivalent of Morita-type with level $2·\mathrm{pd}{(}_{B^e}A/B)+p-1$. Then, the corollary follows from Theorem 3. □

Applying Theorem 3 to idempotents, we get the following corollary.

Corollary 4.

Let $\Lambda$ be a finite dimensional algebra, and let e be an idempotent in $\Lambda$. Then, the following statements hold:

(1) 

If ${\mathrm{pd}}_{{(e\Lambda e)}^o}\Lambda e<\infty$, $e\Lambda \in \mathrm{proj}(e\Lambda e)$ and ${\mathrm{pd}}_{{\Lambda }^e}(\Lambda /\Lambda e\Lambda )<\infty$, then

$$\left|\varphi \dim (\Lambda )-\varphi \dim (e\Lambda e)\right|\le l,$$

where $l=\max \{{\mathrm{pd}}_{{(e\Lambda e)}^o}\Lambda e,{\mathrm{pd}}_{{\Lambda }^e}(\Lambda /\Lambda e\Lambda )\}$.

(2) 

If ${\mathrm{pd}}_{e\Lambda e}e\Lambda <\infty$, $\Lambda e\in \mathrm{proj}{(e\Lambda e)}^o$ and ${\mathrm{pd}}_{{\Lambda }^e}(\Lambda /\Lambda e\Lambda )<\infty$, then

$$\left|\varphi \dim (\Lambda )-\varphi \dim (e\Lambda e)\right|\le l,$$

where $l=\max \{{\mathrm{pd}}_{e\Lambda e}e\Lambda ,{\mathrm{pd}}_{{\Lambda }^e}(\Lambda /\Lambda e\Lambda )\}$.

Proof. 

This follows from [13] (Remark 4.5) and Theorem 3. □

Now, we apply the above discussion in the context of Morita rings with zero bimodule maps; see [27] for a study of their homological properties. Let A and B be algebras, and consider two finitely generated bimodules ${}_{B}M_A$ and ${}_{A}N_B$. We consider the Morita ring $\Lambda =\left(\begin{array}{cc}A& {}_{A}N_B\\ {}_{B}M_A& B\end{array}\right)$ with multiplication given by

$$\left(\begin{array}{cc}a& n\\ m& b\end{array}\right)\left(\begin{array}{cc}{a}^{\prime }& n^{\prime }\\ m^{\prime }& b^{\prime }\end{array}\right)=\left(\begin{array}{cc}a{a}^{\prime }& a{n}^{\prime }+n{b}^{\prime }\\ m{a}^{\prime }+b{m}^{\prime }& b{b}^{\prime }\end{array}\right).$$

Consider the idempotents $\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$ and $\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$, respectively; then, we obtain the following:

Corollary 5.

Let $\Lambda =\left(\begin{array}{cc}A& {}_{A}N_B\\ {}_{B}M_A& B\end{array}\right)$ be a Morita ring. Then, the following statements hold:

(1) 

If ${\mathrm{pd}}_{B^e}B<\infty$, ${\mathrm{pd}}_{A^o}M<\infty$ and $N\in \mathrm{proj}\left(A\right)$, then

$$\left|\varphi \dim (\Lambda )-\varphi \dim \left(A\right)\right|\le l,$$

where $l=\max \{{\mathrm{pd}}_{A^o}M,{\mathrm{pd}}_{B^e}B\}$.

(2) 

If ${\mathrm{pd}}_{A^e}A<\infty$, ${\mathrm{pd}}_{B}M<\infty$ and $N\in \mathrm{proj}\left(B^o\right)$, then

$$\left|\varphi \dim (\Lambda )-\varphi \dim \left(B\right)\right|\le l,$$

where $l=\max \{{\mathrm{pd}}_{B}M,{\mathrm{pd}}_{A^e}A\}$.

Proof. 

This follows from [13] (Example 4.6) and Theorem 3. □

A special case of Morita ring is the triangular matrix algebra; so, we have the following:

Corollary 6.

Let $\Lambda =\left(\begin{array}{cc}A& 0\\ {}_{B}M_A& B\end{array}\right)$ be a triangular matrix algebra, where $A,B$ are algebras and M a finitely generated B-A-bimodule. Then, the following statements hold:

(1) 

If ${\mathrm{pd}}_{B^e}B<\infty$ and ${\mathrm{pd}}_{A^o}M<\infty$, then

$$\left|\varphi \dim (\Lambda )-\varphi \dim \left(A\right)\right|\le l,$$

where $l=\max \{{\mathrm{pd}}_{A^o}M,{\mathrm{pd}}_{B^e}B\}$.

