Morita equivalence

Let $R$ be a ring. Write $\mathcal{M}_{R}$ for the category of right modules over $R$ . Two rings $R$ and $S$ are said to be Morita equivalent if $\mathcal{M}_{R}$ and $\mathcal{M}_{S}$ are equivalent as categories (http://planetmath.org/EquivalenceOfCategories). What this means is: we have two functors

F:\mathcal{M}_{R}\to\mathcal{M}_{S}\qquad\mbox{ and }\qquad G:\mathcal{M}_{S}% \to\mathcal{M}_{R}

such that for any right $R$ -module $M$ and any right $S$ -module $N$ , we have

GF(M)\cong_{R}M\qquad\mbox{ and }\qquad FG(N)\cong_{S}N,

where $A\cong_{R}B$ means that there is an $R$ -module isomorphism between $A$ and $B$ .

Example. Any ring $R$ with $1$ is Morita equivalent to any matrix ring $M_{n}(R)$ over it.

Proof.

Assume $n>1$ . For convenience, we will also say a module to mean a right module.

Let $M$ be an $R$ -module. Set $F(M)=\{(m_{1},\ldots,m_{n})\mid m_{i}\in M\}$ . Then $F(M)$ becomes a module over $M_{n}(R)$ if we adopt the standard matrix multiplication $m A$ , where $m\in F(M)$ and $A\in M_{n}(R)$ . If $f:M_{1}\to M_{2}$ is an $R$ -module homomorphism. Set $F(f):F(M_{1})\to F(M_{2})$ by $F(f)(m_{1},\ldots,m_{n})=(f(m_{1}),\ldots,f(m_{n}))\in F(M_{2})$ . Then $F$ is a covariant functor by inspection.

Next, let $N$ be an $M_{n}(R)$ -module. Write $e(r)$ as the $n\times n$ matrix whose cell $(1,1)$ is $r\in R$ and $0$ everywhere else. For simplicity we write $e:=e(1)$ . Note that $e$ is an idempotent in $M_{n}(R)$ : $e=ee$ , and $e$ commutes with $e(r)$ for any $r\in R$ : $ee(r)=e(r)e$ .

Set $G(N)=\{se\mid s\in N\}$ . For any $r\in R$ , define $se\cdot r:=see(r)=se(r)e$ . Since $se(r)\in N$ , this multiplication turns $G(N)$ into an $R$ -module. If $g:N_{1}\to N_{2}$ is an $M_{n}(R)$ -module homomorphism, define $G(g):G(N_{1})\to G(N_{2})$ by $G(g)(se)=g(s)e$ . If $N_{1}\lx@stackrel{{\scriptstyle g}}{{\longrightarrow}}N_{2}\lx@stackrel{{% \scriptstyle h}}{{\longrightarrow}}N_{3}$ are $M_{n}(R)$ -module homomorphisms, then

G(h\circ g)(se)=(h\circ g)(s)e=h(g(s))e=G(h)[g(s)e]=G(h)[G(g)se]=G(h)\circ G(g% )(se)

so that $G$ is a covariant functor.

If $M$ is any $R$ -module, then $GF(M)=\{(m_{1},\ldots,m_{n})e\mid m\in M\}=\{(m_{1},0,\ldots,0)^{T}\mid m\in M% \}\cong M$ , where $m^{T}$ stands for the transpose of the row vector $m\in M$ into a column vector.

On the other hand, if $N$ is any $M_{n}(R)$ -module, then $FG(N)=\{(s_{1}e,\ldots,s_{n}e)\mid s_{i}\in N\}$ . Before proving that $FG(N)\cong N$ , let’s do some preliminary work.

Denote $e_{ii}$ by the $n\times n$ matrix whose cell $(i,i)$ is 1 and $0$ everywhere else. Then each $e_{ii}$ is idempotent, $e_{ii}e_{jj}=0$ for $i\neq j$ , and $e_{11}+\cdots+e_{nn}=1$ . From this, we see that $N=N_{1}\oplus\cdots\oplus N_{n}$ , where $N_{i}=Ne_{ii}$ , and $N_{i}\cong N_{j}$ as $M_{n}(R)$ -modules. Since $N_{1}=Ne$ has an $R$ -module structure as we had shown earlier, $N_{i}$ are all $R$ -modules. Let $\pi_{i}:N\to N_{i}$ be the projection map, $\psi_{i}:N_{i}\to N$ be the embedding of $N_{i}$ into $N$ , and $\phi_{ij}:N_{i}\to N_{j}$ be the isomorphism from $N_{i}$ to $N_{j}$ given by $\phi_{ij}(se_{ii})=se_{jj}$ . All these are $M_{n}(R)$ -module homomorphisms since $e_{ii}A=Ae_{ii}$ .

Now, take any $s\in N$ , then $s\mapsto(\pi_{1}(s),\ldots,\pi_{n}(s))\mapsto(\phi_{11}\pi_{1}(s),\ldots,\phi_% {n1}\pi_{n}(s))\in FG(N)$ is a homomorphism $\alpha:N\to FG(N)$ . Conversely, $(s_{1}e,\ldots,s_{n}e)\mapsto(\phi_{11}(s_{1}e),\ldots,\phi_{1n}(s_{n}e))% \mapsto\psi_{1}(\phi_{11}(s_{1}e))+\cdots+\psi_{n}(\phi_{1n}(s_{n}e))\in N$ is also a homomorphism $\beta:FG(N)\to N$ . By inspection, $\alpha$ and $\beta$ are inverses of each other, and hence $FG(N)\cong N$ . ∎

Remark. A property $P$ in the class of all rings is said to be Morita invariant if, whenever $R$ has property $P$ and $S$ is Morita equivalent to $R$ , then $S$ has property $P$ as well. By the example above, it is clear that commutativity is not a Morita invariant property.

Title	Morita equivalence
Canonical name	MoritaEquivalence
Date of creation	2013-03-22 16:38:49
Last modified on	2013-03-22 16:38:49
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16D90
Defines	Morita equivalent
Defines	Morita invariance
Defines	Morita invariant