On Weitzenböck’s theorem in positive characteristic

Fauntleroy, A.

doi:10.1090/S0002-9939-1977-0460345-3

On Weitzenböck’s theorem in positive characteristic
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Proc. Amer. Math. Soc. 64 (1977), 209-213 Request permission

Abstract:

Let k be an algebraically closed field and let $f:{G_a} \to {\text {GL}}(V)$ be a finite-dimensional k-rational representation of the additive group ${G_a}$. If the subspace of ${G_a}$-fixed points in V is a hyperplane, then the ring of ${G_a}$-invariant polynomial functions on V is finitely generated over k. This result is an analog of a classical theorem of Weitzenböck, a modern proof of which has been given by C. S. Seshadri.

References

Armand Borel, Linear representations of semi-simple algebraic groups, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 421–440. MR 0372054
Amassa Fauntleroy, Linear $G_{a}$ actions on affine spaces and associated rings of invariants, J. Pure Appl. Algebra 9 (1976/77), no. 2, 195–206. MR 447268, DOI 10.1016/0022-4049(77)90066-4
Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, DOI 10.1016/0001-8708(74)90067-X
Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi, Unipotent algebraic groups, Lecture Notes in Mathematics, Vol. 414, Springer-Verlag, Berlin-New York, 1974. MR 0376696, DOI 10.1007/BFb0070517
Oystein Ore, On a special class of polynomials, Trans. Amer. Math. Soc. 35 (1933), no. 3, 559–584. MR 1501703, DOI 10.1090/S0002-9947-1933-1501703-0

On a theorem of Weitzenböck

1-3

Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0