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Abstract: This paper studies separating subsets of an invariant ring or, more generally, of any set consisting of functions. We prove that a subset of a finitely generated algebra always contains a finite separating subset. We also show that a general version of Noether’s degree bound holds for separating invariants, independently of the characteristic. While the general finiteness result is non-constructive, the Noether bound provides an easy algorithm for computing separating invariants of finite groups. The paper also contains a conceptual investigation of the difference between separating and generating subsets. [Copyright &y& Elsevier]

Abstract: Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes in terms of a so-called colon-operation. From this, generators of can be obtained in finite time if it is finitely generated. Under the additional hypothesis that is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is . Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal. [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]