Kähler differentials of affine monomials curves
Let \(\Gamma\subset\mathbb{N}_0\) be a submonoid generated by an arithmetic progression \({S=\{m_0, m_1, \ldots, m_p\}}\) where \(m_i = m + i d\) for \(i \in \{0, \ldots, p\}\) and \(m \geq 2\), \(d \geq 1\), \(\text{gcd} (m, d) = 1\). Let \(A : = k[\Gamma] \subset k[t]\) be the monoid ring of \(A\) over a field \(k\) of characteristic 0. The authors study the algebra of Kähler differentials \(\Omega^\bullet_{A/k} = \bigoplus_{q \geq 0} \Omega^q_{A/k}\). The ring \(A\) is graded if we consider \(t\) to have weight 1, and the \(A\)-modules \(\Omega^n_{A/k}\) are graded by giving the differential \(d t\) weight 1. In section 1 the authors compute \(\dim_k(\Omega^1_{A/k})\) in terms of partitions of \(n\). In section 4 they determine the index of regularity of \(\Omega^1_{A/k}\) and \(\Omega^1_{A'/k}\) where \(A'= k [t^m, t^{m+1}, \ldots, t^{m+p}]\). In section 6 they completely describe \(\Omega^q_{A/k}\) for \(q \geq 3\), generalizing a result of \textit{L. G. Roberts} [``Kähler differentials of certain cusps'', in: Algebraic \(K\)-theory, Pap. 2nd Great Lakes Conf., Fields Inst. Commun. 16, 279-284 (1997)] to the case \(p \leq m - 1\). This permits the authors to determine the Hilbert function of \(\Omega^2_{A/k}\) in terms of \(\Omega^1_{A/k}\) and the Gaussian polynomials. In each section examples are given to illustrate the results.
scientific article; zbMATH DE number 1428662