Motivic strict ring models for K-theory

arXiv:0907.4121v1 [math.AT] 23 Jul 2009 Motivic strict ring models for K-theory Oliver R¨ondigs, Markus Spitzweck, Paul Arne Østvær July 23, 2009 Abstract It is shown that the K-theory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective ‘strict’ is used to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. Both the categories of motivic functors and motivic symmetric spectra furnish convenient frameworks for constructing the ring models. Analogous topological results follow by running the same type of arguments as in the motivic setting. Contents 1 Introduction 2 2 A strict model 4 3 The homotopy type 10 4 Multiplicative structure 14 1 1 Introduction Motivic homotopy theory can be viewed as an expansion of classical homotopy theory to an algebro-geometric setting. This has enabled the introduction of homotopy theoretic techniques in the study of generalized ring (co)homology theories for schemes, and as in classical algebra one studies these via modules and algebras. From this perspective, motives are simply modules over the motivic Eilenberg-MacLane ring spectrum [8], [9]. The main purpose of this paper is to show that the K-theory of every noetherian base scheme of finite Krull dimension acquires strict ring object models in motivic homotopy theory, and thereby pave the way towards a classification of modules over K-theory. An example of a base scheme of particular interest is the integers. Working with some flabby smash product which only become associative, commutative and unital after passage to the motivic stable homotopy category is inadequate for our purposes. The whole paper is therefore couched in terms of motivic functors [1] and motivic symmetric spectra [3]. Throughout the paper the term ‘K-theory’ is short for homotopy algebraic K-theory. Fix a noetherian base scheme S of finite Krull dimension with multiplicative group scheme Gm. Denote by KGL the ordinary motivic spectrum representing K-theory [13]. As shown in [11], see also [2] and [6], inverting a homotopy class β ∈π2,1Σ∞BGm+ in the motivic suspension spectrum of the classifying space of the multiplicative group scheme yields a natural isomorphism in the motivic stable homotopy category Σ∞BGm+[β−1] ∼ = / KGL. We shall turn the Bott inverted model for K-theory into a commutative monoid KGLβ in the category of motivic symmetric spectra. To start with, the multiplicative structure of Gm induces a commutative monoid structure on the motivic symmetric suspension spectrum of the classifying space BGm+. A far more involved analysis dealing with an actual map rather than some homotopy class allows us to define KGLβ and eventually verify that it is a commutative monoid with the same homotopy type as K-theory. Several of the main techniques employed in the proof are of interest in their own right, and can be traced back to constructions for symmetric spectra, cf. [10]. It also turns out that there exists a strict ring model for K-theory in the category of motivic functors. Here the motivic functor model is constructed in a leisurely way by transporting KGLβ via the strict symmetric monoidal functor relating motivic symmetric spectra to motivic functors. 2 When suitably adopted the motivic argument works also in topological categories. The topological strict ring models appear to be new, even in the case of symmetric spectra. The Bott element considered by Voevodsky in [13] is obtained from the virtual vector bundle [OP1] −[OP1(−1)]. A key step in the construction of KGLβ is to interpret the same element, viewed in the pointed motivic unstable homotopy category, as an actual map between motivic spaces. In order to make this part precise we shall use a lax symmetric monoidal fibrant replacement functor for pointed motivic spaces. Fibrancy is a constant source for extra fun in abstract homotopy theory. The problem resolved in this paper is no exception in that respect. It is also worthwhile to emphasize the intriguing fact that β does not play a role in the definition of the multiplicative structure of KGLβ. However, the Bott element enters in the definition of the unit map 1 →KGLβ, which is part of the monoid structure, and in the structure maps. In fact, up to some fibrant replacement, KGLβ is constructed fairly directly from Σ∞BGm+ by intertwining a map representing β with the structure maps. On the level of homotopy groups this type of intertwining has the effect of inverting the Bott element. As a result, we obtain the desired homotopy type. In [7] it is shown that under a certain normalization assumption the ring structure on KGL in the motivic stable homotopy category is unique over the ring of integers Z. For any base scheme S the multiplicative structure pulls back to give a distinguished monoidal structure on KGL. We show the multiplicative structures on KGLβ and KGL coincide in the motivic

…(Full text truncated)…

This content is AI-processed based on ArXiv data.