The group of units on an affine variety (Q2878800)

Let \(X\) be a normal affine variety over an algebraically closed field \(k\). The author of this paper studies the group \(\mathcal{O}^*(X)\) of units in the ring \(\mathcal{O}(X)\) of regular functions on \(X\). (If \(X\) is projective, \(\mathcal{O}(X)=k\), so the question is not interesting.)NEWLINENEWLINEThe author considers two main cases: cyclic covers of the affine space (Section 2) and restriction of a cyclic cover of the projective space to an open set (Section 3). In the first case, let \(A = k[x_1, \ldots, x_m]\) and \(T = A[z]/(z^n-f)\) for some irreducible square-free \(f \in A\) and an \(n\) invertible in \(k\). The author conjectures that \(T^*=k^*\) and proves it when \(n\) is a prime number or \(n=4\). The principal tool is Galois cohomology, of the Galois group of \(T/A\) (more precisely, of the Galois extension \(T[z^{-1}] / A[f^{-1}]\)). In the second case, let \(\pi: Y \to \mathbb{P}^m\) be a cyclic cover, and \(\pi: X \to U\) be a restriction to an open set. This case is split into two sub-cases, depending on whether \(\pi\) is ramified over \(\mathbb{P}^m \setminus U\). The results obtained in the two sub-cases are applied to affine curves and certain affine hypersurfaces in \(\mathbb{A}^m\).NEWLINENEWLINEThroughout the paper, there are numerous examples, making the paper very friendly to the reader.