## View abstract

### 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

#### University of Sao Paulo (USP), Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak7d311499896965bd45f425befa513963').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy7d311499896965bd45f425befa513963 = 'l&#101;v' + '&#64;'; addy7d311499896965bd45f425befa513963 = addy7d311499896965bd45f425befa513963 + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_text7d311499896965bd45f425befa513963 = 'l&#101;v' + '&#64;' + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloak7d311499896965bd45f425befa513963').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy7d311499896965bd45f425befa513963 + '\'>'+addy_text7d311499896965bd45f425befa513963+'<\/a>';

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.