The classifying topos of a group scheme and invariants of symmetric bundles
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | CASSOU-NOGUÈS, Philippe | |
| hal.structure.identifier | Department of Mathematics [Philadelphia] | |
| dc.contributor.author | CHINBURG, Ted | |
| hal.structure.identifier | Université Toulouse III - Paul Sabatier [UT3] | |
| dc.contributor.author | MORIN, Baptiste | |
| hal.structure.identifier | Mathematical Institute [Oxford] [MI] | |
| dc.contributor.author | TAYLOR, Martin | |
| dc.date.accessioned | 2024-04-04T03:21:20Z | |
| dc.date.available | 2024-04-04T03:21:20Z | |
| dc.date.created | 2013-01-21 | |
| dc.date.issued | 2015-01-05 | |
| dc.identifier.issn | 0024-6115 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194650 | |
| dc.description.abstractEn | Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the étale cohomology ring $A_Y:=H^*(Y_{et},{\bf Z}/2{\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants and $B_{{\bf O}(q)}$ is the classifying topos of ${\bf O}(q)$ as defined by Grothendieck and Giraud. The cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})$ contains canonical classes $\mathrm{det}[q]$ and $[C_q]$ of degree 1 and 2 respectively, which are obtained from the determinant map and the Clifford group of $q$. The classical Hasse-Witt invariants $w_i(q)$ live in the ring $A_Y$. Our main theorem provides a computation of ${det}[q]$ and $[C_{q}]$ as polynomials in $HW_{1}(q)$ and $HW_{2}(q)$ with coefficients in $A_Y$ written in terms of $w_1(q),w_2(q)\in A_Y$. This result is the source of numerous standard comparison formulas for classical Hasses-Witt invariants of quadratic forms. Our proof is based on computations with (abelian and non-abelian) Cech cocycles in the topos $B_{{\bf O}(q)}$. This requires a general study of the cohomology of the classifying topos of a group scheme, which we carry out in the first part of this paper. | |
| dc.language.iso | en | |
| dc.publisher | London Mathematical Society | |
| dc.title.en | The classifying topos of a group scheme and invariants of symmetric bundles | |
| dc.type | Article de revue | |
| dc.subject.hal | Mathématiques [math]/Théorie des nombres [math.NT] | |
| dc.subject.hal | Mathématiques [math]/Géométrie algébrique [math.AG] | |
| dc.identifier.arxiv | 1301.4928 | |
| bordeaux.journal | Proceedings of the London Mathematical Society | |
| bordeaux.page | 1-44 | |
| bordeaux.volume | 108 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-01027776 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01027776v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Proceedings%20of%20the%20London%20Mathematical%20Society&rft.date=2015-01-05&rft.volume=108&rft.spage=1-44&rft.epage=1-44&rft.eissn=0024-6115&rft.issn=0024-6115&rft.au=CASSOU-NOGU%C3%88S,%20Philippe&CHINBURG,%20Ted&MORIN,%20Baptiste&TAYLOR,%20Martin&rft.genre=article |
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