Higher-dimensional arithmetic using <i>p</i>-adic étale Tate twists

Abstract

This paper is a survey on recent researches of the author and his recent joint work with Shuji Saito. We will explain how to construct p-adicétale Tate twists on regular arithmetic schemes with semistable reduction, and state some fundamental properties of those objects. We will also explain how to define cycle class maps from Chow groups toétale cohomology groups with coefficients in p-adicétale Tate twists and state injectivity and surjectivity results on those new cycle class maps. Dedicated to V. Snaith on his 60th birthday

Syntomic complexes and p-adic étale Tate twists

“…On schemes which are smooth or regular with semistable reduction over a discrete valuation ring (DVR), Theorem 1.8 identifies Z/𝑝 𝑛 (𝑖) 𝑋 with the 'p-adic étale Tate twists' considered in [Sat07], and earlier in the smooth case in [Gei04,Sch94]; cf. [Sat05] for a survey. Many special cases of Theorem 1.8 have previously appeared in the literature.…”

Syntomic complexes and $p$-adic étale Tate twists

“…On schemes which are smooth or regular with semistable reduction over a DVR, Theorem 1.8 identifies the objects Z/p n (i) X considered here with the "p-adic étale Tate twists" considered in [Sat07], and earlier in the smooth case in [Gei04,Sch94]; cf. [Sat05] for a survey.…”

A local to global principle for higher zero-cycles

“…By [35] the middle vertical map is an isomorphism and the left vertical map is surjective. This implies that the right vertical map is an isomorphism.…”