A Note on the First Rigid Cohomology Group for Geometrically Unibranch Varieties

Abstract

We prove a vanishing of the first rigid cohomology group for geometrically unibranch varieties with supports in a proper closed subset and apply it to the full faithfulness problem of the restriction functors of overconvergent isocrystals. As an application, we prove that the first rigid cohomology group is pure of weight 1 for proper and geometrically unibranch varieties over a finite field. We also establish a comparison result of rigid cohomology groups between a geometrically unibranch variety and its normalization.