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Free actions of p-groups on affine varieties in characteristic p


Let K be an algebraically closed field and $\mathbb{A}$ nKn affine n-space. It is known that a finite group $\frak{G}$ can only act freely on $\mathbb{A}$ n if K has characteristic p > 0 and $\frak{G}$ is a p-group. In that case the group action is “non-linear” and the ring of regular functions K[ $\mathbb{A}$ n] must be a trace-surjective K $\frak{G}$ -algebra.

Now let k be an arbitrary field of characteristic p > 0 and let G be a finite p-group. In this paper we study the category $\mathfrak{Ts}$ of all finitely generated trace-surjective kG algebras. It has been shown in [13] that the objects in $\mathfrak{Ts}$ are precisely those finitely generated kG algebras A such that AGA is a Galois-extension in the sense of [7], with faithful action of G on A. Although $\mathfrak{Ts}$ is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show that s-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category $\mathfrak{Ts}$ also has “weakly initial objects”, which are closely related to the essential dimension of G over k. Our results yield a geometric structure theorem for free actions of finite p-groups on affine k-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory of p-groups and, potentially, to a new constructive approach to homogeneous invariant theory.

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[1] Auslander, M. and Goldmann, O. The Brauer group of a commutative ring. Trans. Amer. Math. Soc. 97 (1960), 367409.
[2] Berhuy, G. and Favi, G. Essential Dimension: A functorial point of view (after A Merkurjev). Documenta Mathematica 8 (2003), 270330.
[3] Bourbaki, N.. Groupes et Algèbres de Lie, chapters IV,V,VI (Herman, Paris, 1968).
[4] Bruns, W. and Herzog, J. Cohen–Macaulay Rings (Cambridge University Press, 1993).
[5] Buhler, J. and Reichstein, Z. On the essential dimension of a finite group. Compos Math. 106 (1997), 159179.
[6] Kang, M. C. Essential dimensions of finite groups. arXiv.math: 0611673v2 (2006), 1–24.
[7] Chase, S. U., Harrison, D. K. and Rosenberg, A. Galois theory and Galois cohomology of commutative rings. Mem. Amer. Math. Soc. 52 (1965), 1533.
[8] Chu, H., Hu, S. J., Kang, M. C. and Zhang, J. Groups with essential dimension one. Asian J. Math. 12 (2) (2008), 177191.
[9] Benson, D. J.. Polynomial Invariants of Finite Groups (Cambridge University Press, 1993).
[10] Costa, D. L. Retracts of polynomial rings. J. Algebra 44 (1975), 492502.
[11] Duncan, A. and Reichstein, Z. Versality of algebraic group actions and rational points on twisted varieties with an appendix containing a letter from J.–P. Serre. J. Algebraic Geom. 3 (November 2015), 499530.
[12] Eakin, P. A note on finite dimensional subrings of polynomial rings. Proc. Amer. Math. Soc. 31 (1) (1972), 7580.
[13] Fleischmann, P. and Woodcock, C. F. Non-linear group actions with polynomial invariant rings and a structure theorem for modular Galois extensions. Proc. London Math. Soc. 103 (5) (November 2011), 826846.
[14] Fleischmann, P. and Woodcock, C. F. Universal Galois algebras and cohomology of p-groups. J. Pure Appl. Alg. 217 (3) (2013), 530545.
[15] Fleischmann, P. and Woodcock, C. F. Modular group actions on algebras and p-local Galois extensions for finite groups. J. Alg. 442 (2015), 316353.
[16] Gupta, N. On the cancellation problem for the affine space 3 in characteristic p. arXiv:1208.0483v2 (2013), pages 19.
[17] Jensen, C. U., Ledet, A. and Yui, N. Generic Polynomials. (Cambridge University Press, 2002).
[18] Karpenko, N. A. and Merkurjev, A. S. Essential dimension of finite p-groups. Invent. Math. 172 (3) (2008), 491508.
[19] Kraft, H., Lötscher, R. and Schwarz, G. W. Compression of finnite group actions and covariant dimension II. J. Algebra 322 (2009), 94107.
[20] Kraft, H. and Schwarz, G. W. Compression of finnite group actions and covariant dimension. J. Algebra 313 (2007), 268291.
[21] Ledet, A. Finite groups of essential dimension one. J. Algebra 311 (2007), 3137.
[22] Ledet, A. On the essential dimension of p-groups. In Galois theory and modular forms. Dev. Math. Vo. 11 (Kluwer, 2003), pages 159172.
[23] Lorenz, M.. Multiplicative Invariant Theory (Springer, 2005).
[24] Popov, V. L.. Sections in Invariant Theory (Scand. Univ. Press, Oslo, 1994).
[25] Reichstein, Z. On the notion of essential dimension for algebraic groups. Transform. Groups 5 (2000), 265304.
[26] Serre, J. P. Cohomological invariants, Witt invariants and trace forms. In Cohomological invariants in Galois cohomology (Amer. Math. Soc., 2003), pages 1100.
[27] Serre, J.-P. How to use finite fields for problems concerning infinite fields. arXiv:0903.0517v2 (2009), pages 1–12.
[28] Smith, L. Polynomial Invariants of Finite Groups (A. K. Peters, 1995).
[29] Shpilrain, V. and Yu, J. T. Polynomial retracts and the Jacobian conjecture. Trans. Amer. Math. Soc. 352 (1) (1999), 477484.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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