Semisimplicity of affine cellular algebras

TL;DR

This paper characterizes when affine cellular algebras are semisimple, linking algebraic properties to geometric conditions and providing criteria for separability and Jacobson semisimplicity.

Contribution

It establishes necessary and sufficient conditions for the semisimplicity of affine cellular algebras, including geometric and bilinear form criteria, and explores separability and Jacobson semisimplicity.

Findings

Affine cellular algebra is semisimple iff associated scheme is reduced and 0-dimensional.

Semisimplicity is equivalent to all bilinear forms being isomorphisms.

Over a perfect field, semisimplicity coincides with separability.

Abstract

In this note, we prove that an affine cellular algebra $A$ is semisimple if and only if the scheme associated to $A$ is reduced and 0-dimensional, and the bilinear forms with respect to all layers of $A$ are isomorphisms. Moreover, if the ground ring is a perfect field, then $A$ is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple.

Equations

$$\{\mathrm{semisimple}\}\subset \{\mathrm{Jacobson}\mathrm{semisimple}\}\subset \{\mathrm{semiprime}\}$$

$$\begin{array}{ccc}J& \stackrel{\alpha }{\to }& \Delta {\otimes }_B{\Delta }^{\prime }\\ \ast \downarrow & & \downarrow v_1\otimes b_1{\otimes }_B{b}_2\otimes v_2\mapsto v_2\otimes \sigma (b_2){\otimes }_B\sigma (b_1)\otimes v_1\\ J& \stackrel{\alpha }{\to }& \Delta {\otimes }_B{\Delta }^{\prime }\end{array}$$

$$(u\otimes b\otimes v)(u^{\prime }\otimes b^{\prime }\otimes v^{\prime })=u\otimes b\varphi (v,u^{\prime })b^{\prime }\otimes v^{\prime }$$

$$\tilde{a}+\tilde{b}=\widetilde{a+b},a,b\in \Lambda ,$$

$$\tilde{a}\cdot \tilde{b}=\widetilde{a{a}_{0}b},a,b\in \Lambda ,$$

$$\lambda \tilde{a}=\widetilde{\lambda a},\lambda \in K,a\in \Lambda .$$

$$\tilde{a}\mapsto \phi (\tilde{a})=a{a}_0$$

$$Spec[A]:=Spec(\prod _{j=1}^m{B}_j)$$

$$\tilde{a}.(x+N_{\mathfrak{m}})=\tilde{a}.x+N_{\mathfrak{m}}=a{a}_0.x+N_{\mathfrak{m}}=0+N_{\mathfrak{m}},$$

$$0=\tilde{a}.([1]+N_{\mathfrak{m}})=\tilde{a}[1]+N_{\mathfrak{m}}=[a{a}_0]+N_{\mathfrak{m}},$$

$$I_1\supset I_2\supset I_3\supset \cdots \supset I_n\supset \cdots$$

$$V_1\otimes I_1\otimes V_1\supset V_1\otimes I_2\otimes V_1\supset V_1\otimes I_3\otimes V_1\supset \cdots \supset V_1\otimes I_n\otimes V_1\supset \cdots$$

Full text

Corresponding Author: Bowen Sun

The work is supported by the Natural Science Foundation of Hebei Province, China (A2021501002); China Scholarship Council (202008130184) and Natural Science Foundation of China (11871107).

1. Introduction

Affine cellular algebras were introduced by Koenig and Xi in [17], who extended the framework of cellular algebras due to Graham and Lehrer [9]. The isomorphism classes of simple representations of affine cellular algebras are parameterized by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. Many classes of algebras have been found to be affine cellular, including affine Hecke algebras of type A [17], affine Temperley-Lieb algebras [17], Khovanov-Lauda-Rouquier algebras of finite types [14, 15], affine quasi-hereditary algebras [13], affine Brauer algebras [3], affine Birman-Murakami-Wenzl algebras [4], affine Yokonuma-Hecke algebras [7], affine q-Schur algebras [6], BLN algebras [5], affine Hecke algebras of rank two [10] and so on.

In [2], Carvalho, Koenig and Lomp studied ring theoretical structure of affine cellular algebras. The aim of this note is to do some work along this direction. More exactly, we will study the semisimplicity of affine cellular algebras. Let us first clarify some relevant notions.

