Orsatti’s Contribution to Module Theory

Abstract

I met Adalberto for the first time in November 1972: I was a first year student in Mathematics and he was my Algebra Professor. I do remember those tough days pretty well: the word “set” was completely new to me and each every time, at the end of Adalberto’s lesson, I thought I got an Arabic class instead of an Algebra one. Then suddenly, during Christmas time, I felt I could understand everything and that everything was so beautiful that, in the end, I gave up a very fancy interest for cybernetics (!) and decided to get more Algebra courses. Thus, in the next years, I followed two more courses by Adalberto and also got the honour to have him as my thesis advisor. Since then a never-ended collaboration began. Adalberto is not only a very clear teacher; he has the special gift of capturing your attention leading you inside the subject. For me he is the most fascinating person I ever met during my mathematical experience. And not only this. He taught me the most difficult thing: how to do research. It is nice here to remember that each every time I was getting depressed because we were not able to prove something then he began telling me one of his incredible jokes. Also if we found out that something we would have liked to be true, it was not, he was telling me “This is the truth, Claudia, what do you want more then the truth?” Orsatti’s first work in module theory is his first paper on Duality [O1]. The astonishing idea he got was that Pontryagin duality between discrete and compact abelian groups (see [P1] and [P2]) and Lefschetz duality (see [L]) between discrete and linearly compact vector spaces as well as Kaplansky’s (see [K]) and Macdonald’s (see [Mc]) generalizations of this last one could be all thought as a particular case of a much more general situation. Namely, let A be a commutative ring and K an Hausdorff complete topological module over the ring A endowed with its discrete topology. Assume that: P1) The mapping a 7→ μa= multiplication by a in K defines an isomorphism A ∼= ChomA(K, K). P2) K has no small submodules i.e. there is a neighbourhood U of 0 in K such that the only submodule of K contained in U is 0. Let A-TM denote the category of Hausdorff topological modules over the ring A endowed with its discrete topology and let C(KA) be the subcategory of A-TM consisting of those topological modules which are topological isomorphic to closed submodules of topological products of copies of K. Let D(KA) be the subcategory of Mod-A cogenerated by KA. For every M ∈ Mod-A let M∗ be the A-module HomA(M,K) endowed with the topology of pointwise convergence. Clearly M∗ ∈ C(KA). For every M ∈ A-TM, let M∗ be the abstract module ChomA(M, K);