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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2021
- DOI:
- 10.48550/arXiv.2110.08888
- arXiv:
- arXiv:2110.08888
- Bibcode:
- 2021arXiv211008888D
- Keywords:
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- Mathematics - Rings and Algebras;
- 13N15