Recall that a ring $R$ is called semisimple if it is semisimple as a left $R$-module. Every semisimple ring is isomorphic to a finite direct product of matrix rings and is left and right artinian. Denote by $J(R)$ the Jacobson radical of $R$. Then ring $R$ is called Jacobson semisimple if $J(R)=0$. Note that the action of $J(R)$ on a simple $R$-module is zero. A ring $R$ is called semiprime if $R$ has no nonzero nilpotent ideals. We have the following hierarchy of the algebras aforementioned:

[TABLE]

It is helpful to point out that if $R$ is a left artinian ring, then $R$ being semisimple is equivalent to being Jacobson semisimple, and is equivalent to being semiprime, too. Note that a commutative semiprime ring is also called reduced.

The main goal of this note is to prove the following theorem, which gives a sufficient and necessary condition for an affine cellular algebra $A$ to be semisimple. Denote by $Spec[A]$ the associated scheme to $A$ (see Definition 3.3) and denote by $\phi_{j}$ the bilinear form of the $j$-th layer of $A$.

Theorem  Let $K$ be a field and let $A$ be an affine cellular $K$-algebra. Then $A$ is semisimple if and only if

1. (1)

$Spec[A]$* is a reduced [math]-dimensional scheme;* 2. (2)

$\phi_{j}$* are invertible for all $j$.*

Moreover, if the ground ring is a perfect field, then $A$ is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple.

2. Affine cellular algebras

In this section, we give a quick review on the definitions and some known results about an affine cellular algebra which are needed in the next section. The main reference is [17].

Let $K$ be a principal ideal domain. Given two $K$-modules $V$ and $W$, we denote by $\tau$ the swich map: $V\otimes_{K}W\rightarrow W\otimes_{K}V$, $v\otimes_{K}w\rightarrow w\otimes_{K}v$ for $v\in V$ and $w\in W$. A $K$-algebra $B$ is called affine if $B=K[x_{1},\cdots,x_{t}]/I$, where $K[x_{1},\cdots,x_{t}]$ is a polynomial ring in finitely many variables $x_{1},\cdots,x_{t}$ and $I$ an ideal. A $K$-involution $\ast$ on a $K$-algebra $A$ is a $K$-linear anti-automorphism with $(a^{\ast})^{\ast}=a$ for all $a\in A$.

Definition 2.1**.**

[17, Definition 2.1]** Let $A$ be a unitary $K$-algebra with a $K$-involution $\ast$. A two-sided ideal $J$ in $A$ is called an affine cell ideal if and only if the following data are given and the following conditions are satisfied:

1. (1)

The ideal $J$ is fixed by $\ast$: $(J)^{\ast}=J.$ 2. (2)

There exist a free $K$-module $V$ of finite rank, an affine $K$-algebra $B$ with identity and with a $K$-involution $\sigma$ such that $\Delta:=V\otimes_{K}B$ is an $A$-$B$-bimodule, on which the right $B$-module structure is induced by $B_{B}$. 3. (3)

There is an $A$-$A$-bimodule isomorphism $\alpha:J\rightarrow\Delta\otimes_{B}\Delta^{\prime},$ where $\Delta^{\prime}=B\otimes_{K}V$ is a $B$-$A$-bimodule with the left $B$-module induced by ${}_{B}B$ and with the right $A$-module structure defined by $(b\otimes v)a:=\tau(a^{\ast}(v\otimes b))$ for $a\in A$, $b\in B$ and $v\in V$, such that the following diagram is commutative:

[TABLE]

The algebra $A$ with $K$-involution $\ast$ is called affine cellular if and only if there is a $K$-module decomposition $A=J_{1}^{\prime}\oplus J_{2}^{\prime}\oplus\cdots J_{m}^{\prime}$ (for some $m$) with $(J_{j}^{\prime})^{\ast}=J_{j}^{\prime}$ for each $j$ $(j=1,\dots,m)$ and such that setting $J_{j}:=\bigoplus_{l=1}^{j}J_{l}^{\prime}$ gives a chain of two-sided ideals of $A$: $0=J_{0}\subset J_{1}\subset J_{2}\subset\cdots\subset J_{m}=A$ (each of them fixed by $\ast$), and each $J_{j}^{\prime}=J_{j}/J_{j-1}$ is an affine cell ideal of $A/J_{j-1}$ (with respect to the involution induced by $\ast$ on the quotient).

