Brauer-Thrall for totally reflexive modules over local rings of higher dimension

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive $R$-modules of multiplicity $n$.Comment: to appear in Algebras and Representation Theor

Prions, prionoids and protein misfolding disorders

Prion diseases are progressive, incurable and fatal neurodegenerative conditions. The term ‘prion’ was first nominated to express the revolutionary concept that a protein could be infectious. We now know that prions consist of PrPSc, the pathological aggregated form of the cellular prion protein PrPC. Over the years, the term has been semantically broadened to describe aggregates irrespective of their infectivity, and the prion concept is now being applied, perhaps overenthusiastically, to all neurodegenerative diseases that involve protein aggregation. Indeed, recent studies suggest that prion diseases (PrDs) and protein misfolding disorders (PMDs) share some common disease mechanisms, which could have implications for potential treatments. Nevertheless, the transmissibility of bona fide prions is unique, and PrDs should be considered as distinct from other PMDs