Bauer, Tilman; Carlson, Magnus

Tensor Products of Affine and Formal Abelian Groups

Doc. Math. 24, 2525-2582 (2019)
DOI: 10.25537/dm.2019v24.2525-2582
Communicated by Mike Hill


In this paper we study tensor products of affine abelian group schemes over a perfect field \(k\). We first prove that the tensor product \(G_1 \otimes G_2\) of two affine abelian group schemes \(G_1,G_2\) over a perfect field \(k\) exists. We then describe the multiplicative and unipotent part of the group scheme \(G_1 \otimes G_2\). The multiplicative part is described in terms of Galois modules over the absolute Galois group of \(k\). We describe the unipotent part of \(G_1 \otimes G_2\) explicitly, using Dieudonné theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

Mathematics Subject Classification

14L17, 16W30, 14L05


Dieudonné theory, affine group schemes, tensor products, formal groups


  • 1. Bruns, Winfried and Herzog, Jürgen, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39 (1993), Cambridge University Press, Cambridge; zbl 0788.13005; MR1251956.
  • 2. Buchstaber, Victor and Lazarev, Andrey, Dieudonné modules and \(p\)-divisible groups associated with Morava \(K\)-theory of Eilenberg-Mac Lane spaces, Algebr. Geom. Topol., 7, 529-564 (2007); DOI 10.2140/agt.2007.7.529; zbl 1134.55004; MR2308956; arxiv math/0507036.
  • 3. Demazure, Michel, Lectures on \(p\)-divisible groups, Lecture Notes in Mathematics, 302 (1986), Springer-Verlag, Berlin; zbl 0247.14010; MR883960.
  • 4. Demazure, Michel and Gabriel, Pierre, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs (1970), Masson \& Cie, Éditeur, Paris and North-Holland Publishing Co., Amsterdam; zbl 0203.23401; MR0302656.
  • 5. Davis, Christopher and Kedlaya, Kiran S., On the Witt vector Frobenius, Proc. Amer. Math. Soc., 142, 7, 2211-2226 (2014); DOI 10.1090/S0002-9939-2014-11953-8; zbl 1291.13038; MR3195748; arxiv 1409.7530.
  • 6. Fontaine, Jean-Marc, Groupes \(p\)-divisibles sur les corps locaux (1977), Société Mathématique de France, Paris; zbl 0377.14009; MR0498610.
  • 7. Goerss, Paul G., Hopf rings, Dieudonné modules, and \(E_*\Omega^2S^3\). Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., 239, 115-174 (1999), Amer. Math. Soc; zbl 0954.55006; MR1718079.
  • 8. Gille, Philippe and Polo, Patrick(ed.), Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 7 (2011), Société Mathématique de France, Paris; zbl 1241.14002; MR2867621.
  • 9. Görtz, Ulrich and Wedhorn, Torsten, Algebraic geometry. I, Advanced Lectures in Mathematics (2010), Vieweg + Teubner, Wiesbaden; DOI 10.1007/978-3-8348-9722-0; zbl 1213.14001; MR2675155.
  • 10. Hedayatzadeh, S. Mohammad Hadi, Exterior powers of \(\pi \)-divisible modules over fields, J. Number Theory, 138, 119-174 (2014); DOI 10.1016/j.jnt.2013.10.023; zbl 1347.11081; MR3168925.
  • 11. Hopkins, Mike and Lurie, Jacob, Ambidexterity in K(n)-Local Stable Homotopy Theory. (2013);
  • 12. Milne, J. S., Algebraic groups: the theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, 170 (2017), Cambridge Univ. Press; DOI 10.1017/9781316711736; zbl 1390.14004; MR3729270.
  • 13. Milnor, John W. and Moore, John C., On the structure of Hopf algebras, Ann. of Math. (2), 81, 211-264 (1965); DOI 10.2307/1970615; zbl 0163.28202; MR0174052.
  • 14. Newman, Kenneth and Radford, David E., The cofree irreducible Hopf algebra on an algebra, Amer. J. Math., 101, 5, 1025-1045 (1979); DOI 10.2307/2374124; zbl 0422.16003; MR546301.
  • 15. Sweedler, Moss E., Hopf algebras, Mathematics Lecture Note Series, vii+336 pp. (1969), W. A. Benjamin, Inc., New York; zbl 0194.32901; MR0252485.
  • 16. Takeuchi, Mitsuhiro, Tangent coalgebras and hyperalgebras. I, Japan. J. Math., 42, 1-143 (1974); zbl 0309.14042; MR0389896.


Bauer, Tilman
Department of Mathematics, Kungliga Tekniska Högskolan, Lindstedtsvägen 25, 10044 Stockholm, Sweden
Carlson, Magnus
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel