Rigidity of effective motives in equal characteristic case [rigidity_in_p]

Cheni Yuki Yang
Ongoing

We use a motivic tame twist inspired by [Lüd24] to construct a spectrum representing $\mathbb{G}_{m}$ in tame motives. As a consequence, we prove a rigidity result for tame motivic homotopy theory after passing to the logarithmic motivic homotopy category. This allows us in particular to construct a synthetic theory of spectra in $p$-adic setting.

References

Reference. $p$-adic tame Tate twists [lueders2024tame]

2024
Morten Lüders

Recently, Hübner-Schmidt defined the tame site of a scheme. We define $p$-adic tame Tate twists in the tame topology and prove some first properties. We establish a framework analogous to the Beilinson-Lichtenbaum conjectures in the tame topology for $p$-adic tame Tate twists and tame logarithmic deRham-Witt sheaves. Both only differ from their étale counterpart in cohomological degrees above the weight. These cohomology groups can be analysed using the Gersten conjecture which, at least conjecturally, has a nice shape in the tame topology. We prove the Gersten conjecture for tame logarithmic deRham-Witt sheaves for curves in positive characteristic and note that the conjecture in arbitrary dimension would follow from strict $\mathbb{A}^1$-invariance.

@preprint{luderspadicTameTate2024,
  title = {$p$-adic tame Tate twists},
  author = {Lüders, Morten},
  year = 2024,
  number = {arXiv:2407.07979},
  eprint = {2407.07979},
  primaryclass = {math.AG},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2407.07979},
}