The algebra of observables in noncommutative deformation theory

TL;DR

This paper generalizes the Generalized Burnside Theorem for noncommutative deformation theory, proving the algebra of observables is a closure operation without field assumptions, and extends isomorphism results to broader algebra classes.

Contribution

It extends the Generalized Burnside Theorem to cases without field assumptions and shows the observables construction acts as a closure operation for finitely generated algebras.

Findings

Generalized Burnside Theorem holds over arbitrary fields.

The algebra of observables is a closure operation.

Isomorphism of observables persists for broader algebra classes.

Abstract

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field $k$. Secondly, we prove that the $\mathcal O$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of finite dimensional $A$-modules, in the sense that $\eta_B: B \to \mathcal O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf M$ is considered as a family of $B$-modules.

Equations

$$\rho :A\to \underset{i=1}{\overset{r}{\oplus }}{\mathrm{End}}_k(M_i)$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\cong \mathcal{O}(\mathsf{M})=(H_{ij}{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))$$

$${\eta }_R:A\to {\mathrm{End}}_R(M_R)\cong (R_{ij}{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))$$

$${\eta }_R(a)=\sum _i{e}_i\otimes {\rho }_i+\sum _{i,j,l}{r}_{ij}^l\otimes {\varphi }_{ij}^l⟺(e_i\otimes m_i)a=e_i\otimes (m_{i}a)+\sum _{j,l}{r}_{ij}^l\otimes {\varphi }_{ij}^l(m_i)$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\varphi :\mathrm{Mor}(H,-)\to {\mathsf{Def}}_{\mathsf{M}}$$

$$\mathcal{O}(\mathsf{M})={\mathrm{End}}_H(M_H)\cong (H_{ij}{\hat{\otimes }}_k{\mathrm{Hom}}_k(M_i,M_j))$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E=E_r\stackrel{f_r}{\to }{E}_{r-1}\to \cdots \to E_2\stackrel{f_2}{\to }{E}_1\stackrel{f_1}{\to }{E}_0=0$$

$$0\to M_i\to E_i\stackrel{f_i}{\to }{E}_{i-1}\to 0$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$J/J^2\cong ((J(H)/J(H{)}^2{)}_{ij}{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))\cong ({\mathrm{Ext}}_A^1(M_i,M_j{)}^{\ast }{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))$$

$$J(A)/J(A{)}^2\to ({\mathrm{Ext}}_A^1(M_i,M_j{)}^{\ast }{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))$$

$${\dim }_k{\mathrm{Ext}}_A^1(M_i,M_j)\le {\dim }_k{\mathrm{Der}}_k(A,{\mathrm{Hom}}_k(M_i,M_j))\le {\dim }_k{\mathrm{Hom}}_k(M_i,M_j{)}^d$$

$$\overline{B}=B/J(B)\cong (H/J(H){\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))\cong \underset{i}{\oplus }{\mathrm{End}}_k(M_i).$$

$$B/J^2\cong ((H/J(H{)}^2{)}_{ij}{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B/J(B)\cong C/J(C)\cong \underset{i}{\oplus }{\mathrm{End}}_k(M_i)$$

$${\mathrm{Ext}}_{B_n}^1(M_i,M_j)\cong {\mathrm{Ext}}_B^1(M_i,M_j)$$

$$D(s)=\{M\in X:M\stackrel{\cdot s}{\to }M\text{ is invertible}\}\subseteq X.$$

$${\eta }_s:A\to \underset{\mathsf{M}\subseteq D(s)}{\underleftarrow{\lim }}\mathcal{O}(\mathsf{M})$$

$$\underset{\mathsf{M}\subseteq D(s)}{\underleftarrow{\lim }}\mathcal{O}(\mathsf{M})$$

$$A\cong \mathrm{im}(\eta )\subseteq (H_{ij}{\otimes }_k{\mathrm{Hom}}_k(M_i,M_j))=\mathcal{O}(\mathsf{M})$$

$$x^3-1=(x-1)(x^2+x+1)=(x-1)(x-\omega )(x-{\omega }^2)$$

$$A=k[{\mathbb{Z}}_3]\cong k^3=\left(\begin{array}{ccc}k& 0& 0\\ 0& k& 0\\ 0& 0& k\end{array}\right).$$

$$H=\left(\begin{array}{cc}k& 0\\ 0& k\end{array}\right)\Rightarrow A\cong \mathrm{im}(\eta )=\left(\begin{array}{cc}k& 0\\ 0& K\end{array}\right)\subseteq \mathcal{O}(\mathsf{M})=\left(\begin{array}{ccc}k& & 0\\ 0& & {\mathrm{End}}_k(K)\end{array}\right).$$

