A commutative P^1-spectrum representing motivic cohomology over Dedekind domains

Algebraic cobordism of number fields

“…An analogy to Lichtenbaum's conjecture for hermitian K-theory was proven in [KRØ18, Theorem 1.10]. We use the motivic cohomology spectrum of Spitzweck over the ring of S-integers in a number field F [Spi12].…”

“…The results over rings of S-integers relies on the work of Levine [Lev99] and Spitzweck [Spi12] on motivic cohomology of Dedekind domains. Our results over rings of integers are somewhat similar to those of Rognes and Weibel [RW00] on algebraic K-theory of rings of 2-integers.…”

Nilpotence in normed MGL-modules

“…Many of the most important E ∞ -rings in the classical stable homotopy category, such as MU, KU, and HZ, have normed analogs, such as MGL, KGL, and HZ ∈ NAlg(SH(S)). Here, we will be particularly interested in HZ, which denotes Spitzweck's motivic cohomology spectrum [Spi12], and MGL, which denotes the algebraic cobordism spectrum (see, e.g., [BH17,§16]).…”

Hermitian $K$-theory, Dedekind $ζ$-functions, and quadratic forms over rings of integers in number fields

“…In the first main result of this paper we identify the mod 2 hermitian K-groups KQ n (O F,S ; Z/2) up to extensions of motivic cohomology groups of Dedekind domains as defined in [16], [30], [59]. Our method of proof reveals for n ≥ 1 the existence of an 8-fold periodicity isomorphism (1.1) KQ n (O F,S ; Z/2) ∼ = KQ n+8 (O F,S ; Z/2).…”

“…These facts make the slice spectral sequence amenable to calculations over base schemes affording an explicit description of the action of the motivic Steenrod algebra on its motivic cohomology ring. More generally, using Spitzweck's work of motivic cohomology in [59], after localization the isomorphisms in (1.13), (1.14), and (1.15) hold over Dedekind domains of mixed characteristic with no residue fields of characteristic 2, see [53, §2.3]. We investigate the convergence properties of (1.12) for KGL, KQ, and KW.…”