de Rham-Witt

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de Rham-Witt cohomology

Motivation: take $X\in {\mathsf{sm}} \mathsf{Alg} {\mathsf{Var}}{/ {k}} $, and define ${ {H}^{\scriptscriptstyle \bullet}} _\mathrm{dR}(X_{/ {k}} ) \coloneqq { {{\mathbb{H}}}^{\scriptscriptstyle \bullet}} ( { {\Omega}^{\scriptscriptstyle \bullet}} _{X/k})$ using hypercohomology. Over ${\mathbf{C}}$ this will be isomorphic to singular cohomology and take values in ${}{k}{\mathsf{Mod}} $, but if $\operatorname{ch}k = p$ then the cohomology is entirely $p{\hbox{-}}$torsion. So Grothendieck/Berthelot define crystalline cohomology which takes values in ${}_{W(k)}{\mathsf{Mod}} $, the $p{\hbox{-}}$typical Witt vectors. Originally this was defined in terms of the structure sheaf of the crystalline topos of $X$, but a more modern definition realizes it as ${ {H}^{\scriptscriptstyle \bullet}} _{\mathrm{crys}}(X_{/ {k}} ) = { {{\mathbb{H}}}^{\scriptscriptstyle \bullet}} ( { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}})$, the hypercohomology of the de Rham-Witt complex. This is a lift of algebraic de Rham in the following sense: ${\operatorname{cofib}}(p { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}} \to { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}}) \simeq { {\Omega}^{\scriptscriptstyle \bullet}} _{X/k}$ is a quasi-isomorphism of cochain complexes of sheaves.