Clearly, if all the affine algebras $B_{j}$ are equal to the ground ring $K$, we recover the definition of a cellular algebra given by Koenig and Xi in [16]. Note that the original definition of a cellular algebra was given by Graham and Lehrer in [9].

For an affine cell ideal $J$ in an algebra $A$, the following lemma gives the basic structure of $J$ when viewed as an algebra (without unit) in itself.

Lemma 2.2**.**

[17, Proposition 2.2]** Let $J$ be an affine cell ideal in a $K$-algebra $A$ with an involution $\ast$. We identify $J$ with $\Delta\otimes_{B}\Delta^{\prime}=V\otimes_{K}B\otimes_{K}V$. Then:

1. (1)

There is a $K$-linear map $\phi:V\otimes_{K}V\rightarrow B$ such that

[TABLE]

for all $u,u^{\prime},v,v^{\prime}\in V$ and $b,b^{\prime}\in B$. 2. (2)

If $I$ is an ideal in $B$ and $u,v\in V$, then $V\otimes_{K}I\otimes_{K}V$ is an ideal in $A$.

Because of the importance of bilinear form $\phi$, we often write $\Delta\otimes_{B}\Delta^{\prime}$ as $\mathcal{A}(V,B,\phi)$. Let $\{v_{1},\cdots,v_{n}\}$ be a basis of $V$ and identify the bilinear form $\phi$ with the matrix $\phi=(\phi_{ij})=\phi(v_{i},v_{j})$ (we often use the same notation for the bilinear form and its matrix in this note). Then $J$ is isomorphic to a swich algebra $S(M_{n}(B),\,(\phi_{ij}))$ with the definition given as follows.

Definition 2.3**.**

[17, Definition 3.3]** Let $\Lambda$ be a $K$-algebra and fix an element $a_{0}\in\Lambda$. We define a new $K$-algebra $\widetilde{\Lambda}=S(\Lambda,\,a_{0})$, called the swich algebra of $\Lambda$ with respect to $a_{0}$, where as a set $\widetilde{\Lambda}=\{\widetilde{a}\mid a\in\Lambda\}$, and the algebra structure on $\widetilde{\Lambda}$ is given by

[TABLE]

[TABLE]

[TABLE]

A swich algebra of a matrix algebra $M_{n}(B)$ is in fact a generalized matrix algebra in the sense of [1]. By a straightforward computation, one can prove that the map $\varphi:\widetilde{\Lambda}\rightarrow\Lambda$ defined by

[TABLE]

is an algebra homomorphism, and a $\Lambda$-module $M$ will become a $\widetilde{\Lambda}$-module $M^{\varphi}$ via $\varphi$. The following lemma establishes a relationship between the set of all simple modules over $\Lambda$ and that over a swich algebra $\widetilde{\Lambda}$.

Lemma 2.4**.**

[17, Theorem 3.10]** Let $\Lambda$ be a $K$-algebra with identity such that $\Lambda$ is finitely generated over its centre. Then there is a bijection $\omega$ between the set of non-isomorphic simple $\Lambda$-modules $E$ with $\widetilde{\Lambda}.E\neq 0$, and the set of all non-isomorphic simple $\widetilde{\Lambda}$-modules, which is given by $E\mapsto E^{\varphi}/\{x\in E^{\varphi}\mid\widetilde{\Lambda}.x=0\}$.

We conclude this section by a result of [17] which is needed in Section 3.

Lemma 2.5**.**

[17, Theorem 3.12 (2)]** Let $A$ be an affine cellular algebra with a cell chain $0=J_{0}\subset J_{1}\subset J_{2}\subset\cdots\subset J_{m}=A$ such that each layer $J_{j}/J_{j-1}\cong\mathcal{A}(V_{j},B_{j},\phi_{j})$. Then for $1\leq j\leq m$, $\phi_{j}$ is an isomorphism if and only if the determinant $det(\phi^{(j)}_{st})$ of $\phi_{j}$ is a unit in $B_{j}$. In particular, if all $\phi_{j}$ are isomorphisms, then $A$ is isomorphic, as an affine cellular $K$-algebra, to $\bigoplus_{j=1}^{m}M_{n_{j}}(B_{j})$, where $n_{j}$ is the dimension of $V_{j}$.

