Prime characteristic singularities and the deformation problem

Abstract: Let $P$ be a property of local rings (such as regular, Gorenstein, or complete). We say that $P$ deforms if a local ring $R$ enjoys property $P$ provided there exists a nonzerodivisor $x$ such that $R/xR$ is $P$. (For example, the properties of being regular or Gorenstein deform, but the property of being complete does not deform). The deformation problem, as it pertains to the prime characteristic singularity classes of $F$-regular, $F$-rational, $F$-pure, and $F$-injective singularities, has a rich history that dates to work of Fedder in the 1980's and remains an active research area. We will survey the history of the deformation problem of these four prime characteristic singularity classes and discuss a recent solution to the deformation of $F$-purity problem in rings which are $\mathbb{Q}$-Gorenstein. This talk is based on a collaboration with Austyn Simpson.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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