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This paper contains three new algorithms for computing invariant rings. The first two apply to invariants of a finite group acting on a finitely generated algebra over a Euclidean ring. This may be viewed as a first step in “computational arithmetic invariant theory.” As a special case, the algorithms can compute multiplicative invariant rings. The third algorithm computes the invariant ring of a reductive group acting on a vector space, and often performs better than the algorithms known to date. The main tool upon which two of the algorithms are built is a generalized version of an ideal that was already used by Derksen in his algorithm for computing invariants of linearly reductive groups. As a further application, these so-called extended Derksen ideals give rise to invariantization maps, which turn an arbitrary ring element into an invariant. For the most part, the algorithms of this paper have been implemented. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group acting on a polynomial ring A over the field K and let denote the corresponding ring of invariants. Let B be the subalgebra of generated by all homogeneous elements of degree less than or equal to the group order . Then in general B is not equal to if the characteristic of K divides . However we prove that the field of fractions coincides with the field of invariants . We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for . We prove that there is always a nonzero transfer of degree , such that the localization can be generated by fractions of homogeneous invariants of degrees less than . If with finite-dimensional -module V, then c can be chosen in degree one and can be replaced by . Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for . [Copyright &y& Elsevier]

We study the cohomology modules Hi(G,R) of a p‐group G acting on a ring R of characteristic p, for i>0. In particular, we are interested in the Cohen–Macaulay property and the depth of Hi(G,R) regarded as an RG‐module. We first determine the support of Hi(G,R), which turns out to be independent of i. Then we study the Cohen–Macaulay property for H1(G,R). Further results are restricted to the special case that G is cyclic and R is the symmetric algebra of a vector space on which G acts. We determine the depth of Hi(G,R) for i odd and obtain results in certain cases for i even. Along the way, we determine the degrees in which the transfer map TrG R →RG has non‐zero image. [ABSTRACT FROM PUBLISHER]

Let A=BG, where B is a Noetherian algebra over a field K of characteristic p≠0 and G is a finite group such that p divides &z.sfnc;G&z.sfnc;. We give estimates for the depth of A in terms of the codimension of the branch locus of the extension B/A. [Copyright &y& Elsevier]