The algebra $\bigoplus_{j=1}^{m}M_{n_{j}}(B_{j})$ will be called the asymptotic algebra of the affine cellular algebra $A$.

3. Semisimplicity

In this section, we study the semisimplicity of affine cellular algebras. We first need to review some definitions and notations from commutative algebra. The main references here are [12] and [19].

Definition 3.1**.**

Let $R$ be a commutative ring with identity. The spectrum of $R$ is defined to be $Spec(R):=\{\mathfrak{P}\subseteq R\ |\ \mathfrak{P}\ is\ a\ prime\ ideal\}$.

It is well-known that one can put a topology on $Spec(R)$, which is the so-called Zariski topology. Then we can give the definition of an affine scheme.

Definition 3.2**.**

An affine scheme is a pair $(Spec(R),\mathscr{O}_{R})$ consisting a spectrum of $R$ equipped with Zariski topology, and its structure sheaf $\mathscr{O}_{R}$. If $R$ is a reduced ring, then $(Spec(R),\mathscr{O}_{R})$ is called reduced. The dimension of $(Spec(R),\mathscr{O}_{R})$ is defined to be the Krull dimension of $R$.

We refer the reader to [19, Definition 2.20] for the definition of the structure sheaf mentioned in Definition 3.2. By abuse of notations, we will often write simply $Spec(R)$ for the affine scheme $(Spec(R),\mathscr{O}_{R})$.

Based on the above preparation, we can define the associated scheme to an affine cellular algebra, which will play a key role in this section.

Definition 3.3**.**

Let $A$ be an affine cellular algebra. We call

[TABLE]

the associated scheme to $A$.

Given an affine cellular algebra $A$, we will show that some ring theoretical properties of $A$, for example, semisimplicity, are partially determined by $Spec[A]$.

Let us first consider Jacobson semisimplicity. For this goal, we need the following lemma about a swich algebra. For simplicity of description, we stipulate that [math] is a zero-divisor.

Lemma 3.4**.**

Let $R$ be a unital ring such that $R$ is finitely generated over its center and $\widetilde{R}=S(R,a_{0})$, a swich algebra of $R$. Then $\widetilde{R}$ is Jacobson semisimple if and only if $R$ is Jacobson semisimple with $a_{0}$ not a zero-divisor.

Proof.

By Lemma 2.4, there is a bijection between the set of non-isomorphic simple $R$-modules $E$ with $\widetilde{R}.E\neq 0$, and the set of all non-isomorphic simple $\widetilde{R}$ -modules, which is given by $E\mapsto E^{\varphi}/N$, where $N$ is $\{x\in E\mid\widetilde{R}.x=0\}$. For each maximal left ideal ${\mathfrak{m}}$ of $R$, denote the corresponding simple $R$-module $R/{\mathfrak{m}}$ by $E_{\mathfrak{m}}$.

($\Rightarrow$) Suppose that $\widetilde{R}$ is Jacobson semisimple. Then $\widetilde{R}$ is semiprime. Consequently, we have from [2, Lemma 2.1] that $a_{0}$ is not a zero-divisor. Take an element $a$ of $J(R)$. Then $aE_{\mathfrak{m}}=0$ for arbitrary $\mathfrak{m}$. Assume that $\widetilde{R}.E_{\mathfrak{m}}\neq 0$. For $x+N_{\mathfrak{m}}\in E_{\mathfrak{m}}^{\varphi}/N_{\mathfrak{m}}$, we have

[TABLE]

where the last equality holds because $a_{0}.x\in E_{\mathfrak{m}}$ and $aE_{\mathfrak{m}}=0$. This implies that $\widetilde{a}$ annihilates all simple $\widetilde{R}$-modules, that is, $\widetilde{a}\in J(\widetilde{R})$, and hence $\widetilde{a}=0$. As a result, $a=0$ and $R$ is Jacobson semisimple.

