hypercohomology

Taking hypercohomology computes the intersection 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.