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\(\mathbb{A}^1\)-homotopy theory of schemes. (English) Zbl 0983.14007
In this paper the authors develop \(\mathbb{A}^1\)-homotopy theory of schemes – a homotopy theory of algebraic varieties where the affine line plays the role of the unit interval.
In the three chapters and nine paragraphs the authors present:
A homotopy category of a site with interval;
the \(\mathbb{A}^1\)-homotopy category of schemes over a base,
classifying spaces of algebraic groups.
First, they give a number of general results about simplicial sheaves on sites which are latter applied to the study of the homotopy category of schemes. Then, the authors study the basic properties of the \(\mathbb{A}^1\)-homotopy category \({\mathcal H}(S)\) of smooth schemes over a base scheme \(S\) with interval \(((Sm/S)_{Nis},\mathbb{A}^1)\) where \(Sm/S\) is the category of smooth schemes (of finite type) over \(S\) and Nis refers to the Nisnevich topology. They discuss the properties of the homotopy category of simplicial sheaves on \((Sm/S)_{Nis}\), then they prove three theorems with a major role in further applications of their constructions. Finally the authors consider some examples of topological realization functors.
The last chapter is dedicated to applications of the general technique developed above. The main results are: A geometrical construction of a space which represents in \({\mathcal H}(S)\) the functor \(H^1_{et}(-,G)\) for étale group schemes \(G\) of order prime to \(\text{char} (S)\), the second result shows that algebraic \(K\)-theory of a regular scheme \(S\) can be described in terms of morphisms in \({\mathcal H}(S)\) with values in the infinite Grassmannian and the third result shows how one can use \(\mathbb{A}^1\)-homotopy theory together with basic functoriality for simplicial sheaves on smooth sites to give a definition of Quillen-Thomason \(K\)-theory for all Noetherian schemes.

MSC:
14F35 Homotopy theory and fundamental groups in algebraic geometry
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
18G55 Nonabelian homotopical algebra (MSC2010)
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