• Corpus ID: 245837871

The very effective covers of KO and KGL over Dedekind schemes

@inproceedings{Bachmann2022TheVE,
  title={The very effective covers of KO and KGL over Dedekind schemes},
  author={Tom Bachmann},
  year={2022}
}

• Tom Bachmann
• Published 8 January 2022
• Mathematics

We answer a question of Hoyois–Jelisiejew–Nardin–Yakerson regarding framed models of motivic connective K-theory spectra over Dedekind schemes. 1. Statement of results Let S be a scheme. The category PΣ(Cor (S)) of presheaves with framed transfers [5, §2.3] is a motivic analog of the classical category of E∞-monoids. We have the framed suspension spectrum functor Σ∞fr : PΣ(Cor (S))→ SH(S) which was constructed in [6, Theorem 18]. By analogy with the classical situation, one might expect that many interesting motivic spectra can be obtained as framed suspension spectra. This is indeed the case; see [8, §1.1] for a summary. This note concerns the following examples of the above idea. One has framed presheaves [8, §6] Vect,Bil ∈ PΣ(Cor (S)) where Vect(X) is the groupoid of vector bundles on X and Bil(X) is the groupoid of vector bundles with a non-degenerate symmetric bilinear form. There exist Bott elements β ∈ π2,1Σ ∞ fr Vect and β̃ ∈ π8,4Σ ∞ fr Bil and canonical equivalences [7, Proposition 5.1] [8, Proposition 6.7] (Σ∞fr Vect)[β ] ≃ KGL and (Σ∞fr Bil)[β̃ ] ≃ KO. Here KGL is the motivic spectrum representing homotopy algebraic K-theory and KO is the motivic spectrum representing homotopy hermitian K-theory. Again by comparison with the classical situation, this suggests that Σ∞fr Vect and Σ ∞ fr Bil should be motivic analogs of connective K-theory spectra. Another way of producing “connective” versions is by passing to (very) effective covers [12, 11]. It was proved in [8, 7] that these two notions of connective motivic K-theory spectra coincide, provided that S is regular over a field. Our main result is to extend this comparison to more general base schemes. Theorem 1.1. Let S be a scheme. (1) Suppose that f1(HZ) = 0 ∈ SH(S). The canonical map Σ∞fr Vect→ f0KGL ∈ SH(S) is an equivalence. (2) Suppose in addition that 1/2 ∈ S and HW≥2 = 0 ∈ SH(S). The canonical map Σ∞fr Bil→ f̃0KO ∈ SH(S) is an equivalence. These assumptions are satisfied if S is essentially smooth over a Dedekind scheme (containing 1/2 in case (2)). Remark 1.2. That the assumptions are satisfied for Dedekind schemes is proved in [4, Proposition B.4] for (1) and in [3, Lemma 3.8] for (2). They in fact hold for all schemes; this will be recorded elsewhere. Example 1.3. Bott periodicity implies formally that fnKGL ≃ Σ kgl and sn(KGL) ≃ Σ kgl/β. Theorem 1.1(1) implies that kgl/β ≃ HZ (see Lemma 2.1). Hence in this situation the slice filtration for KGL yields a convergent spectral sequence, with E2-page given by (Spitzweck’s) motivic cohomology. Date: January 11, 2022. As a notational convention for this introduction, whenever we mention KO we shall assume that 1/2 ∈ S.