**Around the Poincaré Lemma, After Beilinson **

* Luc Illusie *

**Abstract**

The following notes are an expanded version of talks on Beilinson’s paper as reported by Beilinson (J. AMS **25**, 715–738, 2012) given at the department of mathematics of the university of Padova on October 25 and 26, 2012, at the Séminaire RÉGA (Réseau d’Étudiants en Géométrie Algébrique, Institut Henri Poincaré, Paris) on December 12, 2012, lectures at the KIAS (Seoul) on January 3 and 4, 2013, a course at the Morningside Center of Mathematics (Beijing) on February 18, 25, and March 4, 11, 2013, and a course at the VIASM (Hanoi) on September 4, 5, 9, 10, 11, 2014. I do not discuss Beilinson’s next paper as reported by Beilinson (Camb. J. of Math. **1**, 1–51, 2013), which deals with refinements of the comparison theorem between *p*-adic étale cohomology and de Rham cohomology.