Re: Picard group of a ringoid

[Michael Barr <barr@barrs.org>]

I have never seen a name for this.  I think, if it hasn't been defined
before, I would be inclined to call it the Morita group.  There is a large
groupoid, let me call it the Morita groupoid, whose objects are rings and
for which a morphism R --> S is a left S, right R bimodule M such that
tensoring with M gives an equivalence between the category of left R
modules and left S modules.  This is locally small since M must be a
finitely generated projective left S module and the group you are dealing
with is simply the group of endomorphisms of R in that groupoid.  the
whole theory is due to Morita (and the main theorem, the Morita theorem).
This is for rings, of course.  I assume that a ringoid is a small
preadditive category.  A preadditive category with finitely many objects
is Morita equivalent to a ring so it will be true for them.  Beyond that,
it would have to be examined because I am not sure what corresponds to
finitely generated.  

On Sat, 16 Dec 2000, Bill Rowan wrote:

> Tensor product gives a monoid structure on the class of isomorphism types
> of R,R-bimodules, for a ring or ringoid R.  Restricting to those elements
> for which there is a two-sided inverse yields a group.  I am inclined to call
> this the nonabelian Picard group and denote it by NPic(R).  If we start with
> a commutative ring R, then the usual Picard group of R, Pic(R), can be viewed
> as an abelian subgroup of NPic(R).
> 
> Has anyone seen this before?  Does anyone have some other idea about what this
> should be called?
> 
> Bill Rowan
>