## Theses and Dissertations

#### Title

On Rings of Invariants for Cyclic p-Groups

5-2017

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Department

Mathematical Sciences

Lance E. Miller

Mark Johnson

Paolo Mantero

#### Keywords

Commutative Algebra, Invariant Theory

#### Abstract

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p >0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular.

We also study the a-invariant for a graded ring S, that is, the maximal graded degree of the top local cohomology module of S. We give an upper bound for the a-invariant of R^G and we show for any subgroup H\leq G, we can bound the a-invariant of R^G by the a-invariant of R^H. We extend this result to more general modular rings of invariants where R^{G'} is quasi-Gorenstein and G' has a normal, cyclic, p-Sylow subgroup.

Given a subgroup H \leq G we consider the natural action of H on R and the associated ring of invariants R^H. When G acts in a particular way, we determine the representation underlying the action of H on R. Building on work of Watanabe and Yoshida, we estimate the Hilbert-Kunz multiplicity of R^G in a way that does not depend on finding explicit generators for R^G.

%, that is, we give an upper bound for the asymptotic growth rate of the colength of Frobenius powers of the homogeneous maximal ideal.

We extend this result to modular rings of invariants for groups G' which have a normal, cyclic, p-Sylow subgroup.

Finally, we also consider computations of the norm of x_4 for a cyclic modular action on k[x_1, \ldots, x_n]. This builds on and extends work of Sezer and Shank.

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