Full text

1. Introduction

Let $k$ be a field, let $A$ be a finite dimensional associative algebra over $k$, and let $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ be the family of simple right $A$-modules, up to isomorphism. We consider the algebra homomorphism

[TABLE]

given by right multiplication of $A$ on the family $\mathsf{M}$. By the extended version of the classical Burnside Theorem, $\rho$ is surjective when $k$ is algebraically closed, and if $A$ is semisimple, then it is an isomorphism. We remark that Artin-Wedderburn theory gives a version of the theorem that holds over any field:

Theorem** (Classical Burnside Theorem).**

Let $A$ be a finite dimensional $k$-algebra, and let $\{M_{1},\dots,M_{r}\}$ be the family of simple right $A$-modules. If $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$, then $\rho:A\to\operatorname*{\oplus}_{i}\,\operatorname{End}_{k}(M_{i})$ is surjective.

In Laudal [3], a generalization called the Generalized Burnside Theorem was obtained. This is a structural result for not necessarily semisimple algebras, and the essential idea of Laudal was to replace $\rho$ with the versal morphism $\eta$ defined by noncommutative deformations of modules. Let us recall the construction:

Let $A$ be an arbitrary associative $k$-algebra, let $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ be a family of right $A$-modules, and consider the noncommutative deformation functor $\mathsf{Def}_{\mathsf{M}}$. This functor has a pro-representing hull $H$ and a versal family $M_{H}$ if $\mathsf{M}$ is a swarm. Following Laudal [3], we define the algebra of observables of a swarm $\mathsf{M}$ to be $\mathcal{O}(\mathsf{M})=\operatorname{End}_{H}(M_{H})\cong(H_{ij}\otimes_{k}\operatorname{Hom}_{k}(M_{i},M_{j}))$, and its versal morphism to be the algebra homomorphism $\eta:A\to\mathcal{O}(\mathsf{M})$ given by right multiplication of $A$ on the versal family $M_{H}$. It fits into the commutative diagram

[TABLE]

where $\rho:A\to\operatorname*{\oplus}_{i=1}^{r}\operatorname{End}_{k}(M_{i})$ is the algebra homomorphism given by right multiplication of $A$ on the family $\mathsf{M}$. By Theorem 1.2 in Laudal [3], it follows that $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf{M}$ is the family of simple $A$-modules, and $k$ is algebraically closed. In this paper, we prove a more general version of this result:

Theorem** (Generalized Burnside Theorem).**

Let $A$ be a finite dimensional $k$-algebra, and let $\mathsf{M}$ be the family of simple right $A$-modules, up to isomorphism. The versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ is injective. If $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$, then $\eta$ is an isomorphism. In particular, $\eta$ is an isomorphism if $k$ is algebraically closed.

In case $D_{i}=\operatorname{End}_{A}(M_{i})$ is a division algebra with $\dim_{k}D_{i}>1$ for some simple module $M_{i}$, it is often not difficult to describe the image of $\eta$ as a subalgebra of $\mathcal{O}(\mathsf{M})$, and we shall give examples. As an application of the theorem, we introduce the standard form of any finite dimensional algbra $A$, given as

[TABLE]

when $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$, or as a subalgebra of $\mathcal{O}(\mathsf{M})$ in general.

