The generalized slices of Hermitian K ‐theory

Abstract

We compute the generalized slices (as defined by Spitzweck-Østvaer) of the motivic spectrum KO (representing Hermitian K-theory) in terms of motivic cohomology and (a version of) generalized motivic cohomology, obtaining good agreement with the situation in classical topology and the results predicted by Markett-Schlichting. As an application, we compute the homotopy sheaves of (this version of) generalized motivic cohomology, which establishes a version of a conjecture of Morel. 1125which is (at best) considered an analog of S 1 , and yet the layers are already given at double speed, something seems amiss. (2) In classical topology there is another version of K-theory, namely the K-theory of real (not complex) vector bundles, denoted KO. There is also Bott periodicity, this time resulting in the computation that the homotopy groups of KO are given by Z, Z/2, Z/2, 0, Z, 0, 0, 0 and then repeating periodically. There is an analog of topological KO in algebraic geometry, namely Hermitian K-theory [6] and (also) denoted KO ∈ SH(k). It satisfies an appropriate form of Bott periodicity, but this is not captured accurately by its slices, which are also very different from the topological analog [21]. (3) The slice filtration does not always converge. Thus just considering slices is not enough, for example, to determine if a morphism of spectra is an isomorphism.Problem (3) has lead Spitzweck-Østvaer [25] to define a refined version of the effectivity condition yielding the slice filtration which they call being 'very effective'. In this article we will argue that their filtration also solves issues (1) and (2).To explain the ideas, recall that the category SH(k) eff is the localizing (so triangulated!) subcategory generated by objects of the form Σ ∞ X + for X ∈ Sm(k) (that is, no desuspension by G m ). Then one defines SH(k) eff (n) = SH(k) ∧ T ∧n and for E ∈ SH(k) the n-effective cover f n E ∈ SH(k) eff (n) is the universal object mapping to E. (Note that since SH(k) eff is triangulated, we have SH(k) eff ∧ T ∧n = SH(k) eff ∧ G ∧n m .) In contrast, Spitzweck-Østvaer define the subcategory of very effective spectra SH(k) veff to be the subcategory generated under homotopy colimits and extensions by Σ ∞ X + ∧ S n where X ∈ Sm(k) and n 0. This subcategory is not triangulated! As before we put SH(k) veff (n) = SH(k) veff ∧ T ∧n . (Note that now, crucially, SH(k) veff ∧ T ∧n = SH(k) veff ∧ G ∧n m .) Then as before the very n-effective cover f n E ∈ SH(k) veff (n) is the universal object mapping to E. The cofibresf n+1 E →f n E →s n E are called the generalized slices of E.As pointed out by Spitzweck-Østvaer, the connectivity off n E in the homotopy t-structure increases with n, so the generalized slice filtration automatically converges. Moreover, it is easy to see thatf n KGL = f n KGL (that is, the n-effective cover of KGL is 'accidentally' already very n-effective) and thuss n KGL = s n (KGL). This explains how the generalized slice filtration solves problem (1): we see that the 'G m -slices' (that is, ordinary slices)...

On equivariant and motivic slices

“…Since the odd slices of Re(E) are contractible, it is even by definition, and by Proposition 5.4 we must have π 2q−1,q Re(E) = 0. This proves (2).…”

“…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.…”

“…For the converse, recall again that Σ s,t 1 is connective for s ≥ t. Suppose now we are given a cellular spectrum E with π a,b E = 0 for a − b < 0. Then, by the method of killing cells, as described in the proof of Proposition 4(2) of [2] (see also the 'Details on killings cells' after the proposition), one can construct a cellularly connective spectrum Z along with a map Z → E inducing an isomorphism on bigraded homotopy groups. Since both Z and and E are cellular, this map is an equivalence [ ] -the proof is in the same spirit as the one given above.…”

Remarks on motivic Moore spectra

“…The slice computation is slightly more complicated for the motivic spectrum KQ representing hermitian K-theory. For comparison purposes, the case of kq, the very effective cover of KQ [1], [2], is more convenient. Set kq nh := kq ∧ C nh , and similarly cKW nh := cKW ∧ C nh = cKW ∨ Σ 1,0 cKW.…”

“…Theorem 2.1 implies that for every motivic spectrum E and for every integer s, π s+(⋆) E is a graded K MW -module. As a first instance besides the motivic sphere spectrum 1, consider the very effective cover kq → KQ of the motivic spectrum representing hermitian K-theory [1], [2]. Using kq instead of the effective cover f 0 KQ → KQ leads to a slight improvement on the computation [21, Theorem 5.5].…”