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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]

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]