Spectral sequence theory of graded modules and its application to the Buchsbaum property and Segre products (Q1803856)

For a finitely generated graded module \(M=\oplus_{n\in\mathbb{Z}}\) \(M_ n\) over a noetherian graded ring \(R=\oplus_{n\geq 0}R_ n=\) \(k[R_ 1]\) over a field \(R_ 0=k\) such that \(\text{depth}(M)\geq 2\), the author, in an earlier paper, constructs a spectral sequence \(\{E_ r^{p,q}(M)\}\) such that \[ E_ 1^{p,q}(M)=\oplus_{n\in\mathbb{Z}}H^ q(\text{Proj}(R),\widetilde M(n))\otimes K^ p(m,R), \] and characterized the ``\((1,r)\)-Buchsbaum property'' of \(M\) in terms of this spectral sequence. In this paper, the author gives two further applications: (1) If the local cohomology modules \(H^ i_ m(M)\), \(i<\dim(M)\) are of finite length, duality relations between \(\{E_ r^{p,q}(M)\}\) and \(\{E_ r^{p,q}(M^ \lor)\}\) are established, where \(M^ \lor\) is the canonical dual module of \(M\). (2) Let \(N\) be another graded module over a graded ring \(S\). Then, under certain conditions, a cohomological criterion is given for the Buchsbaum property of the graded module \(M\#N=\oplus_{n\in\mathbb{Z}}\) \(M_ n\otimes_ kN_ n\) over the ring \(R\#S\) (the Segre product of \(M\) and \(N)\).