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Session S07 - Differential operators in algebraic geometry and commutative algebra

Friday, July 16, 12:00 ~ 12:30 UTC-3

A survey on $d$-simplicity

Daniel Levcovitz

Let $k$ be a field of characteristic zero and $d$ a $k$-derivation of a commutative $k$-algebra $R$. We say that $d$ is a simple derivation of $R$ (or just that $R$ is $d$-simple) if $R$ does not have any proper non-zero ideal $I$ such that $d(I)\subseteq I$. Such an ideal is called a $d$-invariant ideal, a $d$-stable ideal or simply a $d$-ideal. Research on simple derivations of commutative $k$-algebras has increased significantly in the past years. This was motivated by several connections of $d$-simple algebras with other branches of mathematics such as noncommutative noetherian ring theory; D-modules; holomorphic foliations and also with the difficult question of algebraic independence of formal solutions of differential equations. In this talk we will present a survey on $d$-simplicity including some examples, main results and some conjectures about the isotropy group of simple derivations.