Morita equivalence lemma for arbitrary algebras

1 Morita equivalence lemma for arbitrary algebras

Let us consider first an example of Morita equivalence; thus, for an integer n ≥ 1, let Mat_n(A) be the algebra of n × n-matrices with entries in an algebra A. The following is a typical example of Morita equivalence that involves noncommutative algebras.

Important Notes:

• Even if A is a commutative algebra, the algebra Mat_n(A) is of course not commutative for any n > 1 because the matrix multiplication is generally non-commutative.
• In general, the algebra A cannot be recovered from its corresponding abelian category A-mod. Therefore, in order for a concept in 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”.