Rigid Cohomology and de Rham-Witt complexes

Pierre Berthelot (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to $W_n$ with coefficients in a crystal that is only supposed to be flat over $W_n$.

Mots clés: crystal; crystalline cohomology; rigid cohomology; Witt vectors; de Rham-Witt complex; slopes of Frobenius

Domaines: Géométrie algébrique [math.AG]

Pierre Berthelot. Rigid Cohomology and de Rham-Witt complexes. Rendiconti del Seminario Matematico della Università di Padova, 2012, 128, pp.287-344. ⟨10.4171/RSMUP/128-8⟩. ⟨hal-00699900⟩