Around $\mathrm{\ell }$-independence

In this article we study various forms of $\mathrm{\ell }$-independence (including the case $\mathrm{\ell }=p$) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\mathrm{\ell }$-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 $\ell$-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 $\ell$-independence results for the cohomology of semistable varieties from the well-known results on $\ell$-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 characteristic $p$ we show a similar weak version of $\mathrm{\ell }$-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.