Let $A$ be any finitely generated $k$-algebra and let $\mathsf{M}$ be any family of finite dimensional right $A$-modules. In this more general situation, the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ is not necessarily an isomorphism. However, we may consider the algebra $B=\mathcal{O}(\mathsf{M})$ of observables, and $\mathsf{M}$ as a family of right $B$-modules, and iterate the process. We prove that the operation $(A,\mathsf{M})\mapsto(B,\mathsf{M})$ has the following closure property:

Theorem** (Closure Property).**

Let $A$ be a finitely generated $k$-algebra, let $\mathsf{M}$ be a family of finite dimensional $A$-modules, and let $B=\mathcal{O}(\mathsf{M})$. Then the versal morphism $\eta^{B}:B\to\mathcal{O}^{B}(\mathsf{M})$ of $\mathsf{M}$, considered as a family of right $B$-modules, is an isomorphism.

One may consider a noncommutative algebraic geometry where the closed points are represented by simple modules; see for instance Laudal [4]. With this point of view, one may use versal morphisms $\eta:A\to\mathcal{O}(\mathsf{M})$ for families $\mathsf{M}$ of $A$-modules to construct noncommutative localization homomorphisms $\eta_{s}:A\to A_{s}$ for any $s\in A$. We explain this construction in Section 6. These localization maps are universal $S$-inverting localization maps, where $S=\{1,s,s^{2},\dots\}$, and can be used as an essential building block for structure sheaves on noncommutative schemes.

2. Noncommutative deformations of modules

Let $A$ be an associative algebra over a field $k$. For any right $A$-module $M$, there is a deformation functor $\mathsf{Def}_{M}:\mathsf{l}\to\mathsf{Sets}$ defined on the category $\mathsf{l}$ of commutative Artinian local $k$-algebras $R$ with residue field $k$. We recall that $\mathsf{Def}_{M}(R)$ is the set of equivalence classes of pairs $(M_{R},\tau_{R})$, where $M_{R}$ is an $R$-flat $R$-$A$ bimodule on which $k$ acts centrally, and $\tau_{R}:k\otimes_{R}M_{R}\to M$ is an isomorphism of right $A$-modules. Deformations in $\mathsf{Def}_{M}(R)$ are called commutative deformations since the base ring $R$ is commutative.

Noncommutative deformations were introduced in Laudal [3]. The deformations considered by Laudal are defined over certain noncommutative base rings instead of the commutative base rings in $\mathsf{l}$. In what follows, we shall give a brief account of noncommutative deformations of modules. We refer to Laudal [3], Eriksen [2] and Eriksen, Laudal, Siqveland [1] for further details.

For any positive integer $r$ and any family $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ of right $A$-modules, there is a noncommutative deformation functor $\mathsf{Def}_{\mathsf{M}}:\mathsf{a}_{r}\to\mathsf{Sets}$, defined on the category $\mathsf{a}_{r}$ of noncommutative Artinian $r$-pointed $k$-algebras with exactly $r$ simple modules (up to isomorphism). We recall that an $r$-pointed $k$-algebra $R$ is one fitting into a diagram of rings $k^{r}\to R\to k^{r}$, where the composition is the identity. The condition that $R$ has exactly $r$ simple modules holds if and only if $\overline{R}\cong k^{r}$, where $\overline{R}=R/J(R)$ and $J(R)$ denotes the Jacobson radical of $R$.

The noncommutative deformations in $\mathsf{Def}_{\mathsf{M}}(R)$ are equivalence classes of pairs $(M_{R},\tau_{R})$, where $M_{R}$ is an $R$-flat $R$-$A$ bimodule on which $k$ acts centrally, and $\tau_{R}:k^{r}\otimes_{R}M_{R}\to M$ is an isomorphism of right $A$-modules with $M=M_{1}\oplus\dots\oplus M_{r}$. In concrete terms, an algebra $R$ in $\mathsf{a}_{r}$ is a matrix ring $R=(R_{ij})$ with $R_{ij}=e_{i}Re_{j}$. By abuse of notation, we write $e_{i}$ for the idempotent $e_{i}=(0,0,\dots,i,\dots,0)$ in $k^{r}$, and also for its image in $R$ via the structural map $k^{r}\to R$. As left $R$-modules, we have that $M_{R}\cong(R_{ij}\otimes_{k}M_{j})$ and its right $A$-module structure is given by an algebra homomorphism

