文摘
Let $M$ be a finitely generated bigraded module over the standard bigraded polynomial ring $S=K[x_1,\ldots ,x_m, y_1,\ldots ,y_n]$ , and let $Q=(y_1,\ldots ,y_n)$ . The local cohomology modules $H^k_Q(M)$ are naturally bigraded, and the components $H^k_Q(M)_j=\bigoplus _iH^k_Q(M)_{(i,j)}$ are finitely generated graded $K[x_1,\ldots ,x_m]$ -modules. In this paper we study the regularity of $H^k_Q(M)_j$ , and show in several cases that $\mathrm reg H^k_Q(M)_j$ is linearly bounded as a function of $j$ .