Mara D. Neusel

The Lasker-Noether theorem for *-invariant ideals

Abstract This article is motivated by the study begun in [8] of modular invariants of finite groups using as tools, the Steenrod algebra and the Dickson algebra. The ring of invariants [V] G of a representation ϱ: G ↪ GL(n, ) of a finite group G over a Galois field of characteristic p is an unstable graded connected commutative Noetherian algebra over the Steenrod algebra *. We adopt this more general point of view and study *-invariant ideals in unstable graded connected commutative Noetherian algebras H* over a Galois field . (An ideal I⊂H* is called *-invariant if it is closed under the action of the Steenrod algebra.) Our goal is to show that *-invariant ideals have a *-invariant primary decomposition.

Cryptanalysis of a system using matrices over group rings

Abstract In several recent works of D. Kahrobaei, C. Koupparis, and V. Shpilrain, public-key protocols have been proposed which depend on the difficulty of computing discrete logarithms in matrix rings over group rings. In particular, the specific ring of 3×3 matrices over 𝔽 7 S 5

The Noether Map I

{\mathbb {F}_7S_5}

The noether map II

has been proposed for use in some of these protocols. In this paper, we show that the discrete logarithm problem in this matrix ring can be solved on a modern PC in seconds, and we give a solution to the challenge problem over 𝔽 2 S 5

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

{\mathbb {F}_2S_5}

Degree bounds—An invitation to postmodern invariant theory

proposed in one of the aforementioned works.

Separating invariants for modular

Abstract Let æ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G. In this paper we study the image of the associated Noether map . It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure . This is true without any restrictions on the group, representation, or ground field. Moreover, we show that the extension is a finite p-root extension if the characteristic of the ground field is p. Furthermore, we show that the Noether map is surjective, if V = 𝔽 n is a projective 𝔽G-module. We apply these results and obtain upper bounds on the degrees of a minimal generating set of 𝔽[V] G and the Cohen-Macaulay defect of 𝔽[V] G . We illustrate our results with several examples.

P

Let ρ: G → GL(n, F) be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map η G G : F[V(G)] G → F[V] G . In our 2005 paper it has been shown that the Noether map is surjective if V is a projective FG-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for p-groups (where p is the characteristic of the ground field F) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of V.

-groups and groups acting diagonally

We consider purely inseparable extensions H → P* √H of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group G ≤ GL(V) and a vector space decomposition V = W 0 ⊕ W 1 ⊕ ··· ⊕ W e such that H = (F[W 0 ] ⊗ F[W 1 ] p ⊗ ··· ⊗ F[We]p e ) G and P* √H = F[V] G , where (-) denotes the integral closure. Moreover, H is Cohen-Macaulay if and only if P* √H is Cohen-Macaulay. Furthermore, H is polynomial if and only if P* √H is polynomial, and P* √H = F[h 1 ,... h n ] if and only if .H=F[h 1 ,...,h n0 ,h p n1 ,h p2 n1+1 ,...,h pe ne ], where n e = n and n i = dim F (W 0 ⊕···⊕ W i ).