1 Morita equivalence lemma for arbitrary algebras
Let us consider first an example of Morita equivalence; thus, for an integer n ≥ 1, let Matn(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.
Theorem 1.1. Morita equivalence Lemma for arbitrary algebras
For any algebra A and any integer n ≥ 1, the algebras A and Matn(A) are Morita equivalent.
Important Notes:
- Even if A is a commutative algebra, the algebra Matn(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”.
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