### The relative Heller operator and relative cohomology for the Klein 4-group

Preprint submitted: pdf

We determine the relative cohomology groups H^i_X(G,N), where G is the Klein 4-group, X the set of all proper subgroups of G, and N is any representation of G over a field of characterstic 2. This generalises some results of Pamuk and Yalcin, who computed H^i_&xi(G,k) for every i. We also show that all cup products in H^i_X(G,k). Our methods are quite direct in that we compute for each N an explicit X-relatively projective resolution and rely on the classification of KG-modules.

### Degree bounds for modular covariants (with Müfit Sezer)

Published version: Forum Math. 32 issue 2, pages 905-910 (2020)

Preprint submitted: pdf

Using some ideas developed in the paper below, we determine the Noether bound (minimum degree d for which there is a generating set in degrees at most d) for the module of covariants k[V,W]^G over k[V]^G when G is a cyclic group of order p.

### Modular covariants of cyclic groups of order p

Preprint submitted: pdf

For G a group and V,W kG-modules, the module of covariants is the set of G-equivariant polynomial maps from V to W. This is k[V]^G-module. We give a generating set for this module when G is a cyclic group of order p, V is an indecomposable module of dimension 2 or 3, and W is an arbitrary indecomposable G-module.

### On the depth of quotients of modular invariant rings by transfer ideals (with Müfit Sezer)

Preprint submitted: pdf

We prove that for a p-group, the depth of K[V]^G/I where I is a sum of images of transfers is bounded below by dim(V^G). We compute the depth of K[V]^G/I^G_1 for every indecomposable representation of the Klein four-group.

### Symmetric powers and modular invariants of elementary abelian p-groups

Published version: Journal of Algebra 492,pages 157-184 (2017).

Preprint version: pdf

Let G be an elementary abelian p-group of order q and V a faithful modular representation of dimension 2. We describe how to decompose S(S^m(V)*) into indecomposable summands. We use this decomposition to prove, among other things, that the rings of invariants K[S^m(V)]^G are generated by invariants of degree at most 2q-3.

### Locally finite derivations and modular coinvariants (with Müfit Sezer)

Published version: Quarterly Journal of Mathematics 69 issue 3, pages 1053-1062 (2018).

Preprint version: pdf

We prove that the algebra of coinvariants of a modular representation of a cyclic-p-group is free as a module over the subalgebra generated by so-called “terminal” variables. We also prove some weaker results for arbitrary p-groups.

### On separating fixed points from zero by invariants (with Martin Kohls)

Published version: Communications in Algebra 45 issue 1, pages 371–375 (2017).

Preprint version: pdf

Let G be any group and V a representation of G over a field of characteristic p. We prove that if v is a G-fixed point, then the invariant f of smallest degree such that f(v)≠0 has degree a power of p.

### Zero-separating invariants for linear algebraic groups (with Martin Kohls)

Published version: Proceedings of the Edinburgh Mathematical Society 49, issue 4, pages 911-924 (2016).

Preprint version: pdf

We extend the investigation of the paper below to infinite algebraic groups.

### Zero-separating invariants for finite groups (with Martin Kohls)

Published version: Journal of Algebra 411, pages 92-113 (2014).

Preprint version: pdf

Let G be a finite group and K a field of characteristic p dividing |G|. We investigate the numbers δ(G) and σ(G). Here, for a kG-module V, δ(G,V) is the minimum degree such that every fixed point in V can be separated from zero by an invariant of degree at most δ(G,V), and δ(G) is the supremum taken over all finitely-generated kG-module, while σ(G,V) is the minimum degree such that every point in V (outside the nullcone) can be separated from zero by an invariant of degree at most σ(G,V), and σ(G) is defined analogously.

### Separating invariants for arbitrary linear actions of the additive group (with Emilie Dufresne and Müfit Sezer)

Published version: Manuscripta Mathematica 143, pages 207-219 (2014).

Preprint version: pdf

We extend the results of the paper below to decomposable representations.

### Separating invariants for the basic G_a actions (with Martin Kohls)

Published version: Proceedings of the American Mathematical Society 140, issue 1, pages 135-146 (2012).

Preprint version: pdf

We describe an explicit separating algebra for the classical covariants of binary forms in characteristic zero.

### On the depth of separating algebras of finite groups

Published version: Contributions to Algebra and Geometry 53, issue 1, pages 31-39 (2012).

Preprint version: pdf

Extending ideas in the paper below, we show that, under certain conditions, the depth of a separating subalgebra is bounded above by the depth of the ring of invariants. In particular this always holds for a cyclic group of order p.

### On the Cohen-Macaulay property of separating algebras (with Emilie Dufresne and Martin Kohls)

Published version: Transformation Groups 14, pages 771-785 (2009).

Preprint version: pdf

We show that in the modular case, there are representations of of finite groups which not only have non-Cohen-Macaulay rings of invariants, but whose ring of invariants do not contain a Cohen-Macaulay separating subalgebra, answering negatively a question of G. Kemper.

### Depth and detection in modular invariant theory

Published version: Journal of Algebra 322, issue 5, pages 1653-1666 (2009).

Preprint version: pdf

We find conditions for the depth of a modular ring of invariants to attain its minimum which are both necessary and sufficient. We apply these to classify the representations of C_2 x C_2 whose rings of invariants have minimal depth.

### Associated primes for cohomology modules

Published version: Archiv der Math 91, issue 6, pages 481-485 (2008).

Preprint version: pdf

We show that the associated primes of the modules H*(G,R), where G is a finite group and R=S(V*) for V a modular representation of G, are radicals of certain transfer ideals.

### On the depth of modular invariant rings for C_p x C_p (with Peter Fleischmann)

Published version: In *Symmetry and Spaces: In honour of Gerry Schwarz, *pages 45-63.

Preprint version: pdf

Contains the most significant results of my thesis. We apply known sufficient conditions (due to Fleischmann, Kemper and Shank) for the depth of a modular ring of invariants to attain its minimum value to groups of the form C_p x C_p. In particular we show that the minimum is attained for all but one indecomposable representation of C_2 x C_2.