[TABLE]

that lifts $\rho:A\to\operatorname*{\oplus}_{i}\,\operatorname{End}_{k}(M_{i})$. Explicitly, we interpret $\eta_{R}(a)$ as a right action of $a$ on $M_{R}$ via

[TABLE]

where $\rho_{i}:A\to\operatorname{End}_{k}(M_{i})$ is the algebra homomorphism given by the right action of $A$ on $M_{i}$, such that $\rho=(\rho_{1},\dots,\rho_{r})$, and where $r_{ij}^{l}\in R_{ij}$ and $\phi_{ij}^{l}\in\operatorname{Hom}_{k}(M_{i},M_{j})$. Deformations in $\mathsf{Def}_{\mathsf{M}}(R)$ can therefore be represented by commutative diagrams

[TABLE]

These deformations are called noncommutative deformations since the base ring $R$ is noncommutative.

For any $r$-pointed algebra $R$, with structural maps $k^{r}\to R\to k^{r}$, we write $I(R)=\ker(R\to k^{r})$. Recall that the pro-category $\widehat{\mathsf{a}}_{r}$ is the full subcategory of the category of $r$-pointed algebras consisting of algebras $R$ such that $R/I(R)^{n}$ is Artinian for all $n$ and such that $R$ is complete in the $I(R)$-adic topology.

The family $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ is called a swarm if $\dim_{k}\operatorname{Ext}^{1}_{A}(M,M)$ is finite. In this case, the noncommutative deformation functor $\mathsf{Def}_{\mathsf{M}}$ has a pro-representing hull $H$ in the pro-category $\widehat{\mathsf{a}}_{r}$ and a versal family $M_{H}\in\mathsf{Def}_{\mathsf{M}}(H)$; see Theorem 3.1 in Laudal [3]. The defining property of the miniversal pro-couple $(H,M_{H})$ is that the induced natural transformation

[TABLE]

on $\mathsf{a}_{r}$ is smooth (which implies that $\phi_{R}$ is surjective for any $R$ in $\mathsf{a}_{r}$), and that $\phi_{R}$ is an isomorphism when $J(R)^{2}=0$. The miniversal pro-couple $(H,M_{H})$ is unique up to (non-canonical) isomorphism.

Let $\mathsf{M}$ be a swarm of right $A$-modules, and let $(H,M_{H})$ be the miniversal pro-couple of the noncommtutative deformation functor $\mathsf{Def}_{\mathsf{M}}:\mathsf{a}_{r}\to\mathsf{Sets}$. We define the algebra of observables of $\mathsf{M}$ to be

[TABLE]

where $\widehat{\otimes}$ is the completed tensor product (the completion of the tensor product), and write $\eta:A\to\mathcal{O}(\mathsf{M})$ for the induced versal morphism, giving the right $A$-module structure on $M_{H}$. By construction, it fits into the commutative diagram

[TABLE]

Remark 1**.**

Notice that the diagram extends the right action of $A$ on the family $\mathsf{M}$ to a right action of $\mathcal{O}(\mathsf{M})$, such that $\mathsf{M}$ is a family of right $\mathcal{O}(\mathsf{M})$-modules.

Remark 2**.**

For any $R$ in $\mathsf{a}_{r}$ and any deformation $M_{R}\in\mathsf{Def}_{\mathsf{M}}(R)$, there is a morphism $u:H\to R$ in $\widehat{\mathsf{a}}_{r}$ such that $\mathsf{Def}_{\mathsf{M}}(u)(M_{H})=M_{R}$ by the versal property, and the deformation $M_{R}$ is therefore given by the composition $\eta_{R}=u^{*}\circ\eta$ in the diagram

[TABLE]

In this sense, the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ determines all noncommutative deformations of the family $\mathsf{M}$.

