Geodesic
Free summands of stably free modules
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e85

Voir la notice de l'article provenant de la source Cambridge University Press

Let R be a commutative ring. One may ask when a general R-module P that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb {A}^n)$ denotes the variety $\operatorname {GL}(n) / \operatorname {GL}(n-r)$ over a field k, then the projection $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section if and only if the following holds: any module P over any k-algebra R with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$. Using techniques from $\mathbb {A}^1$-homotopy theory, we characterize those n for which the map $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field.We conclude that if $P \oplus R \cong R^{24m}$ and R contains the field of rational numbers, then P contains a free summand of rank $2$. If R contains a quadratically closed field of characteristic $0$, or the field of real numbers, then P contains a free summand of rank $3$. The analogous results hold for schemes and vector bundles over them. 
Gant, Sebastian; Williams, Ben. Free summands of stably free modules. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e85. doi: 10.1017/fms.2025.39
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[1] Adams, J. F. and Walker, G., ‘On complex Stiefel manifolds’, Math. Proc. Cambridge Philos. Soc. 61(1) (1965), 81. issn: 0305-0041, 1469-8064. doi: 10.1017/S0305004100038688 (cit. on p. 1). Google Scholar | DOI

[2] Asok, A., Bachmann, T. and Hopkins, M. J., ‘On -stabilization in unstable motivic homotopy theory’, Preprint, 2024, doi: 10.48550/arXiv.2306.04631. arXiv: 2306.04631 [math] (cit. on pp. 1, 2). Google Scholar | DOI

[3] Asok, A. and Doran, B., ‘On unipotent quotients and some -contractible smooth schemes’, International Mathematics Research Papers 2007 (2007), 51. issn: 1687-3017. doi: 10.1093/imrp/rpm005 (cit. on p. 1). Google Scholar

[4] Asok, A. and Fasel, J., ‘A cohomological classification of vector bundles on smooth affine threefolds’, Duke Math. J. 163(14) (2014), 2561–2601. issn: 0012-7094. doi: 10.1215/00127094-2819299. arXiv: 1204.0770 (cit. on p. 2). Google Scholar | DOI

[5] Asok, A. and Fasel, J., ‘An explicit -degree map and applications’, J. Topol. 10(1) (2017), 268–300. issn: 1753-8416. doi: 10.1112/topo.12007 (cit. on p. 2). Google Scholar | DOI

[6] Asok, A., Fasel, J. and Williams, B., ‘Motivic spheres and the image of the Suslin–Hurewicz map’, Invent. Math. 219(1) (2020), 39–73. issn: 0020-9910, 1432-1297. doi: 10.1007/s00222-019-00907-z (cit. on p. 2). Google Scholar | DOI

[7] Asok, A., Hoyois, M. and Wendt, M., ‘Affine representability results in -homotopy theory, II: principal bundles and homogeneous spaces’, Geom. Topol. 22(2) (2018), 1181–1225. issn: 1465-3060. doi: 10.2140/gt.2018.22.1181 (cit. on p. 1). Google Scholar | DOI

[8] Asok, A., Wickelgren, K. and Williams, B., ‘The simplicial suspension sequence in -homotopy’, Geom. Topol. 21(4) (2017), 2093–2160. issn: 1465-3060. doi: 10.2140/gt.2017.21.2093 (cit. on p. 2). Google Scholar | DOI

[9] Atiyah, M. F. and Todd, J. A., ‘On complex Stiefel manifolds’, Proc. Cambridge Philos. Soc. 56 (1960), 342–353. issn: 0008-1981. doi: 10.1017/s0305004100034642 (cit. on p. 1). Google Scholar | DOI

[10] Dugger, D. and Isaksen, D. C., ‘Motivic Hopf elements and relations’, New York J. Math. 19 (2013), 823–871. issn: 1076-9803 (cit. on p. 1). Google Scholar

[11] Dugger, D. and Isaksen, D. C., ‘Topological hypercovers and -realizations’, Math. Z. 246(4) (2004), 667–689. issn: 0025–5874. doi: 10.1007/s00209-003-0607-y (cit. on p. 1). Google Scholar | DOI

[12] Eisenbud, D. and Harris, J., The Geometry of Schemes (Graduate Texts in Mathematics) vol. 197 (Springer, New York, 2000). isbn: 978-0-387-98637-1 978-0-387-98638-8 (cit. on p. 1). Google Scholar