($\Leftarrow$) Let $R$ be a Jacobson semisimple ring. Suppose that $\widetilde{a}\in J(\widetilde{R})$ and $\mathfrak{m}$ an arbitrary maximal left ideal of $R$. Then $\widetilde{a}.(E_{\mathfrak{m}}^{\varphi}/N_{\mathfrak{m}})=0$. Note that $R$ is a unital ring, and it is clear that $1_{R}\notin\mathfrak{m}$. For arbitrary $x\in R$, denote $x+\mathfrak{m}\in E_{\mathfrak{m}}$ by $[x]$. We have

[TABLE]

and we deduce $[aa_{0}]\in N_{\mathfrak{m}}$, or $\widetilde{R}.[aa_{0}]=0$ in $E_{\mathfrak{m}}$. In particular, $\widetilde{1}.[aa_{0}]=0$, and thus $a_{0}aa_{0}\in\mathfrak{m}$. As a result, $a_{0}aa_{0}\in J(R)$ because of $\mathfrak{m}$ being arbitrary. Now the Jacobson semisimplicity of $R$ forces $a_{0}aa_{0}$ to be zero. Combining this result with the fact that $a_{0}$ is not a zero-divisor yields $a=0$. Consequently, the radical of $\widetilde{R}$ is zero and this completes the proof. ∎

Now we can give a sufficient condition for an affine cellular algebra being Jacobson semisimple.

Theorem 3.5**.**

Let $A$ be an affine cellular algebra. Then $A$ is Jacobson semisimple if

1. (1)

$Spec[A]$* is a reduced scheme;* 2. (2)

all $\phi_{j}$ are not zero-divisors.

Proof.

The affine cellularity of $A$ implies that a layer $J_{j}/J_{j-1}$ is isomorphic to a swich algebra $\widetilde{M}_{n_{j}}(B_{j})=S(M_{n_{j}}(B_{j}),\,\phi_{j})$. Since $Spec[A]$ is a reduced scheme, each $B_{j}$ is a reduced ring. Note that a reduced affine algebra is Jacobson semisimple. Then $M_{n_{j}}(B_{j})$ is Jacobson semisimple because the matrix algebra over a Jacobson semisimple ring is Jacobson semisimple too. Note that $\phi_{j}$ is not a zero-divisor. Then by Lemma 3.4, we have $\widetilde{M}_{n_{j}}(B_{j})$ is semisimple.

Take an element $r\in J(A)$ and assume that $r=\sum_{j=1}^{m}r_{j}$, where $r_{j}\in J_{j}^{\prime}$. Then $r$ annihilates all simple modules. According to the representation theory of affine cellular algebras [17, Theorem 3.12], the actions of $r_{i},\,\,i=1,2,\dots m-1$ on simple modules belonging to the top layer are all zeros. Thus $r_{m}$ annihilates all simple modules of the top layer. Now the Jacobson semisimplicity of each layer proved above forces $r_{m}$ being zero. By continuing this process finitely many times, we obtain $r_{i}=0$, for $i=1,2,\dots,m$, that is, $r=0$. This completes the proof. ∎

Corollary 3.6**.**

For a unital affine cellular algebra $A$, if $Spec[A]$ is a reduced scheme and $\phi_{j}$ are not zero-divisors for all $j$, then A is semiprime.

The following is an example where the sufficient criterion given in Theorem 3.5 is not necessary. The example also implies that it is likely far away from a characterisation of Jacobson semisimplicity.

Example 3.7**.**

Let $F$ be a field and let $K$ be the formal power series ring $F[[x]]$. Then the Jacobson radical of $K$ is the ideal generated by $x$. This implies that $K$ is not Jacobson semisimple. Let $A=K[x]$. Then $A$ is Jacobson semisimple ([11] Page 433 Exercise 14 (c)). We claim that $A$ is an affine cellular algebra. In fact, we can take a cell chain of $A$ to be $0\subset(x)\subset A$ and define $B_{1}=A$, $V_{1}$ to be the free $K$-module with basis $\{x\}$ and $B_{2}=K$, $V_{2}=K$. Since $B_{2}=K$ is not a reduced ring, $Spec[A]$ is not a reduced scheme.

