MoritaUniquenessTheorem 2009-06-15 14:41:26 2009-06-15 15:27:04 Theorem The main representation result underlying Morita theory is the Eilenberg--Watts theorem. \begin{theorem}[Eilenberg--Watts] Let $A$ and $B$ be rings, and let \[ F:A\text{-}\mathrm{Mod}\longrightarrow B\text{-}\mathrm{Mod} \] be an additive, right-exact functor that preserves arbitrary direct sums. Then there exists a $(B,A)$-bimodule $Q$, unique up to isomorphism, such that $F$ is naturally isomorphic to the tensor-product functor \[ Q\otimes_A(-): A\text{-}\mathrm{Mod}\longrightarrow B\text{-}\mathrm{Mod}. \] One may take \[ Q=F(A), \] with its right $A$-module structure induced by the endomorphisms of the left regular $A$-module. \end{theorem} This theorem yields the standard bimodule characterization of Morita equivalence. \begin{corollary} Two rings $A$ and $B$ are Morita equivalent if and only if there exist an $(A,B)$-bimodule $P$ and a $(B,A)$-bimodule $Q$ such that \[ P\otimes_B Q \cong A \] as $(A,A)$-bimodules and \[ Q\otimes_A P \cong B \] as $(B,B)$-bimodules. In this situation the functors \[ Q\otimes_A(-): A\text{-}\mathrm{Mod}\longrightarrow B\text{-}\mathrm{Mod} \] and \[ P\otimes_B(-): B\text{-}\mathrm{Mod}\longrightarrow A\text{-}\mathrm{Mod} \] are quasi-inverse equivalences. Moreover, \[ \operatorname{End}_A(P)\cong B^{\mathrm{op}}, \qquad \operatorname{End}_B(Q)\cong A^{\mathrm{op}}, \] and $P$ and $Q$ are finitely generated projective generators on the appropriate sides. \end{corollary} \emph{Proof sketch.} If the module categories are equivalent, choose quasi-inverse equivalences $F$ and $G$. By the Eilenberg--Watts theorem, \[ F\cong Q\otimes_A(-), \qquad G\cong P\otimes_B(-) \] for suitable bimodules $Q$ and $P$. Since \[ G\circ F\cong \operatorname{Id}_{A\text{-}\mathrm{Mod}} \quad\text{and}\quad F\circ G\cong \operatorname{Id}_{B\text{-}\mathrm{Mod}}, \] evaluating at the regular modules gives \[ P\otimes_B Q\cong A, \qquad Q\otimes_A P\cong B. \] Conversely, these bimodule isomorphisms immediately show that the two tensor functors above are quasi-inverse equivalences. \begin{corollary} If $A$ and $B$ are Morita equivalent, then: \begin{enumerate} \item the categories of right modules \[ \mathrm{Mod}\text{-}A \quad\text{and}\quad \mathrm{Mod}\text{-}B \] are equivalent; \item the categories of bimodules \[ A\text{-}\mathrm{Mod}\text{-}A \quad\text{and}\quad B\text{-}\mathrm{Mod}\text{-}B \] are equivalent. \end{enumerate} With bimodules $P$ and $Q$ as above, an equivalence on bimodules is given by \[ M \longmapsto Q\otimes_A M\otimes_A P. \] It sends the regular $(A,A)$-bimodule $A$ to the regular $(B,B)$-bimodule $B$, since \[ Q\otimes_A A\otimes_A P \cong Q\otimes_A P \cong B. \] A quasi-inverse is \[ N \longmapsto P\otimes_B N\otimes_B Q. \] \end{corollary} $(B A)$-bimodule Morita theorem % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} The main result for Morita equivalent algebras is provided by the following proposition. \begin{theorem}{\em Morita theorem}. Let $A$ and $B$ be two arbitrary rings, and also let $F : A-mod \to B-mod$ be an additive, right exact functor. Then, there is a $(B,A)$-bimodule $\mathcal{Q}$, which is unique up to isomorphism, so that $F$ is isomorphic to the functor $G$ given by $$A-mod \mapsto B-mod,$$ $$M \mapsto Q \bigotimes {}_A M.$$ \end{theorem} There are also two important and fairly straightforward corollaries of the Morita (uniqueness) theorem. \begin{theorem} {\bf Corollary 1.} Two rings, $A$ and $B$, are Morita equivalent if and only if there is an $(A,B)$-bimodule $M_b$ and a $(B,A)$-bimodule $N_b$ so that $$M_B \bigotimes {}B N_B \simeq A$$ as $A$-bimodules and $$N_B \bigotimes{}_A M_b \simeq B$$ as $B$-bimodules. With these assumptions, one obtains: $$End_{A-mod}(M_b) = B^{op},$$ $$End_{B-mod}(N_b) = A^op$$. Also $M_b$ is projective as an $A$-module, whereas $N_B$ is projective as a $B$-module. \end{theorem} {\em Proof}. All equivalences of categories are exact functors, and therefore they preserve projective objects as required by {\em Corollary 1.} \begin{theorem}{\em Corollary 2.} \begin{itemize} \item (i). If $A$ and $B$ are Morita equivalent rings, then the corresponding categories ${\bf mod-A}$ and ${\bf mod-B}$ are also equivalent. \item (ii). Furthermore, there exists a natural equivalence of categories $${\bf A-bimod} \to {\bf B-bimod}$$ which takes $A$ to $B$, of course along with their natural bimodule structures. \end{itemize} \end{theorem} {\em Proof.} Let $M_b$ and $N_b$ be the bimodules already defined in {\em Corollary 1}. For proposition (i), one utilizes the functors $(− \bigotimes{}_A M_b$ and $(− \bigotimes{}_B N_b)$ to prove the equivalence of the two categories. For the second proposition (ii), one needs to employ the functor $$N_b \bigotimes{}_A - \bigotimes{}_A M_b : {\bf A-bimod} \longrightarrow {\bf B-bimod}$$ to prove the natural equivalence of the latter two categories.