3. Iterated extensions and injectivity of the versal morphism

Let $E$ be a right $A$-module and let $r\geq 1$ be a positive integer. If $E$ has a cofiltration of length $r$, given by a sequence

[TABLE]

of surjective right $A$-module homomorphisms $f_{i}:E_{i}\to E_{i-1}$, then we call $E$ an iterated extension of the right $A$-modules $M_{1},M_{2},\dots M_{r}$, where $M_{i}=\ker(f_{i})$. In fact, the cofiltration induces short exact sequences

[TABLE]

for $1\leq i\leq r$. Hence $E_{1}\cong M_{1}$, $E_{2}$ is an extension of $E_{1}$ by $M_{2}$, and in general, $E_{i}$ is an extension of $E_{i-1}$ by $M_{i}$.

Let $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ be a swarm of right $A$-modules, and let $\mathsf{Def}_{\mathsf{M}}:\mathsf{a}_{r}\to\mathsf{Sets}$ be its noncommutative deformation functor. Then $\mathsf{Def}_{\mathsf{M}}$ has a miniversal pro-couple $(H,M_{H})$, and we consider the induced versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ and its kernel $K=\ker(\eta)$.

We note that Theorem 3.2 in Laudal [3] holds without assumptions on the base field $k$, since the construction that precedes this theorem works over any field. From this observation, we obtain the following lemma:

Lemma 3**.**

Let $\mathsf{M}$ be a swarm of right $A$-modules. For any iterated extension $E$ of the family $\mathsf{M}$, we have that $E\cdot K=0$.

Let $A$ be a finite dimensional $k$-algebra and let $\mathsf{M}$ be the family of all simple right $A$-modules, up to ismorphism. Then $\mathsf{M}$ is a swarm, and we may consider the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$. If $k$ is algebraically closed, then the versal morphism $\eta$ is injective by Corollary 3.1 in Laudal [3]. Using Lemma 3, we generalize this result:

Proposition 4**.**

If $A$, considered as a right $A$-module, is an iterated extension of a swarm $\mathsf{M}$, then the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ is injective. In particular, $\eta$ is injective when $A$ is a finite dimensional algebra and $\mathsf{M}$ is the family of simple right $A$-modules.

Proof.

If $A$ is an iterated extension of $\mathsf{M}$, then $1\cdot K=0$ by Lemma 3, and this implies that $K=0$. If $A$ is finite dimensional, then the right $A$-module $A$ has finite length, and it is an iterated extension of the simple modules. ∎

We remark that our proof, based on Lemma 3, holds whenever there is an element $e\in E$ such that $a\mapsto e\cdot a$ defines an injective right $A$-module homomorphism $A\to E$. This means that $\eta:A\to\mathcal{O}(\mathsf{M})$ is injective if there is an iterated extension $E$ of $\mathsf{M}$ such that $E$ contains a copy of $A_{A}$.

4. The Generalized Burnside Theorem

Let $A$ be a finite dimensional $k$-algebra, and let $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ be the family of simple right $A$-modules, up to isomorphism. Then $\mathsf{M}$ is a swarm, and we consider the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ and the commutative diagram

[TABLE]

Clearly, $\rho$ factors through $A/\operatorname{J}(A)$, and if $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$, then $A/\operatorname{J}(A)\to\operatorname*{\oplus}_{i}\,\operatorname{End}_{k}(M_{i})$ is an isomorphism by the Artin-Wedderburn theory for semisimple algebras. This proves the Classical Burnside Theorem mentioned in the introduction. By Theorem 3.4 in Laudal [3], the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ is an isomorphism when $k$ is algebraically closed. We generalize this result:

Theorem 5**.**

Let $A$ be a finite dimensional $k$-algebra and let $\mathsf{M}$ be the family of simple right $A$-modules, up to isomorphism. Then $\eta:A\to\mathcal{O}(\mathsf{M})$ is injective, and it is an isomorphism if $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$. In particular, the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ is an isomorphism if $k$ is algebraically closed.

Proof.

