Kähler differentials for points in \(\mathbb{P}^n\)
Let \(K\) be an algebraically closed field of arbitrary characteristic, and let \({\mathbb X}\subset{\mathbb P}_K^n\) be a reduced zero-dimensional subscheme consisting of \(s\) points. Let \(I_{\mathbb X}\subset K[X_0,\ldots,X_n]=A\) be the homogeneous ideal of \({\mathbb X}\) and let \(R=A/I_{\mathbb X}\) be the homogeneous coordinate ring of \({\mathbb X}\). Many authors have studied the Hilbert function \(H_R\) and the graded Betti numbers of the graded \(A\)-module \(R\) [cf., e.g., \textit{A. V. Geramita, M. Kreuzer} and \textit{L. Robbiano}, Trans. Am. Math. Soc. 339, No. 1, 163-189 (1993; Zbl 0793.14002), \textit{A. V. Geramita} and \textit{P. Maroscia}, J. Algebra 90, 528-555 (1984; Zbl 0547.14001), \textit{M. Kreuzer}, Can. J. Math. 46, No. 2, 357-379 (1994; Zbl 0826.14030), \textit{P. Maroscia, U. Nagel} and \textit{W. Vogel}, Rend. Sem. Mat., Torino 48, No. 4, 491-510 (1991; Zbl 0782.14043) and C. R. Math. Acad. Sci., Soc. R. Can. 12, No. 1, 27-32 (1990; Zbl 0718.13001), \textit{U. Nagel}, J. Pure Appl. Algebra 96, No. 1, 23-38 (1994; Zbl 0829.14024) and Trans. Am. Math. Soc. 351, No. 11, 4381-4409 (1999), \textit{F. Orechia}, J. Lond. Math. Soc., II. Ser. 24, 85-96 (1981; Zbl 0492.14017)]. In this paper the authors study the graded \(R\)-module of Kähler differentials \(\Omega^1_{R/K}\) and, in particular, its torsion submodule \({\mathcal T}_{R/K}\). A description of \(\Omega^1_{R/K}\) in terms of generators and relations is given in proposition 1.3, and a presentation of \({\mathcal T}_{R/K}\) as a quotient of the syzygy module \(\text{Syz}^1_R(x_0,\ldots,x_n)\) is given in proposition 2.4 (here \(x_i\) is the image of \(X_i\) in \(R\), \(i\in\{0,\ldots,n\}\)). This implies that \({\mathcal T}_{R/K}\) is, in case \(K\) has characteristic zero, generated by homogeneous elements of degree 2. In section 3 the authors study the Hilbert function \(H_\Omega\) of \(\Omega^1_{R/K}\). They show that \(H_\Omega(i)=s\) for all large enough \(i\); the smallest integer \(\rho_{\mathbb X}\) with \(H_\Omega(i)=s\) for all \(i\geq\rho_{\mathbb X}\) is called the regularity of \(\Omega^1_{R/K}\). Furthermore, they show that \(\rho_{\mathbb X}\leq 2\sigma_{\mathbb X}+3\), and that this bound is sharp [cf. proposition 3.5 and example 3.7; here \(\sigma_{\mathbb X}\) is the smallest integer with \(H_R(i)=s\) for all \(i\geq\sigma_{\mathbb X}\)]. In proposition 3.8 the authors give an explicit description of \(({\mathcal T}_{R/K})_{2\sigma_{\mathbb X}+2}\) and thus a criterion for maximal regularity \(\rho_{\mathbb X}=2\sigma_{\mathbb X}+3\). The last two sections are concerned with calculations of \(\Omega^1_{R/K}\) in special cases.
\(0\)-dimensional subschemes of \({\mathbb P}^n\)
scientific article; zbMATH DE number 1340432