Seminar
| Parent Program: | |
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
On the (non-)existence of Ulrich modules
An "Ulrich module" for a local ring R is a non-zero maximal Cohen-Macaulay module whose number
of generators equals its multiplicity, the largest value possible. An "Ulrich sheaf" for a projective scheme X is
a coherent sheaf whose cohomology table looks like the cohomology table of the structure sheaf on projective
space. The existence of Ulrich modules and Ulrich sheaves implies a collection of desirable results. For instance,
if R admits an Ulrich module then Lech's conjecture holds for faithfully flat extensions of R. It has been asked if
every Cohen-Macaulay ring admits an Ulrich module. In this talk, I'll explain the connection between Ulrich modules
and Ulrich sheaves, and use it to prove that there exist complete local Gorenstein normal domains of dimension two
that do not have any Ulrich modules. This result is joint work with Srikanth Iyengar, Linquan Ma, and Ziquan Zhuang.