Around -independence. (17th October 2017) - Chiarellotto, Bruno; Lazda, Christopher
Abstract : In this article we study various forms of -independence (including the case=p ) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of -independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain 'spreading out' result we are able to deduce a much weaker form of -independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce -independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on -independence for smooth and proper varieties over finite fields. As another consequence of this 'spreading out' result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristicp, we show a similar weak version of -independence for the unipotent fundamental group of a semistable curve in mixed characteristic.