Department of Mathematics
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This series of talks covers some 'first steps' towards the study of a very complicated problem. The 'Ziegler Spectrum' of a ring classifies the first-order (model-theoretic) complexity of the category of modules; it has applications in general module theory as well. It is easily seen (but for very abstract reasons) that as one passes from a ring R to the polynomial ring R[x], the complexity of the spectrum 'blows up'. For instance, the Ziegler spectrum of a field K is a single point; the spectrum of K[x] is complicated but describable; and the spectrum of K[x,y] is "wild" (essentially unclassifiable).

Yet the source of this wildness is not understood in concrete terms. For instance, the Ziegler spectrum of the ring of ring Z of integers is similar in form to that of K[x]; yet only a few (obvious) points of the spectrum of Z[x] are widely known in any way. Surely for a ring as well known and explicitly given as this, we should be able to produce uncountably many points of the spectrum in some explicit way, even if we cannot classify them.

This series of talks will start with a review/introduction to the elements of the model theory of modules: systems of linear equations, positive-primitive formulas, pure-injective modules, totally transcendental (tt) modules, the Ziegler spectrum, relation to other concepts from algebra. Then I will continue by constructing uncountable families of tt points of the spectrum, and with some complicated relations between them. Most of these points can be regarded as "unfamiliar". The constructions I have are fairly general, but still one or two steps short of being able to provide real insight into the general problem.