By Proposition 4, the versal morphism $\eta$ is injective, and it is enough to prove that $\eta$ is surjective when $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$. Note that $\eta$ maps the Jacobson radical $\operatorname{J}(A)$ of $A$ to the Jacobson radical $J=(\operatorname{J}(H)_{ij}\otimes_{k}\operatorname{Hom}_{k}(M_{i},M_{j}))$ of $\mathcal{O}(\mathsf{M})$. Moreover, $A$ is $\operatorname{J}(A)$-adic complete since it is finite dimensional, and $\mathcal{O}(\mathsf{M})$ is clearly $J$-adic complete. By a standard result for filtered algebras, it is therefore sufficient to show that $\operatorname{gr}_{1}(\eta):J(A)/J(A)^{2}\to J/J^{2}$ is surjective, since $\operatorname{gr}_{0}(\eta):A/J(A)\to\operatorname*{\oplus}_{i}\,\operatorname{End}_{k}(M_{i})$ is an isomorphism by the Classical Burnside Theorem. We notice that

[TABLE]

since $J(H)/J(H)^{2}$ is the dual of the tangent space $(\operatorname{Ext}^{1}_{A}(M_{i},M_{j}))$ of $\mathsf{Def}_{\mathsf{M}}$. We note that Lemma 3.7 in Laudal [3] holds over any field. Hence the map

[TABLE]

induced by $\eta$ is an isomorphism, and this completes the proof. ∎

5. The closure property

Let $A$ be a finitely generated $k$-algebra of the form $A=k\langle x_{1},\dots x_{d}\rangle/I$, and let $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ be a family of finite dimensional right $A$-modules. Then $\mathsf{M}$ is a swarm, since

[TABLE]

The last inequality follows from the fact that any derivation $D:A\to\operatorname{Hom}_{k}(M_{i},M_{j})$ is determined by $D(x_{l})\in\operatorname{Hom}_{k}(M_{i},M_{j})$ for $1\leq l\leq d$. We consider the algebra of observables $B=\mathcal{O}(\mathsf{M})$ of the swarm $\mathsf{M}$, and write $\eta:A\to B$ for its versal morphism. In general, $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ is a family of right $B$-modules via $\eta$.

Lemma 6**.**

The family $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ of right $B$-modules is the simple right $B$-modules, and it is swarm of $B$-modules.

Proof.

It follows from the Artin-Wedderburn theory that $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ is the family of simple modules over

[TABLE]

Since $B$ and $\overline{B}=B/J(B)$ have the same simple modules, it follows that $\mathsf{M}$ is the family of simple right $B$-modules. We have that $\operatorname{Ext}^{1}_{B}(M_{i},M_{j})$ is a quotient of $\operatorname{Der}_{k}(B,\operatorname{Hom}_{k}(M_{i},M_{j}))$, and any derivation $D:B\to\operatorname{Hom}_{k}(M_{i},M_{j})$ satisfies $D(J^{2})=JD(J)+D(J)J=0$ when $J=J(B)$ since $\mathsf{M}$ is the family of simple $B$-modules. From the fact that

[TABLE]

is finite dimensional, and in particular a finitely generated $k$-algebra, it follows from the argument preceding the lemma that $\mathsf{M}$ is a swarm of $B$-modules. ∎

In this situation, we may iterate the process. Since $\mathsf{M}$ is a swarm of right $B$-modules, the noncommutative deformation functor $\mathsf{Def}_{\mathsf{M}}^{B}$ of $\mathsf{M}$, considered as a family of right $B$-modules, has a miniversal pro-couple $(H^{B},M_{H}^{B})$. We write $\mathcal{O}^{B}(\mathsf{M})=\operatorname{End}_{H^{B}}(M_{H}^{B})\cong(H_{ij}^{B}\otimes_{k}\operatorname{Hom}_{k}(M_{i},M_{j}))$ for its algebra of observables and $\eta^{B}:B\to\mathcal{O}^{B}(\mathsf{M})$ for its versal morphism.

Theorem 7**.**

Let $A$ be a finitely generated $k$-algebra, let $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ be a family of finite dimensional $A$-modules, and let $B=\mathcal{O}(\mathsf{M})$. Then the versal morphism $\eta^{B}:B\to\mathcal{O}^{B}(\mathsf{M})$ of $\mathsf{M}$, considered as a family of right $B$-modules, is an isomorphism.