[13] Hartshorne, R., Algebraic Geometry (Graduate Texts in Mathematics) vol. 52. (Springer-Verlag, New York, 1977). isbn: 0-387-90244-9 (cit. on p. 1). Google Scholar | DOI

[14] Hovey, M., Model Categories (Mathematical Surveys and Monographs) vol. 63 (American Mathematical Society, Providence, RI, 1999). isbn: 0-8218-1359-5 (cit. on p. 1). Google Scholar

[15] Isaksen, D. C., ‘Etale realization on the -homotopy theory of schemes’, Adv. Math. 184(1) (2004), 37–63. issn: 0001–8708. doi: 10.1016/S0001-8708(03)00094-X (cit. on p. 1). Google Scholar | DOI

[16] James, I. M., ‘Cross-sections of Stiefel manifolds’, Proc. London Math. Soc. 8(4) (1958), 536–547. issn: 0024–6115. doi: 10.1112/plms/s3-8.4.536 (cit. on pp. 1, 2). Google Scholar | DOI

[17] James, I. M., ‘Whitehead products and vector-fields on spheres’, Proc. Cambridge Philos. Soc. 53 (1957), 817–820. issn: 0008-1981. doi: 10.1017/s0305004100032928 (cit. on p. 2). Google Scholar | DOI

[18] Lindel, H., ‘On the Bass-Quillen conjecture concerning projective modules over polynomial rings’, Invent. Math. 65 (1981/0082), 319–324. issn: 0020-9910; 1432-1297/e. doi: 10.1007/BF01389017 (cit. on p. 1). Google Scholar | DOI

[19] Morel, F., -Algebraic Topology over a Field(Lecture Notes in Mathematics) vol. 2052 (Springer, Heidelberg, 2012). isbn: 978-3-642-29513-3 (cit. on pp. 1, 2). Google Scholar

[20] Morel, F. and Voevodsky, V., ‘-homotopy theory of schemes’, Publ. Math. Inst. Hautes Sci. 90(1) (1999), 45–143. issn: 0073–8301. doi: 10.1007/BF02698831 (cit. on p. 1). Google Scholar | DOI

[21] Paechter, G. F., ‘The groups . I’, Quart. J. Math. Oxford Ser. (2) 7 (1956), 249–268. issn: 0033-5606,1464-3847. doi: 10.1093/qmath/7.1.249 (cit. on p. 2). Google Scholar | DOI

[22] Raynaud, M., ‘Modules projectifs universels’, Invent. Math. 6 (1968), 1–26. issn: 0020-9910,1432-1297. doi: 10.1007/BF01389829 (cit. on pp. 1, 2). Google Scholar | DOI

[23] Röndigs, O., Spitzweck, M. and Østvær, P. A., ‘The first stable homotopy groups of motivic spheres’, Ann. of Math. 189(1) (2019), 1–74. issn: 0003-486X. doi: 10.4007/annals.2019.189.1.1 (cit. on pp. 1, 2). Google Scholar | DOI

[24] Röndigs, O., Spitzweck, M. and Østvær, P. A., The second stable homotopy groups of motivic spheres’, Duke Math. J. 173(6) (2024), 1017–1084. issn: 0012-7094. doi: 10.1215/00127094-2023-0023 (cit. on pp. 1, 2). Google Scholar | DOI

[25] Schlichting, M. and Tripathi, G. S., ‘Geometric models for higher Grothendieck–Witt groups in -homotopy theory’, Math. Ann. 362(3–4) (2015), 1143–1167. issn: 0025-5831, 1432-1807. doi: 10.1007/s00208-014-1154-z (cit. on p. 2). Google Scholar | DOI

[26] Toda, H., ‘Quelques tables des groupes d’homotopie des groupes de Lie’, Comptes Rendus Hebdomadaires des S é ances de l’Acad é mie des Sciences 241 (1955), 922–923. issn: 0001-4036 (cit. on p. 1). Google Scholar

[27] Walter, C., Grothendieck–Witt Groups of Triangulated Categories, 2003, https://www.maths.ed.ac.uk/~v1ranick/papers/trigw.pdf (visited on 07/25/2024) (cit. on p. 2). Google Scholar

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