Title: Ulrich modules
Abstract: This talks concerns the question of existence of certain objects over a commutative noetherian local ring. A prime example of such a ring is the local ring of functions at a point on an algebraic variety. In this case the structural properties of the ring reflect the geometric properties of the variety at that point. As for groups, one can study rings through their representations, which in this context are called modules. I will focus on a class of modules called maximal Cohen-Macaulay modules whose properties often reflect the singuarity type of the ring.
Around 1984 Bernd Ulrich identified a family of maximal Cohen-Macaulay modules with special numerial properties, and conjectured that they exist over Cohen-Macaulay rings. In the decades since, it was discovered that many open conjectures in commutative algebra and algebra geometry are simple consequences of the existence of Ulrich modules. I will explain what these “Ulrich modules” are, how they have been used in the literaure, and lead up to recent work of Yhee, and joint work of Ma, Walker, Zhuang, and myself where we construct examples of rings for which Ulrich modules do not exist.