Algebraic Geometry Seminar

Kiran Kedlaya (Berkeley)

"F-isocrystals and p-adic cohomology"

Like local systems in ordinary cohomology and lisse sheaves in $l$-adic cohomology, $F$-isocrystals act as coefficients in the $p$-adic cohomology of algebraic varieties in characteristic $p$. However, the theory of these objects is immature: to date, it has not been possible to prove the finite dimensionality of the $p$-adic cohomology of an arbitrary variety with coefficients in an overconvergent $F$-isocrystal. We will sketch how recent results on $p$-adic differential equations can be used to prove this finiteness.