Motivic twistedK–theory

Abstract

This paper sets out basic properties of motivic twisted K -theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K -theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG m -bundle for the classifying space of the multiplicative group scheme G m . We show a Künneth isomorphism for homological motivic twisted K -groups computing the latter as a tensor product of K -groups over the K -theory of BG m . The proof employs an Adams Hopf algebroid and a trigraded Tor-spectral sequence for motivic twisted K -theory. By adapting the notion of an E 1 -ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted K -groups. It generalizes various spectral sequences computing the algebraic K -groups of schemes over fields. Moreover, we construct a Chern character between motivic twisted K -theory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems. 14F42, 55P43, 19L50; 14F99, 19D99 MotivationTopological K -theory has many variants which have all been developed and exploited for geometric purposes. Twisted K -theory or "K -theory with coefficients" was introduced by Donovan and Karoubi in [11] using Wall's graded Brauer group. More general twistings of K -theory arise from automorphisms of its classifying space of Fredholm operators on an infinite dimensional separable complex Hilbert space. Of particular geometric interest are twistings given by integral 3-dimensional cohomology classes. The subject was further developed in the direction of analysis by Rosenberg in [41].Twisted K -theory resurfaced in the late 1990's with Witten's work on classification of D-brane charges in type II string theory [54]. Fruitful interactions between algebraic topology and physics afforded by twisted K -theory continues today; see eg Oberwolfach Rep. 3, no. 4 [9], Atiyah and Segal [4;5], Bouwknegt et al [8] and Tu, Xu

On equivariant and motivic slices

“…There is an alternative to Voevodsky's slice filtration known as the very effective slice filtration [37,2]. In Section 4 we introduce the cellular version of this slice filtration, and give conditions on when this agrees with the effective slice filtration.…”

“…Of fundamental importance for us is the following result [37, Theorem 5.7]. Proof (1) [37]. We will study this category, denoted SH(S) veff cell in more detail in the next section.…”

“…To deal with the localizations we appeal to work of Levine It is remarked in [37], and then proved in detail in [2] that the very effective slice filtration is the positive part of a t-structure on the very effective motivic stable homotopy category. In the case that S is the spectrum of a perfect field F , Bachmann used the homotopy t-structure to give a description of the very effective motivic category in terms of homotopy sheaves.…”

On very effective hermitian K-theory

“…[7, §8.3]. Recall that MGL is a very effective motivic spectrum[24, Theorem 5.7]. The lemma follows from the canonically induced motivic weak equivalences in the commutative diagram…”

Existence and uniqueness of $$E_{\infty }$$ E ∞ structures on motivic $$K$$ K -theory spectra

“…The uniqueness part of the theorem rules out the existence of exotic E ∞ structures on KGL. We note that related motivic E ∞ structures are used in the recent construction of the Atiyah-Hirzebruch spectral sequence for motivic twisted K-theory [20].…”