Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T17:45:04.497Z Has data issue: false hasContentIssue false

1 - Finite group schemes

Published online by Cambridge University Press:  21 November 2024

By
Michel Brion
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal  and
D. I. Stewart
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
D. I. Stewart
Affiliation:
University of Manchester
Get access

Summary

These extended notes give an introduction to the theory of finite group schemes over an algebraically closed field, with minimal prerequisites. They conclude with a brief survey of the inverse Galois problem for automorphism group schemes.

Keywords

group schemeinverse Galois problem
Type
Chapter
Information
Groups St Andrews 2022 in Newcastle , pp. 1 - 52
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Blanc, J., Brion, M., “Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics,” Ann. Fac. Sci. Toulouse Math. (6) 32 (2023), 607–622.CrossRefGoogle Scholar
[2] Bragg, D., “Automorphism groups of curves over arbitrary fields,” preprint, https://arxiv.org/abs/2304.02778Google Scholar
[3] Brion, M., “Automorphism groups of almost homogeneous varieties,” in: Facets of algebraic geometry. A collection in honor of William Fulton’s 80th birthday. Volume 1, London Math. Soc. Lecture Note Series 472, pp. 54–76, Cambridge Univ. Press, 2022.Google Scholar
[4] Brion, M., Schröer, S., “The inverse Galois problem for connected algebraic groups,” preprint, https://arxiv.org/abs/2205.08117Google Scholar
[5] Cushing, D., Stagg, G., Stewart, D., “A Prolog assisted search for new simple Lie algebras,” Math. Comp. 93 (2024), 1473–1495.Google Scholar
[6] Darda, R., Yasuda, T., “Inverse Galois problem for semicommutative finite group schemes,” preprint, https://arxiv.org/abs/2210.01495Google Scholar
[7] Demazure, M., Gabriel, P., “Groupes algébriques,” Masson, Paris, 1970.Google Scholar
[8] Demazure, M., Grothendieck, A., “Séminaire de Géométrie Algébrique du Bois Marie, 1962–64, Schémas en groupes (SGA3),” Tome I. Propriétés générales des schémas en groupes, Doc. Math. 7, Soc. Math. France, Paris, 2011.Google Scholar
[9] Dinh, T., Oguiso, K., “A surface with discrete and nonfinitely generated automorphism group,” Duke Math. J. 168 (2019), 941–966.CrossRefGoogle Scholar
[10] Eisenbud, D., Harris, J., “The geometry of schemes,” Graduate Texts Math. 197, Springer, New York, 2000.Google Scholar
[11] Eisenbud, D., “Commutative algebra with a view towards algebraic geometry,” Graduate Texts Math. 150, Springer, New York, 1996.Google Scholar
[12] Florence, M., “Realization of Abelian varieties as automorphism groups,” preprint, https://arxiv.org/abs/2102.02581Google Scholar
[13] Greenberg, L., “Maximal groups and signatures,” in: Discontinuous groups and Riemann surfaces, pp. 207–226. Princeton Univ. Press, Princeton, N.J., 1974.CrossRefGoogle Scholar
[14] Grothendieck, A., “Techniques de construction et théorèmes déxistence en géométrie algébrique IV: les schémas de Hilbert,” Sém. Bourbaki, Vol. 6 (1960–1961), Exp. 221, 249–276.Google Scholar
[15] Hartshorne, R., “Algebraic geometry,” Graduate Texts Math. 52, Springer, New York, 1977.Google Scholar
[16] Lesieutre, J., “A projective variety with discrete, non-finitely generated automorphism group,” Invent. Math. 212 (2018), 189–211.Google Scholar
[17] Liedtke, C., “A McKay correspondence in positive characteristic,” preprint, https://arxiv.org/abs/2207.06286Google Scholar
[18] Lombardo, D., Maffei, A., “Abelian varieties as automorphism groups of smooth projective varieties,” Int. Math. Res. Not. (2020), 1942—1956.CrossRefGoogle Scholar
[19] Madan, M., Rosen, M., “The group of automorphisms of a function field,” Proc. Amer. Math. Soc. 115 (1992), 923–929.CrossRefGoogle Scholar
[20] Madden, D., Valentini, R., “The group of automorphisms of algebraic function fields,” J. Reine Angew. Math. 343 (1983), 162–168.Google Scholar
[21] Malle, G., Matzat, B., “Inverse Galois theory. 2nd edition,” Springer Monographs in Mathematics, Springer, Berlin, 2018.CrossRefGoogle Scholar
[22] Martin, G., “Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces,” Algebra Number Theory 16 (2022), 1655–1704.CrossRefGoogle Scholar
[23] Martin, G., “Automorphism group schemes of bielliptic and quasibielliptic surfaces,” Épiother Géom. Algébrique 6 (2022), Article no. 9.Google Scholar
[24] Matsumura, H., Oort, F., “Representability of group functors, and automorphisms of algebraic schemes,” Invent. Math. 4 (1967), 1–25.CrossRefGoogle Scholar
[25] Milne, J. S., “Algebraic groups. The theory of group schemes of finite type over a field,” Cambridge Stud. Adv. Math. 170, Cambridge Univ. Press, Cambridge, 2017.Google Scholar
[26] Mumford, D., “Abelian varieties,” Oxford Univ. Press, Oxford, 1970.Google Scholar
[27] Neukirch, J., Schmidt, A., Wingberg, K., “Cohomology of number fields. 2nd ed.,” Grundlehren Math. Wiss. 323, Springer, Berlin, 2008.Google Scholar
[28] Oguiso, K., “A surface in odd characteristic with discrete and non-finitely generated automorphism group,” Adv. Math. 375 (2020) 107397, 20 pp.CrossRefGoogle Scholar
[29] Pink, R., “Finite group schemes,” course notes available at https://people.math.ethz.ch/˜pink/ftp/FGS/CompleteNotes.pdfGoogle Scholar
[30] S. Schröer, Tziolas, N., “The structure of Frobenius kernels for automorphism group schemes,” Algebra Number Theory 17 (2023), 1637–1680,Google Scholar
[31] Serre, J.-P., “Topics in Galois theory. Notes written by Henri Darmon. 2nd ed.,” Research Notes in Math. 1, Wellesley, MA, 2007.Google Scholar
[32] Silverman, J. H., “The arithmetic of elliptic curves. Second edition,” Graduate Texts Math. 106, Springer, New York, 2009.Google Scholar
[33] Strade, H., “Simple Lie algebras over fields of positive characteristic. I, II, III,” de Gruyter Expositions in Mathematics 38, 42, 57, de Gruyter, Berlin, 2013–2017.Google Scholar
[34] Tate, J., “Finite flat group schemes,” in: Modular forms and Fermat’s last theorem, pp. 121–154, Springer, New York, 1997.CrossRefGoogle Scholar
[35] Tziolas, N., “Automorphisms of smooth canonically polarized surfaces in positive characteristic,” Adv. Math. 310 (2017), 235–289; corrigendum ibid., 585–593.CrossRefGoogle Scholar
[36] Viviani, F., “ Simple finite group schemes and their infinitesimal deformations,” Rend. Sem. Mat. Univ. Politec. Torino 68 (2010), 171–182.Google Scholar
[37] Waterhouse, W., “Introduction to affine group schemes,” Graduate Texts Math. 66, Springer, New York, 1979.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×