Author: Eiko

Tags: category theory, morita equivalence, algebra

Time: 2024-11-11 00:23:57 - 2024-11-11 00:31:19 (UTC)

Morita Equivalence

Can be viewed as a baby version of derived equivalence, when we say two algebras A and B are Morita equivalence, they are equivalent in a deep sense. Precisely speaking it means their module categories are equivalent.

ModAModB

Full-idempotent Give Morita Equivalence

When you have a full idempotent e in A, i.e. an idempotent e2=e that satisfy AeA=A, here the left side accepts linear extending. Then A is automatically Morita equivalent to B=eAe, the functor is given by

E:ModAModB,MMe

with reverse functor given by

F:ModBModA,NNBA.

Note that here Me is a right B=eAe module because meae=(mea)e.