The p-completed cyclotomic trace in degree 2

Absolute prismatic cohomology

“…In this case, the Breuil-Kisin twist A{1} has a canonical generator e A on which the Frobenius acts by the formula ϕ(e A ) = e A /[p] q (Notation 2.6.3). Moreover, if u ∈ A is a rank one element satisfying u p ≡ 1 (mod I), then the prismatic logarithm log ∆ satisfies the identity log ∆ (u p ) = log q (u)e A where log q (u) is the q-logarithm studied in [3] (Corollary 2.6.11). We will be particularly interested in the case where the prism (A, I) is crystalline, in which case we have log ∆ (u p ) = log(u)e A , where log(u) = n>0 (−1) n−1 (u−1) n n is the usual logarithmic series (Corollary 2.6.12).…”

“…Transversal Prisms. In this section, we review the definition of a transversal prism, introduced by Anschütz and Le Bras in [3]. For purposes of giving examples, it will be convenient to introduce a slightly more general notion.…”

“…Our goal in this section is to construct the Breuil-Kisin twist A{1} in the case of a transversal prism (A, I). We begin by reviewing some elementary properties of transversal prisms (see [3] for a closely related discussion). Lemma 2.2.1 ([3], Lemma 3.3).…”

“…It follows that Notation 2.6.3 supplies a canonical generator e A ∈ A{1} which satisfies the identity ϕ A{1} (e A ) = e A p . We now compare the prismatic logarithm of Definition 2.5.11 with the q-logarithm of [3]. Proposition 2.6.5.…”

“…]], the elements γ n,q (u − 1) are automatically unique if they exist. For existence, we refer to Proposition 4.9 of [3]. Note that reduction modulo q − 1 carries each γ n,q (u − 1) to the divided power (u−1) n n!…”