In mathematics, studying objects through their representations is a well-tested path; the area representation theory is so named because it represents the study of a group, a ring, or any other algebraic structure via linear transformations, which are sometimes easier to work with. In algebraic geometry and commutative algebra, where the objects in question are varieties and their coordinate rings, the representations take the form of modules and sheaves.

Image for theorem about 9 point on cubic curve, the special case of Cayley–Bacharach theorem.

In the 1980s, mathematician Bernd Ulrich, now a professor at Purdue University, raised a question regarding the existence of modules with rather special properties over commutative rings that satisfy a fairly mild condition, called the Cohen–Macaulay condition. These modules were subsequently called Ulrich modules. Ulrich’s question motivated much work in the field in the four decades since and has inspired similar conjectures in other fields of mathematics. Now, Ulrich’s conjecture has finally been resolved, thanks to work done by several mathematicians attending the commutative algebra semester program held in spring 2024 at the Simons Laufer Mathematical Sciences Institute (SLMath).

In the four decades since Ulrich modules were introduced, researchers have discovered that many open conjectures in commutative algebra and algebraic geometry are simple consequences of the existence of Ulrich modules and sheaves. That is, if mathematicians know that a ring has an Ulrich module, they know that it is easier to deal with than a ring that doesn’t have one, so researchers have been trying to find ways to prove that different types of rings have Ulrich modules.

Along with his coauthors Joseph Brennan and Jürgen Herzog, Ulrich established the existence of Ulrich modules over many classes of rings. There has been even more progress on the geometric side, with Ulrich sheaves being constructed in many special cases.

It came as something of surprise when, in 2021, Farrah Yhee, then a graduate student at the University of Michigan and an NSF GRFP recipient, discovered rings that do not possess any Ulrich modules. Her examples were not Cohen–Macaulay, and so they did not answer Ulrich’s question directly — but her work opened the door to the possibility of resolving the question.

At SLMath, Srikanth Iyengar, Linquan Ma, and Mark Walker discovered some obstructions that precluded the existence of Ulrich models in certain rings arising from algebraic geometry. However, finding rings for which these obstructions existed proved to be a challenge. They turned to Ziquan Zhuang, an algebraic geometer at Johns Hopkins University, for help. Within a few days, Zhuang produced a ring with the desired property. Building on those examples, the four of them joined forces and constructed many families of rings that satisfy the Cohen–Macaulay property but do not admit Ulrich modules.

The group’s work settles Ulrich’s question in the negative, producing examples of Cohen–Macaulay rings that do not have Ulrich modules, but research continues on the analogous question related to Ulrich sheaves. This leaves the door open for a more subtle understanding of precisely which rings admit Ulrich modules, perhaps to be determined at a future SLMath program.

Evelyn Lamb