Extension of Scalars for Torsion Free Modules over Dedekind Domains

Symposia Mathematica (INDAM) 23(1979), pp.287 - 305.

In 1977, just as I was starting my teaching career here in Hawaii, I was invited to attend a conference on abelian group theory at Rome. The Italian Academy of Sciences offered me hundreds of thousands of lira to help defray my travel expenses. When converted to American money, it came out to about $500.

As part of the invitation, I was requested to present a paper at the meeting. In any case, presenting a paper would be an essential prerequisite for extracting money from the University of Hawaii to cover the rest of my expenses.

Coming up with an idea for a new paper has always been the most frightening part of research for me. But after I thought about it for awhile, I realized that there were some ideas which had been knocking around in my head for quite some time, which amounted to taking the results which I had proved in my previous two papers and generalizing them from the local case to get the same results for Butler groups as well.

It wasn't clear that this idea was substantial enough to actually base a new piece of research on, but the nice thing about this invitation was that I wouldn't have a referee to argue with, so I could get away with something that might not have passed muster at a journal.

In fact, as I thought about it over the next few weeks, I realized that the paper I had in mind wouldn't have to be completely junky. Even if it just amounted to putting some of my previous research into a new framework, it would still give me a chance to explain things the way I now wished I had explained them in my previous two papers. Furthermore, with no referee counting the cost of every word, I could be a little more expansive than I usually allowed myself to be, and maybe produce something that would satisfy all the objections of people who said that my papers were overly terse and difficult to read.

Furthermore, I could recast my existing results in the envirnoment of modules over dedekind domains instead of abelian group theory. Of course ``everyone knows'' that results for abelian groups automatically carry over to this wider context, but I liked the idea of being able to use the language of commutative ring theory instead of that of group theory.

In fact, I went overboard and even used the term ``flat'' instead of ``torsion free'' throughout the paper.

In the event, I finally realized that going to Rome was starting to seem much more like a burden than an adventure or pleasure, and with much relief I made the decision not to attend the conference.

The paper was included in the conference proceedings anyway, though, and it turned out to be much less routine than I had anticipated.

In my very sketchy paper for the Las Cruces conference in 1976, I had shown that the category under quasi-homomorphisms of certain types of modules over a discrete valuation rings (namely those whose splitting field K was a given finite algebraic extension of Q), was equivalent to a full subcategory of the category of finite-length modules over a certain not very complicated finite-dimensional algebra (depending on the splitting field K). Now, I presented the mechanism of this equivalence more simply and in somewhat more detail, for a much more general class of modules over a Dedekind domain. And then I showed that one could also construct an equivalence between the category of such modules under homomorphisms (rather than quasi-homomorphisms) and a full subcategory of the category of finitely generated modules over a certain noetherian ring.

The two main theorems were actually major results (at least in my opinion). I wound up having to do some hard work to get some of the proofs.

I think that the final paper would have been readily accepted by any number of reputable journals, perhaps including the Journal of Algebra. In any case, for me it was an essential publication, since it laid down the framework for my ideas in the form which I needed for my subsequent work.

I consider this one of my key papers and I've always been rather disappointed that other abelian group theorists never seemed all that interested in it. I've never quite understood why.