\(\mathbb A^1\)-algebraic topology over a field.

*(English)*Zbl 1263.14003
Lecture Notes in Mathematics 2052. Berlin: Springer (ISBN 978-3-642-29513-3/pbk; 978-3-642-29514-0/ebook). x, 259 p. (2012).

This book deals with \(\mathbb{A}^1\)-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field, in which the affine line is imposed to be contractible. It is a natural sequel to to the paper by F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)].

A main technical result contained in this text is the precise nature of the \(\mathbb{A}^1\)-homotopy sheaves of \(\mathbb{A}^1\)-connected pointed spaces. Using these results the author introduces the \(\mathbb{A}^1\)-homology sheaves of a space and then proves several results, which are the analogues of classical theorems in algebraic topology such as the Brouwer degree theorem and Hurewicz theorem.

Let \(Sm_k\) be the category of smooth schemes over a field \(k\). A sheaf of groups \({\mathcal G}\) on \(Sm_k\) in the Nisnevich topology is said to be strongly \(\mathbb{A}^1\)-invariant if for any \(X\in Sm_k\) the map \[ H^i_{Nis}(X,{\mathcal G})\to H^i_{Nis}(X\times \mathbb{A}^1,{\mathcal G}) \] induced by the projection \(\mathbb{A}^1\times X\to X\) is a bijection , for \(i= 0,1\).

\({\mathcal G}\) is said to be strictly \(\mathbb{A}^1\)-invariant if the above map is a bijection for all \(i\in\mathbb{N}\). A basic example of strictly \(\mathbb{A}^1\)-invariant sheaves are Voevodsky’s \(\mathbb{A}^1\)-invariant sheaves with transfers.

The notion of strong \(\mathbb{A}^1\)-invariance given here is new: there are important examples of non-commutative strongly \(\mathbb{A}^1\)-invariant sheaves groups , e.g., \(\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\). The computation given in 7.3 of the book \(\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\) shows that there is a central extension of sheaves of groups \[ 0\to \mathbb{K}^{MW}_2\to \pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\to \mathbb{G}_m\to 0, \] where, for a field \(F\), \(K^{MW}_*(F)\) is the Milnor-Witt \(K\)-theory of \(F\) and \(\mathbb{K}^{MW}_n\) is the sheaf of Milnor-Witt \(K\)-theory in weight \(n\), so that, if \(X\) is irriducible with function field \(F\), one has \(\mathbb{K}^{MW}_n(X)\subset K^{MW}_n(F)\).

The \(\mathbb{A}^1\)-fundamental group sheaves play a fundamental role in the understanding of \(\mathbb{A}^1\)-connected projective smooth varieties over \(k\). In this context a version of Hurewicz theorem is the following

Theorem 0.1. Let \({\mathcal X}\) be a pointed \(\mathbb{A}^1\)-connected space. Then the Hurewicz morphism \[ \pi^{\mathbb{A}^1}_1({\mathcal X})\to H^{\mathbb{A}^1}_1({\mathcal X}) \] is the initial morphism from the sheaf of groups \(\pi^{\mathbb{A}^1}_1({\mathcal X})\) to a strictly \(\mathbb{A}^1\)-invariant sheaf of Abelian groups. This means that, given a strictly \(\mathbb{A}^1\)-invariant sheaf of Abelian groups \(M\) and a morphism of sheaves of groups \(\pi^{\mathbb{A}^1}_1({\mathcal X})\to M\), it factors uniquely trough \(\pi^{\mathbb{A}^1}_1({\mathcal X})\to H^{\mathbb{A}^1}_1({\mathcal X})\).

A main technical result contained in this text is the precise nature of the \(\mathbb{A}^1\)-homotopy sheaves of \(\mathbb{A}^1\)-connected pointed spaces. Using these results the author introduces the \(\mathbb{A}^1\)-homology sheaves of a space and then proves several results, which are the analogues of classical theorems in algebraic topology such as the Brouwer degree theorem and Hurewicz theorem.

Let \(Sm_k\) be the category of smooth schemes over a field \(k\). A sheaf of groups \({\mathcal G}\) on \(Sm_k\) in the Nisnevich topology is said to be strongly \(\mathbb{A}^1\)-invariant if for any \(X\in Sm_k\) the map \[ H^i_{Nis}(X,{\mathcal G})\to H^i_{Nis}(X\times \mathbb{A}^1,{\mathcal G}) \] induced by the projection \(\mathbb{A}^1\times X\to X\) is a bijection , for \(i= 0,1\).

\({\mathcal G}\) is said to be strictly \(\mathbb{A}^1\)-invariant if the above map is a bijection for all \(i\in\mathbb{N}\). A basic example of strictly \(\mathbb{A}^1\)-invariant sheaves are Voevodsky’s \(\mathbb{A}^1\)-invariant sheaves with transfers.

The notion of strong \(\mathbb{A}^1\)-invariance given here is new: there are important examples of non-commutative strongly \(\mathbb{A}^1\)-invariant sheaves groups , e.g., \(\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\). The computation given in 7.3 of the book \(\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\) shows that there is a central extension of sheaves of groups \[ 0\to \mathbb{K}^{MW}_2\to \pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\to \mathbb{G}_m\to 0, \] where, for a field \(F\), \(K^{MW}_*(F)\) is the Milnor-Witt \(K\)-theory of \(F\) and \(\mathbb{K}^{MW}_n\) is the sheaf of Milnor-Witt \(K\)-theory in weight \(n\), so that, if \(X\) is irriducible with function field \(F\), one has \(\mathbb{K}^{MW}_n(X)\subset K^{MW}_n(F)\).

The \(\mathbb{A}^1\)-fundamental group sheaves play a fundamental role in the understanding of \(\mathbb{A}^1\)-connected projective smooth varieties over \(k\). In this context a version of Hurewicz theorem is the following

Theorem 0.1. Let \({\mathcal X}\) be a pointed \(\mathbb{A}^1\)-connected space. Then the Hurewicz morphism \[ \pi^{\mathbb{A}^1}_1({\mathcal X})\to H^{\mathbb{A}^1}_1({\mathcal X}) \] is the initial morphism from the sheaf of groups \(\pi^{\mathbb{A}^1}_1({\mathcal X})\) to a strictly \(\mathbb{A}^1\)-invariant sheaf of Abelian groups. This means that, given a strictly \(\mathbb{A}^1\)-invariant sheaf of Abelian groups \(M\) and a morphism of sheaves of groups \(\pi^{\mathbb{A}^1}_1({\mathcal X})\to M\), it factors uniquely trough \(\pi^{\mathbb{A}^1}_1({\mathcal X})\to H^{\mathbb{A}^1}_1({\mathcal X})\).

Reviewer: Claudio Pedrini (Genova)