PlanetPhysics/Morita Equivalence Lemma for Arbitrary Algebras

Morita equivalence lemma for arbitrary algebras

Let us consider first an example of [[../MoritaEquivalentAlgebras2/|Morita equivalence]]; thus, for an integer $n\ge 1$, let $Ma{t}_n(A)$ be the algebra of $n\times n$-matrices with entries in an algebra $A$. The following is a typical example of Morita equivalence that involves [[../AbelianCategory3/|noncommutative]] algebras.

\begin{theorem}{\mathbf Morita equivalence Lemma for arbitrary algebras}

For any algebra $A$ and any integer $n\ge 1$, the algebras $A$ and $Ma{t}_n(A)$ are Morita equivalent. \end{theorem}

{\mathbf Important Notes:}

• Even if $A$ is a commutative algebra, the algebra $Ma{t}_n(A)$ is of course not commutative for any $n>1$ because the [[../Matrix/|matrix multiplication]] is generally [[../AbelianCategory3/|non-commutative]].
• In general, the algebra $A$ cannot be recovered from its corresponding [[../AbelianCategory2/|abelian category]] $A$-mod. Therefore, in order for a [[../PreciseIdea/|concept]] in [[../NoncommutativeGeometry4/|noncommutative geometry]] to have or retain an intrinsic meaning, such a concept must be Morita invariant that is, to remain within the same Morita equivalence class. This raises the important question: what properties of an algebra are Morita invariant ? The answer to this question is provided by the Uniqueness Morita Theorem.