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Prisms and deformations of de Rham cohomology

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Prisms and deformations of de Rham cohomology
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23
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Prisms are generalizations of perfectoid rings to a setting where "Frobenius need not be an isomorphism". I will explain the definition and use it to construct a prismatic site for any scheme. The resulting prismatic cohomology often gives a one-parameter deformation of de Rham cohomology. For instance, it recovers the recently constructed A_{inf}-cohomology for smooth schemes over perfectoid rings (and thus crystalline cohomology when in characteristic p). A relative variant yields cohomological Breuil-Kisin modules, and related ideas also give a co-ordinate free construction of q-de Rham cohomology.