Search

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, in [11], holds in R[[X]], then it will holds in R.

Let M be a (2 × n) non-generic matrix of linear forms in a polynomial ring. For large classes of such matrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal I_2(M) generated by the 2-minors of M. Over an algebraically closed field, any (2×n)-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. Badescu and Valla computed ara(I_2 (M)) when M is a concatenation of scroll blocks. In this case we compute cd(I2 (M)) and extend these results to concatenations of Jordan blocks. Eventually we compute ara(I_2(M)) and cd(I_2 (M)) in an interesting mixed case, when M contains both Jordan and scroll blocks. In all cases we show that ara(I_2(M)) is less than the arithmetical rank of the determinantal ideal of a generic matrix.

Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $mathcal S$ be a Serre subcategory of the category of $R$--modules and $n > pd M$ be an integer such that $lc^{i}_{fa}(M,N)$ belongs to $mathcal S$ for all $i> n$. Then, for any ideal $fbsupseteq fa$, it is also shown that the module $lc^{n}_{fa}(M,N)/{fb}lc^{n}_{fa}(M,N)$ belongs to $mathcal S$.