Proof.

Since $\mathsf{M}$ is a swarm of $A$-modules and of $B$-modules, we may consider the commutative diagram

[TABLE]

The algebra homomorphism $\eta^{B}$ induces maps $B/\operatorname{J}(B)^{n}\to C/\operatorname{J}(C)^{n}$ for all $n\geq 1$, and it is enough to show that each of these induced maps is an isomorphism. For $n=1$, we have

[TABLE]

so it is clearly an isomorphism for $n=1$. For $n\geq 2$, we have that $B_{n}=B/J(B)^{n}$ is a finite dimensional algebra with the same simple modules as $B$ since $M_{i}J^{n}=0$. We may therefore consider the versal morphism of the swarm $\mathsf{M}$ of right $B_{n}$-modules, which is an isomorphism by the Generalized Burnside Theorem since $\operatorname{End}_{B}(M_{i})=k$ for $1\leq i\leq r$. Finally, any derivation $D:B\to\operatorname{Hom}_{k}(M_{i},M_{j})$ satisfies $D(J^{n})=0$ when $n\geq 2$. Therefore, we have that

[TABLE]

and this implies that $B/J(B)^{n}\to C/J(C)^{n}$ coincides with the versal morphism of the swarm $\mathsf{M}$ of right $B_{n}$-modules. It is therefore an isomorphism. ∎

Theorem 7 implies that the assignment $(A,\mathsf{M})\mapsto(B,\mathsf{M})$ is a closure operation when $A$ is a finitely generated $k$-algebra and $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ is a family of finite dimensional right $A$-modules. In other words, the algebra $B=\mathcal{O}(\mathsf{M})$ has the following properties:

1. (1)

The family $\mathsf{M}$ is the family of simple right $B$-modules. 2. (2)

The family $\mathsf{M}$ has exactly the same module-theoretic properties, in terms of extensions and matric Massey products, considered as a family of $B$-modules and as a family of $A$-modules.

Moreover, these properties characterize the algebra of observables $B=\mathcal{O}(\mathsf{M})$.

Remark 8**.**

Assume that $k$ is a field that is not algebraically closed. When $A$ is a finite dimensional $k$-algebra and $\mathsf{M}$ is the family of simple right $A$-modules, it could happen that the division algebra $D_{i}=\operatorname{End}_{A}(M_{i})$ has dimension $\dim_{k}D_{i}>1$ for some simple $A$-modules $M_{i}$. In this case, $\eta:A\to\mathcal{O}(\mathsf{M})$ is not necessarily an isomorphism. However, if the subfamily $\mathsf{M}^{\prime}=\{M_{i}:\operatorname{End}_{A}(M_{i})=k\}\subseteq\mathsf{M}$ is non-empty, we may consider the algebra $B=\mathcal{O}(\mathsf{M}^{\prime})$, and it follows from the closure property that $\eta:B\to\mathcal{O}^{B}(\mathsf{M}^{\prime})$ is an isomorphism. This means that the Generalized Burnside Theorem holds for the family $\mathsf{M}^{\prime}$ of right $B$-modules.

6. Noncommutative localizations via the algebra of observables

Let $A$ be a finitely generated $k$-algebra, and denote by $X=\operatorname{Simp}(A)$ the set of (isomorphism classes of) simple finite dimensional right $A$-modules. For any $s\in A$, we write

[TABLE]

We note that $\{D(s)\}_{s\in A}$ is a base for a topology on $X$, since $D(s)\cap D(t)=D(st)$, which we call the Jacobson topology on $X=\operatorname{Simp}(A)$.

For any inclusion $\mathsf{M}\subseteq\mathsf{M}^{\prime}$ of finite subsets of $D(s)$, there is a surjective algebra homomorphism $\mathcal{O}(\mathsf{M}^{\prime})\to\mathcal{O}(\mathsf{M})$. We may consider the algebra homomorphism

[TABLE]

where the projective limit is taken over all finite subsets $\mathsf{M}\subseteq D(s)$. Notice that $\eta_{s}(s)$ is a unit, since it is a unit in $\mathcal{O}(\mathsf{M})$ for any finite subset $\mathsf{M}\subseteq D(s)$. We define $A_{s}$ to be the subring of the projective limit

[TABLE]

generated by $\eta_{s}(A)$ and $\eta_{s}(s)^{-1}$. By abuse of notation, we write $\eta_{s}$ for the algebra homomorphism $\eta_{s}:A\to A_{s}$ into the subring $A_{s}$.