Let us begin to study semisimplicity of an affine cellular algebra over a field $K$ now. As is well-known, a semisimple algebra is left artinian. So we first give a sufficient and necessary condition for an affine cellular algebra to be left artinian as follows.

Lemma 3.8**.**

Let $A$ be an affine cellular algebra. Then $A$ is a left artin ring if and only if $Spec[A]$ is a [math]-dimensional scheme.

Proof.

If $Spec[A]$ is a [math]-dimensional scheme, then the Krull dimension of every $B_{j}$ is zero. This is equivalent to that all $B_{j}$ are finite dimensional $K$-algebras by [12, Theorem 5.11], and hence the affine cellular algebra $A$ is a finite dimensional $K$-algebra. So $A$ is left artinian.

Conversely, assume that $A$ is a left artin ring. We claim that for arbitrary $i\in\{1,\cdots,m\}$, $B_{i}$ is an artin ring. In fact, we have from $A$ being left artinian that $A/J_{i-1}$ is a left artin ring and $J_{i}^{\prime}$ is a left artin $A/J_{i-1}$-module. If $B_{i}$ is not artinian, then there exists an infinite descending chain of ideals of $B_{i}$

[TABLE]

As a result, we obtain an infinite descending chain of submodules of $J_{i}^{\prime}$

[TABLE]

due to Lemma 2.2. It is a contradiction. Then $Spec[A]$ is a [math]-dimensional scheme, which follows from all $B_{i}$ being artinian. ∎

Employing [12, Theorem 5.11] again, we get a direct corollary of Lemma 3.8 as follows.

Corollary 3.9**.**

Let $A$ be an affine cellular algebra. Then the following statements are equivalent.

1. (1)

$A$* is a left artin algebra.* 2. (2)

$Spec[A]$* is a [math]-dimensional scheme.* 3. (3)

$A$* is a finite dimensional $K$-algebra.* 4. (4)

Each $B_{j}$ is a finite dimensional $K$-algebra.

Now we can give a sufficient and necessary condition for an affine cellular algebra being semisimple.

Theorem 3.10**.**

Let $K$ be a field and let $A$ be an affine cellular $K$-algebra. Then $A$ is semisimple if and only if

1. (1)

$Spec[A]$* is a reduced [math]-dimensional scheme;* 2. (2)

$\phi_{j}$* is invertible for each $j\in\{1,2,\dots m\}$.*

Proof.

$``\Rightarrow"$ Let $A$ be semisimple. Then $A$ is both left artinian and semirpime, and thus $Spec[A]$ is a [math]-dimensional scheme by Corollary 3.9. As is well-known, the quotients of a semisimple ring are semisimple too. This implies the semisimplicity of $A/J_{j-1}$. On the other hand, an ideal of a semisimple algebra is semisimple. This gives that $J_{j}^{\prime}$ is semisimple since $J_{j}^{\prime}$ is an ideal in $A/J_{j-1}$. As a result, $B_{j}$ is a reduced ring with $\phi_{j}$ not a zero-divisor by [2, Proposition 2.5]. Moreover, we also have from Corollary 3.9 that every $B_{j}$ is a finite dimensional $K$-algebra. Recall that a finite dimensional unital algebra enjoy a special property [8, Theorem 1.2.1]: every element is either invertible or a zero-divisor. This forces $\phi_{j}$ to be invertible.

$``\Leftarrow"$ If $Spec[A]$ is a reduced 0-dimensional scheme and all $\phi_{j}$ are invertible, then combining Corollary 3.6 with Corollary 3.9 implies that $A$ is a finite dimensional semiprime algebra, and consequently, $A$ is semisimple. ∎

Note that if $B_{j}$ is a finite dimensional affine $K$-algebra, then $\phi_{j}$ is invertible if and only if $det(\phi_{j})$ is a unit in $B_{j}$. Moreover, employing Lemma 2.5 yields an easy result as follows.

Corollary 3.11**.**

Let $A$ be an affine cellular algebra $A$. Then $A$ is semisimple if and only if it is isomorphic to its asymptotic algebra and $Spec[A]$ is a reduced 0-dimensional scheme.

