Algebraic K -theory, A ¹ -homotopy and Riemann-Roch theorems

Abstract

In this paper, we show that the combination of the constructions done in SGA 6 and the A 1 -homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.Contents 230 JOËL RIOU Theorem 0.2. Let S be a regular scheme. We let K 0 (−) be the presheaf of sets on Sm/S which maps X to K 0 (X). Then the map induced by Theorem 0.1 is a bijection:where Sm/S opp Sets is the category of presheaves of sets on Sm/S. It follows that the operations defined in [7] at the level of K 0 (for example, λ n , Ψ k ) uniquely lift in H(S). From there, using Theorem 0.1, we can make them act on higher algebraic K-theory. This principle also works for operations involving several operands (for example, products) and in a sense that will be made precise in Section 2, we obtain a machinery that takes as an input the algebraic structures on K 0 and outputs such a structure on Z × Gr inside H(S). Thus, Z × Gr is equipped with a structure of a special λ-ring with duality.Structures of a (special) λ-ring had already been obtained on higher K-theory, with different scales of generality. We may mention constructions of products, λ-operations or Adams' operations by Loday [29], Waldhausen [47], Kratzer [25], Soulé [42], Grayson [18], Lecomte [26] and Levine [27]. We compare the structures on K (X) for X regular obtained by our method to these previous constructions in Section 3. The comparison with Waldhausen's product (see Proposition 3.2.1) may seem surprisingly straightforward, but it is a typical use of Theorem 0.2 and its variants involving several operands (see Theorem 1.1.4).Section 4 relates our results to virtual categories, an insight of Deligne [10]. We show that, after inverting 2, constructions done at the level of K 0 refine to these virtual categories, which embodies both K 0 and K 1 . This theory was used by Dennis Eriksson in his thesis [12] in order to refine Riemann-Roch theorems at the level of these virtual categories.In Section 5, we focus on operations τ :We compute them using the splitting principle. We show that the datum of τ is equivalent to the datum of an element in K 0 (S) [[U ]]. Then we construct, up to a unique isomorphism in the stable homotopy category SH(S), the P 1 -spectrum BGL which represents algebraic K-theory and study its endomorphisms (it is somewhat related but quite different from the methods of [1, Chapter 6; 2, 3]). After tensoring with Q, we show that this spectrum decomposes in SH(S) as the direct sums of 'eigenspaces' for the Adams operations. Alternate interesting descriptions of stable operations on algebraic K-theory (and more general oriented theory) have been obtained by very different methods by Naumann, Østvaer and Spitzweck [32].We prove in Section 6 that these ideas can be used to obtain a homotopical variant of some Riemann-Roch theorems in the case of a smooth and projective morphism f :X → S. Basically, we prove that certain Riemann-Roch formulas are satisfied on zeroth K-groups ...

On the Motivic Spectral Sequence

“…For any X ∈ Sm/k and any p ∈ Z the induced map ζ * : 23 is an isomorphism by construction of f 0 (KGL). On the other hand, Theorem 7.8 implies the induced map…”

Riemann–Roch for homotopy invariant K-theory and Gysin morphisms

“…Following [Rio10,5.3], the Q-localization of the K-theory spectrum admits a decomposition induced by the Adams operations, i.e., KGL Q = i∈Z KGL (n)…”

“…(see [Rio10,5.3.17]). We call this morphism the Chern character since for any regular scheme X it induces the classical higher Chern characters ch r,n :…”

Motivic cohomology spectral sequence and Steenrod operations

“…This is left to the reader. (Here we use the notation of [Rio10].) Since the group K −ε (k) is the algebraic K-group of the base field k and, obviously, vanishes for ε > 0, then so does lim ← − (K −ε (k)) Ω .…”