October 21
Philipp Rothmaler,
CUNY
Generalized Bass modules
Over half a century ago Hyman Bass proved that all flat left modules are projective precisely when the underlying ring satisfies the descending chain condition on right principal ideals. He called such rings left perfect. Gena Puninski noticed that this can be given a model theoretic proof. Every infinite descending chain of principal right ideals gives rise to a descending chain of (pp) formulas which, in turn, gives rise to a direct limit of finitely generated projective modules that is not projective. Such a module is flat and not projective, and called a Bass module.
I demonstrate how this construction is elementary model theory and at the same time generalizes to other classes of (pp) formulas and modules, which, among other things, yields a new proof of the late Daniel Simson’s result that all left modules are Mittag-Leffler iff the ring is left pure-semisimple (which, to model theorists, means that all left modules are totally transcendental).
I will emphasize the model theoretic ideas and explain the connection with the algebraic concepts. This is part of ongoing work with Anand Pillay.