Kanex写点吧
Morita等价到底是什么?
我问了一些人,似乎不是你满意的答案

我一窍不通。谢谢。

The terminology "Morita equivalence" appears in various contexts: C*
algebras, string theory, ring theory,etc.Two rings A and B are Morita equivalent if there exist bimodules
X and Y that are inverses of each other with respect to tensor
product. I.e., X is a left A module and a right B module,
Y is the other way around, and

X tensor_B Y = A
Y tensor_A X = B

You can use the module X to convert any left B-module to a left
A-module, and vice versa for Y. The relation between
them establishes that the categories of (left) A-modules and (left)
B-modules are equivalent.
For example, if A is any ring and M_n(A) is the ring of n x n matrices,
then M_n(A) and M_k(A) are Morita equivalent via the bimodules of n x
k and k x n matrices. The bimodule structures are given by left and
right multiplication.

The rings \Z (integers) and \Z[i] (Gaussian integers) are not Morita
equivalent. \Z-modules are just abelian groups, and the irreducible ones
are \Z/p for primes p. On the other hand, the 9-element \Z[i]-module
\Z[i]/3 is irreducible. This module has 8 automorphisms, but no
irreducible abelian group has 8 automorphisms since 9 isn't prime.
The categories of modules are fundamentally different and hence there
is no Morita equivalence.
Actually,the term "Morita equivalence" appears also in the Algebraic Geometry and Algebraic group.

谁能不能举个例子关于 两个范畴的导出范畴等价,我不了解这个方面.