Morita equivalence and moduli problems Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules).
Ex: $M_n(R)$ (the algebra of matrices over a ring $R$) is morita equivalent to $R$.
In fact more generally whenever $A$ is a ring and $e$ is an idempotent in $A$ and $AeA = A$ then the follwing functor is a morita equivalence:
$A$  modules $\rightarrow$  $eAe$ modules
$M$ $\mapsto$ $eM$
Now under nice conditions the categories of $A$ modules (resp $B$ modules) might have a moduli description. Then can you say anything about the induced map on the moduli spaces? I'm asking for situations in which this is known.
Also is there a nice book or paper which talks about Morita equivalence and has lots of examples?
 A: This may be too elementary, but Anderson, Fuller Rings and Categories of Modules has chapter 6 giving basic properties of Morita equivalence. The Morita equivalence section of McConnell, Robson Noncommutative Noetherian Rings goes into more depth; e.g. they show that Morita equivalent rings have isomorphic lattices of (2-sided) ideals, which I don't believe is covered in Anderson-Fuller.
I realize this is not your original question, but it may help with question #2.
A: In the setting you are interested in, that is, finitely generated k-algebras and GIT-quotients of closed or semistable orbits, Morita equivalence induces isomorphism on the moduli spaces provided one scales dimension(vectors) and stability structures accordingly. That is, if B=M_n(A) one should compare Mod(A,k) to Mod(B,nk) moduli. If A and B have a complete set of orthogonal idempotents e_i resp. f_i(that is, a quiver-like situation) and if the Morita equivalence induces rank(f_i) = n_i rank(e_i), then one should compare A-reps of dimension vector alpha=(alpha_i) to B-reps with dimension vector beta=(n_i alpha_i).
Geometrically, the module varieties of Morita-equivalent algebras are related via associated fibre bundle constructions. In the example above, Mod(B,kn) = GL(nk) x^GL(k) Mod(A,k). In general,one had such a description locally in the Zariski topology (coming roughly from the fact that a vectorbundle (or projective) is locally free). Anyway, this gives a natural and geometric one-to-one correspondence between GL(nk)-orbits (isoclasses) of B-reps and GL(k)-orbits of A-reps inducing the desired isos on the quotient variety level (isos of semi-simples). In quiver-like situations one should adjust the dim vectors as above.
Now as to semi-stability. Observe that semi-stable reps are just ordinary reps of a universal localization of your algebra(s), so one can reduce to the closed case by covering the variety of semi-stables by Zariski open (affine) pieces. In quiver-like situations when the Morita-data is as above and your stability structure mu for A is given by the vector (mu_i), then the corresponding stability structure for B is mu'=(1/n_i x mu_i) (or multiply it with a common factor if you want it to have integral components.
A: This is not an example of the kind you asked for, but you might find this interesting anyway - it does involve both moduli and equivalences. In Bridgeland's paper Flops and Derived Categories he constructs certain derived equivalences (which are Fourier-Mukai transforms so have a Morita type flavour) by building a moduli space for certain objects in a derived category which come from a t-structure.
It has been a while since I have looked at it but from memory "Morita Equivalence and Duality" by Cohn is quite a nice book and I think it contains several examples (I hope that I am remembering correctly).
A: There are some nice Morita equivalences arising from Hecke algebras in representation theory - they arise as algebras of bi-invariant functions on a locally compact group under convolution.  Good, workable examples arise from subgroups of finite groups.
If you like categories, there is a rather high-level treatment in this paper (Ben-Zvi, Francis, Nadler).  It is probably not the best place to start.