To conclude the investigation of semisimplicity of an affine cellular algebra, we enhance the condition “reduced” in Theorem 3.10 to “geometrically reduced”, which corresponds to a strengthened version of semisimple algebras: separable algebras.

Let us recall the definitions of a geometrically reduced ring and a separable algebra first.

Definition 3.12**.**

Let $K$ be a field and $\overline{K}$ the algebraic closure of $K$. An affine $K$-algebra $A$ is called geometrically reduced if $A\otimes\overline{K}$ is a reduced ring. A $K$-algebra $A$ is said to be separable if for arbitrary finite extension field $F$ over $K$, $A\otimes F$ is a semisimple $F$-algebra.

The definition of a separable algebra can be considered as a certain property which can be preserved under base change. Especially, we will find that the base change of an affine cellular algebra is actually the base change of affine algebras $B_{j}$. In order to study separable affine cellular algebras, we recall the definition of Étale algebras.

Definition 3.13**.**

[18, Definition 1.5.3]** A finite dimensional $K$-algebra is said to be Étale if it is isomorphic to a finite direct sum of separable extensions of $K$.

The following lemma can be viewed as an equivalent definition of an Étale algebra, which implies that an Étale algebra is in fact a finite dimensional commutative separable algebra.

Lemma 3.14**.**

[18, Proposition 1.5.6]** Let $A$ be an finite dimensional commutative $K$-algebra. Then the following statements are equivalent.

1. (1)

$A$* is Étale.* 2. (2)

$A\otimes_{K}\overline{K}$* is reduced.*

We can give some necessary and sufficient conditions for an affine cellular algebra being separable.

Corollary 3.15**.**

Let $A$ be an affine cellular $K$-algebra. Then the following statements are equivalent.

1. (1)

A is a separable algebra. 2. (2)

$Spec[A]$* is a geometrically reduced [math]-dimensional scheme and $det(\phi_{j})$ invertible for all $j$.* 3. (3)

$\prod_{j=1}^{m}B_{j}$* is an Étale algebra algebra and for all $j$, $det(\phi_{j})$ invertible.* 4. (4)

For all $j$, $B_{j}$ are Étale algebra algebra and $det(\phi_{j})$ invertible.

Proof.

It follows from Lemma 3.14 that (2) is equivalent to (3), and the equivalence between (3) and (4) is clear. Then we only need to prove $(1)\Leftrightarrow(2)$.

$(1)\Rightarrow(2)$ Let $A$ be separable. Then by Definition 3.12, $A$ is semisimple. As a result, $Spec[A]$ is a reduced 0-dimensional scheme with $det(\phi_{j})$ invertible by Theorem 3.10. Then we only need to prove $Spec[A]$ is geometrically reduced, or $\prod_{j=1}^{m}B_{j}\otimes\overline{K}$ is reduced. This is clear from the semisimplicity of $A\otimes\overline{K}$.

$(2)\Rightarrow(1)$ Assume that $Spec[A]$ is a geometrically reduced scheme with $det(\phi_{j})$ invertible. We deduce by Lemma 3.14 that every $B_{j}$ is an Étale algebra and hence a separable algebra. This implies that $B_{j}\otimes F$ is semisimple for arbitrary finite extension field $F$ of $K$, and so $Spec[A\otimes F]$ is a reduced [math]-dimensional scheme. In addition, it is clear that the swich matrices of $A\otimes F$ are the same as those of $A$ and thus all of their determinants are invertible. Therefore, $A\otimes F$ is semisimple and this completes the proof. ∎

Note that when $K$ is a perfect field, a finite dimensional affine $K$-algebra is an Étale algebra if and only if it is reduced (see [18, Remark 1.5.8]). This leads to a direct result as follows.

Theorem 3.16**.**

Let $K$ be a perfect field and $A$ an affine cellular algebra. Then $A$ is semisimple if and only if $A$ is separable.

Acknowledgement. The authors are grateful to Zeren Zheng for some helpful conversations. Part of this work was done when Li visited Institute of Algebra and Number Theory at University of Stuttgart from August 2021 to September 2022. He takes this opportunity to express his sincere thanks to the institute and Prof. S. Koenig for the hospitality during his visit.