Let $S$ be the multiplicative subset $S=\{1,s,s^{2},\dots\}\subseteq A$. Then $\eta_{s}:A\to A_{s}$ is an $S$-inverting algebra homomorphism, and it has the following universal property: If $\phi:A\to B$ is any $S$-inverting algebra homomorphism, then there is a unique algebra homomorphism $\phi_{s}:A_{s}\to B$ such that $\phi_{s}\circ\eta_{s}=\phi$. We remark that $A_{s}$ is a finitely generated $k$-algebra, generated by the images of the generators of $A$ and $\eta_{s}(s)^{-1}$. In general, it is not a (left or right) ring of fractions.

7. Applications

Let $A$ be a finite dimensional $k$-algebra. We consider the family $\mathsf{M}=\{M_{1},\dots,M_{r}\}$ of simple right $A$-modules. By the Generalized Burnside Theorem, $A$ can be written in standard form as

[TABLE]

If $\operatorname{End}_{A}(M_{i})=k$ for $1\leq i\leq r$, then the standard form of $A$ is $A\cong\mathcal{O}(\mathsf{M})$, and in general, it is a subalgebra of $\mathcal{O}(\mathsf{M})$.

The standard form can, for instance, be used to compare finite dimensional algebras and determine when they are isomorphic. Let us illustrate this with a simple example. Let $k$ be a field, and let $A=k[G]$ be the group algebra of $G=\mathbb{Z}_{3}$. In concrete terms, we have that $A\cong k[x]/(x^{3}-1)$, and over a fixed algebraic closure $\overline{k}$ of $k$, we have that

[TABLE]

with $\omega\in\overline{k}$. If $\operatorname{char}(k)\neq 3$ and $\omega\in k$, then the simple $A$-modules are given by $\mathsf{M}=\{M_{0},M_{1},M_{2}\}$, where $M_{i}=A/(x-\omega^{i})$. Furthermore, a calculation shows that $\operatorname{Ext}^{1}_{A}(M_{i},M_{j})=0$ for $0\leq i,j\leq 2$. Hence, the noncommutative deformation functor $\mathsf{Def}_{\mathsf{M}}$ has a pro-representing hull $H=k^{3}$ (it is rigid), and the versal morphism $\eta:A\to\mathcal{O}(\mathsf{M})$ is an isomorphism. The standard form of $A$ is therefore given by

[TABLE]

If $\operatorname{char}(k)=3$, then $M_{0}$ is the only simple $A$-module since $x^{3}-1=(x-1)^{3}$, and we find that $\operatorname{Ext}^{1}_{A}(M_{0},M_{0})=k$. In this case, it turns out that $H\cong k[[t]]/(t^{3})$, and the standard form of $A$ is given by $A=k[\mathbb{Z}_{3}]\cong k[t]/(t^{3})$. In both cases, it follows from the Generalized Burnside Theorem that $\eta$ is an isomorphism, since $\operatorname{End}_{A}(M)=k$ for all the simple $A$-modules $M$.

If $\operatorname{char}(k)\neq 3$ and $\omega\not\in k$, then the simple $A$-modules are given by $\mathsf{M}=\{M,N\}$, where $M=M_{0}=A/(x-1)$ is $1$-dimensional, and $N=A/(x^{2}+x+1)\cong k(\omega)=K$ is $2$-dimensional. In this case, we have that $\operatorname{End}_{A}(M)=k$ and $\operatorname{End}_{A}(N)=K$, and we find that the standard form of $A$ is given by

[TABLE]

It follows from Proposition 4 that $\eta:A\to\mathcal{O}(\mathsf{M})$ is injective. However, it is not an isomorphism in this case.