In this paper we study the local cohomology of finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.

Let K be a field, S = K[x 1 , . . . , x m , y 1 , . . . , y n ] be a standard bigraded polynomial ring and M a finitely generated bigraded S-module. In this paper we study sequentially Cohen-Macaulayness of M with respect to Q = (y 1 , . . . , y n ). We characterize the sequentially Cohen-Macaulayness of L ⊗ K N with respect to Q as an S-module when L and N are non-zero finitely generated graded modules over K[x 1 , . . . , x m ] and K[y 1 , . . . , y n ], respectively. All hypersurface rings that are sequentially Cohen-Macaulay with respect to Q are classified.

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.Let R be a graded ring. Recall that a graded R-module N is tame, if there exists an integer j 0 such that N j = 0 for all j ≤ j 0 , or else N j = 0 for all j ≤ j 0 . Brodmann and Hellus [4] raised the question whether for a finitely generated, positively graded algebra R with R 0 Noetherian, the local cohomology modules H i R + (M) for a finitely generated graded R-module M are all tame. Here R + = i>0 R i is the graded irrelevant ideal of R. See [1] for a survey on this problem.In this paper we consider only rings defined by monomial relations. We first consider the squarefree case, as, from a combinatorial point of view, this is the more interesting case, and also because here the formula we obtain is simpler. So let K be a field and ∆ a simplicial complex on V = {v 1 , . . . , v n }. In Section 1, as a generalization of Hochster's formula [3], we compute (Theorem 1.3) the Hilbert series of local cohomology of the Stanley-Reisner ring K [∆] with respect to a monomial prime ideal. With the choice of the monomial prime ideal, the Stanley-Reisner ring K [∆], and hence also all the local cohomology of it, can be given a natural bigraded structure. In Proposition 1.7 we give a formula for the K -dimension of the bigraded components of the local cohomology modules. Using this formula we deduce that the local cohomology of K [∆] withrespect to a monomial prime ideal is always tame.In [6] Takayama generalized Hochster's formula to any graded monomial ideal that is not necessarily squarefree. In Section 2, as a generalization of Takayama's result, we compute the Hilbert series of local cohomology of monomial ideals with respect to monomial prime ideals and observe that again all these modules are tame. The result proved here is surprising because in a recent paper, Cutkosky and Herzog [5] gave an example which shows that in general not all local cohomology modules are tame.

The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.