Doing away with the Transpose, Revisited

Alex Martsinkovsky (Boston, MA)

Date: April 11, 2025

Abstract: A few years ago, I gave a series of talks under a common title ``Doing away with the transpose'', dealing with some of the famous formulas of Maurice Auslander. All of those formulas rely on the notion of transpose. This operation is the same as the usual transpose of matrices, only stated in terms of modules. The use of the transpose restricts the applicability of Auslander's formulas to finitely presented modules. Moreover, all of those formulas fail in the infinite case. My goal was to find a new formalism that would avoid the transpose and, for finitely presented modules, specialize to the classical results. To this end, I introduced special four-term sequences associated with additive functors, called fundamental sequences, and those tools solved the problem.

Over the last few months, it has transpired that the original fundamental sequences were only the tip of an iceberg. In this talk, I will extend that notion and make such sequences infinite. They tie together the derived functors, the satellites, and the stabilizations. One of the most remarkable features of these constructs is their ubiquity, which I will demonstrate with examples. Time permitting, I will also show applications to the universal coefficient theorems.

This is an expository talk; unfamiliar concepts will be defined and explained. I will only assume the audience has seen the definition of derived functor. In particular, familiarity with functor categories is neither assumed nor needed.