Positive characteristic Poincaré Lemma

Abstract

Let $K$ be a field of characteristic $p>0$ and $\omega$ be an $r$-form in $K^n$. In this case, differently of fields of characteristic zero, the Poincaré Lemma is not true because there are closed $r$-forms that are not exact. We present here a definition of a $p$-closed $r$-forms and a version of the Poincaré Lemma that is valid for $p$-closed polynomial or rational $r$-forms on $K^n$ and, as a consequence, the de Rham cohomology modules of $K^n$ are not trivial.