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Can anybody give me an example of a "naturally-occurring" algebraic category C in which:

  1. C has two non-isomorphic objects A and B which are bi-embeddable via monic maps; but

  2. C does NOT have any infinite collection A_1, A_2, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category C, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?

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2 Answers 2

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I don't know if this counts as a "naturally-occuring" algebraic category but as I scanned through my internal list of categories of modules over a ring, I spotted that the following has (1) and (2).

Take G to be the "p-adic dihedral group" that is a semidirect product of the p-adic integers and a cyclic group of order 2 where the latter acts by -1 on the former. Then form the completed group algebra (Iwasawa algebra) with coefficients in the field with p elements, R=F_p[[G]].

Now there are precisely two isomorphism classes of indecomposable projective modules [P_ 1] and [P_ 2] and P_ 1 and P_ 2 are bi-embeddable via monic maps (see section 9.6 of http://www.dpmms.cam.ac.uk/~sjw47/char.pdf for the proof of both these claims).

It follows that if we take the category of all finitely generated projective modules over R then it consists of objects that are direct sums of m copies of P_ 1 and n copies of P_ 2, say. Two such objects will be bi-embeddable if and only if they have the same value of m+n thus whilst there are arbitrarily large finite collections of pairwise bi-embeddable but pairwise nonisomorphic objects in this category there are no infinite ones.

I am sure there will be many more examples along these lines, I just happen to spend a lot of time thinking about Iwasawa algebras.

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Thanks! This is exactly the kind of example I was looking for. Abstracting a bit, it seems that if you ever have a "reasonable" category of algebraic objects with two "indecomposables" A and B which are bi-embeddable but nonisomorphic, then you ought to be able to take the subcategory of all objects "finitely generated by A and B" and get a similar thing satisfying both 1 and 2. I'm too lazy at the instant to think what's the optimal way to make this precise -- maybe it should be a statement about abelian categories? –  John Goodrick Oct 15 '09 at 22:54
    
If by finitely generated you mean under direct sum in an additive category then I think that will be true. –  Simon Wadsley Oct 16 '09 at 7:05
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Natural example: the category of function fields of supersingular elliptic curves over a fixed algebraically closed field of characteristic $p$ (you pick $p$, say bigger than $13$ or so to ensure more than one isomorphism class). Note that this is -- up to translations -- the opposite category to the Brandt module category, in which the objects are the ss elliptic curves themselves and the morphisms are isogenies.

As $p$ varies, this gives a family of (essentially) finite categories such that any two objects are mutually embeddable and the number of isomorphism classes of objects tends to infinity with $p$.

If I may be so bold, I spent much of a paper talking about the relation between two function fields that each is embeddable in the other, which I called (borrowing from the theory of elliptic curves) "isogeny".

PLC, On elementary equivalence, isomorphism and isogeny.
J. Théor. Nombres Bordeaux 18 (2006), no. 1, 29--58.

http://math.uga.edu/~pete/logic.pdf

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