Asymptotic linear bounds of Castelnuovo–Mumford regularity in multigraded modules

Let A   be a Noetherian standard NN-graded algebra over an Artinian local ring A0A0. Let I1,…,ItI1,…,It be homogeneous ideals of A and M   a finitely generated NN-graded A  -module. We prove that there exist two integers k,k′k,k′ such thatreg(I1n1⋯ItntM)⩽(n1+⋯+nt)k+k′for all  n1,…,nt∈N.