Constructing arbitrary torsion elements for a local cohomology module
Let \(R\) be a Noetherian ring and \(I\) an ideal. If \(\dim R/I > 0\), then local cohomology module \(H_I^p(R)\) has many strange properties. For example, \textit{A. K. Singh} [Math. Res. Lett. 7, No. 2--3, 165--176 (2000; Zbl 0965.13013)] showed the following: Let \[ R = \mathbb{Z}[U, V, W, X, Y, Z]/(UX + VY + WZ) \] and \(I = (X, Y, Z)\). Then, for any prime integer \(p\), the abelian group \(\mathbb{Z}/p \mathbb{Z}\) embeds into \(H_I^3(R)\). Consequently, \(H_I^3(R)\) has infinitely many associated primes. In the present paper, the author enlarges Singh's result. He shows that we can find arbitrary finitely generated abelian group in \(H_I^3(R)\).
scientific article; zbMATH DE number 6162678
scientific article