(2) 

If ${\mathrm{pd}}_{A^e}A<\infty$ and ${\mathrm{pd}}_{B}M<\infty$, then

$$\left|\varphi \dim (\Lambda )-\varphi \dim \left(B\right)\right|\le l,$$

where $l=\max \{{\mathrm{pd}}_{B}M,{\mathrm{pd}}_{A^e}A\}$.

Proof. 

This follows from Corollary 5. □

Now, we give the following concrete examples to illustrate how our theorem can be used to compute $\varphi$-dimension. Recall that a quiver $Q=\left(Q_0,Q_1,s,t\right)$ is a quadruple consisting of two sets: $Q_0$ (whose elements are called vertices) and $Q_1$ (whose elements are called arrows), and two maps $s,t:Q_1\to Q_0$, which associate to each arrow $\alpha \in Q_1$, its source $s\left(\alpha \right)\in Q_0$, and its target $t\left(\alpha \right)\in Q_0$, respectively. For any $l\ge 1$, a path of length l is an oriented sequences of arrows $p={\alpha }_1{\alpha }_2\cdots {\alpha }_l$ such that $t\left({\alpha }_i\right)=s\left({\alpha }_{i-1}\right)$ for all $i=2,\cdots ,l$. We also associate with each vertex $a\in Q_0$ a path of length $l=0$, called the trivial path at a and denoted by ${\epsilon }_a$. A path algebra $kQ$ is a k-algebra whose underlying k-vector space has its basis the set of all paths in Q such that the product of two basis vectors p and $p^{\prime }$ is defined by ${\delta }_{t\left(p\right)s\left(p^{\prime }\right)}p{p}^{\prime }$, where $\delta$ is the Kronecker delta. For more facts about path algebras, we refer the readers to [28] (Chapter II).

Example 1.

Let $A=kQ/I$ be the algebra, where Q is the quiver

and $I=\langle {\alpha }^2,\beta \alpha \rangle$. Let $B=k\left[x\right]/\langle x^2\rangle$. By [20] (Example 7.5), there exist two bimodules M and N such that $(M,N)$ defines a singular equivalence of Morita-type with level 1 between A and B. Applying Theorem 3, we have that $\left|\varphi \dim \left(A\right)-\varphi \dim \left(B\right)\right|\le 1$. On the other hand, we get $\varphi \dim \left(B\right)=0$, since the algebra B is selfinjective, see [4]. As a result, we obtain that $\varphi \dim \left(A\right)\le 1$, and then, we deduce that $\varphi \dim \left(A\right)=1$, since the $\varphi$-dimension of an algebra is zero if and only if it is selfinjective; see [4].

Example 2.

For any $n\ge 1$, let $A_n=k{Q}_n/I$, where $Q_n$ is the quiver

and $I=\left\{{\alpha }_1\beta ,\beta \gamma ,\gamma {\alpha }_1\right\}$. Then, the algebras $A_1$ and $A_n$ are related by by arrow removal for all $n\ge 2$; see [29] (Example 4.5). It follows from [30] (Proposition 4.4) and [15] (Proposition 5.4) that $A_1\subset A_n$ is a bounded extension with ${(A_n/A_1)}^{\otimes 2}=0$ and $\mathrm{pd}{(}_{A_1^e}(A_n/A_1))=0$. Applying Theorem 3, we have that $\left|\varphi \dim \left(A_n\right)-\varphi \dim \left(A_1\right)\right|\le 1$, and then, we deduce that $\varphi \dim \left(A_n\right)=1$ by the same argument as in Example 1.

5. Conclusions and Future Work

We mainly investigate the behavior of the $\varphi$-dimension, a homological invariant arising from the Igusa–Todorov function, under singular equivalences of Morita-type with level. Our findings contribute to the study of the $\varphi$-dimension and can be applied to bounded extensions, Morita rings, triangular matrix algebras, and so on. In future work, we will consider other related homological invariants (such as the $\psi$-dimension and the delooping level) in the setting of singular equivalences of Morita-type with level.