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Generically split octonion algebras and \(\mathbb{A}^1\)-homotopy theory. (English) Zbl 1430.14051
The paper studies classification of the ‘generically split’ octonion algebras over smooth affine \(k\)-varieties with the methods of \(\mathbb{A}^1\)-homotopy theory. The results thus obtained give full classification when the dimension of the variety is not greater than 2, and partial answers in dimensions up to 5, where new explicit invariants are constructed related to the arithmetic of the base field and Chow groups of the variety.
Octonion algebras over a scheme \(X\) can be considered as the étale-local forms of the split octonion algebra, the latter on an affine scheme \(\mathrm{Spec\ }R\) is a specific associative algebra structure on the free rank 8 module \(\mathrm{Mat}_2(R)\oplus \mathrm{Mat}_2(R)\). Equivalently, octonion algebras are étale torsors under the group of automorphisms of the split octonion algebra, which is \(\mathrm{G}_2\). ‘Generically split’ octonion algebras are those which become split after restricting to generic points of the scheme, and are classified by torsors in the Nisnevich topology under \(\mathrm{G}_2\). The results of the previous papers of the authors [Duke Math. J. 166, No. 10, 1923–1953 (2017; Zbl 1401.14118); Geom. Topol. 22, No. 2, 1181–1225 (2018; Zbl 1400.14061)] show that for an affine scheme \(X\) (at least, but not necessarily) over a field the pointed set \(\mathrm{H}^1_{\mathrm{Nis}}(X; \mathrm{G}_2)\) is isomorphic to the set of maps \([X, \mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2]\) in the Morel-Voevodsky unstable motivic category \(\mathcal{H}(k)\) from \(X\) to the Nisnevich-local replacement \(\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2\) of the simplicial bar construction on \(\mathrm{G}_2\), which is called the classifying space. Thus, the study of octonion algebras over \(X\) can be done with the use of the homotopical study of \(\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2\), especially the \(\mathbb{A}^1\)-Postnikov tower and corresponding obstruction theory.
Let us give a general idea behind the study of the space \(\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2\). Under certain conditions on the closed embedding of algebraic groups \(H\subset G\) [Section 4.2, Zbl 1400.14061] there is a fiber sequence in \(\mathcal{H}_{\mathbb{A}^1}(k)\) \[G/H\rightarrow \mathrm{B}_{\mathrm{Nis}} H\rightarrow \mathrm{B}_{\mathrm{Nis}} G.\] In the case of \(G_2\), there are two useful closed embeddings: \(\mathrm{SL}_3~\hookrightarrow~\mathrm{G}_2\) corresponding to the Zorn’s construction of an octonion algebra from an oriented rank 3 vector bundle and \(G_2 \hookrightarrow Spin_7\) which is a refinement of the morphism \(\mathrm{G}_2 \hookrightarrow \mathrm{O}_8\) defined via the norm of an octonion algebra. These embeddings satisfy necessary conditions for the fiber sequence above and the quotients have a very easy geometric description \(\mathrm{G}_2/\mathrm{SL}_3 \cong Q_6\), \(\mathrm{Spin}_7/\mathrm{G}_2\cong Q_7\) where \(Q_n\) is a split affine quadric over the base field \(k\). Moreover, these affine quadrics admit a very nice homotopical description, \(Q_6\) is \(\mathbb{A}^1\)-weakly equivalent to \(S^3\wedge\mathbb{G}_m^{\wedge 3}\) and \(Q_7\) to \(S^3\wedge\mathbb{G}_m^{\wedge 4}\).
The \(\mathbb{A}^1\)-Postnikov tower of \(\mathrm{B}_{\mathrm{Nis}} G\) is a sequence of spaces which allows to glue the the classifying space from Eilenberg-Maclane spaces of strictly \(\mathbb{A}^1\)-invariant sheaves of the \(\mathbb{A}^1\)-homotopy groups of \(\mathrm{B}_{\mathrm{Nis}} G\). It is proved that in the case of \(G=\mathrm{G}_2\), the sheaf \(\pi_{1}^{\mathbb{A}^1}(\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2)\) is trivial (as well as for any sheaf of groups by the \(\mathbb{A}^1\)-connectivity theorem of Morel-Voevodsky), \(\tau_{2} (\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2)\cong \mathrm{K}(\mathbb{K}_2^M, 2)\) (Proposition 4.1.4) and, thus, the next one fits into the fiber sequence \[ \tau_{\le 3} (\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2) \rightarrow \tau_{2} (\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2) \rightarrow \mathrm{K}(\pi_3^{\mathbb{A}^1}(\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2), 4).\]
This allows to provide a classification of the octonion algebras over a smooth affine variety \(X\) of dimension \(\le 3\) in terms of the second Chern class of the associated vector bundle and the cohomology of \(X\) with coefficients in Milnor K-theory sheaves (Theorem 4.1.6). In particular, if the base field is algebraically closed, the octonion algebra over \(X\) is uniquely determined by its universal second Chern class in \(\mathrm{CH}^2\) (which is constructed via the second layer of the Postnikov tower above).
A similar approach also allows to answer when the octonion algebra is produced from an oriented rank 3 vector bundle (via the Zorn construction), when the octonion algebra has a trivial associated spinor bundle and when it is determined by its norm form. For example, the first of these questions can be reformulated homotopically as the existence of the lift of the map from \(X\) to \(\mathrm{B}_{\mathrm{Nis}} \mathrm{G}_2\) to \(\mathrm{B}_{\mathrm{Nis}} \mathrm{SL}_3\). The “primary” obstruction to this problem lies in \(\mathrm{H}^4(X,\boldsymbol{\pi}_3^{\mathbb{A}^1}(Q_6))\) which is isomorphic to \(\mathrm{H}^4(X, \mathbf{K}_3^{MW})\) and the next one lies in the group \(\mathrm{H}^5(X,\boldsymbol{\pi}_4^{\mathbb{A}^1}(Q_6))\) which may vanish under some conditions on \(X\), e.g. \(\dim X\le 5\) and \(k=\bar{k}\) (under some conditions on the base field, the sheaf \(\boldsymbol{\pi}_4^{\mathbb{A}^1}(Q_6)\cong\boldsymbol{\pi}_4^{\mathbb{A}^1}(S^3\wedge\mathbb{G}_m^{\wedge 3})\) was calculated in [A. Asok et al., Geom. Topol. 21, No. 4, 2093–2160 (2017; Zbl 1365.14027), Theorem 5.2.5].
All the general results in the paper are provided with interesting geometric examples showing whether the obtained restrictions to the classification are ‘full’ or ‘sharp’. And even though the homotopical methods of the paper are quite different from the classical topology, many of the fiber sequences have a ‘classical’ analogue which is graciously explained by the authors (see e.g. the beginning of Section 3 or Remark 4.2.5). Moreover, some of the classical results about the homotopy groups of the complex compact Lie groups can be re-obtained from the algebraico-geometric results of the paper (Theorem 3.4.8).
14F42 Motivic cohomology; motivic homotopy theory
14L30 Group actions on varieties or schemes (quotients)
20G41 Exceptional groups
57T20 Homotopy groups of topological groups and homogeneous spaces
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