We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation δwδw log det(∆).

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Let a, b be ideals of a commutative Noetherian ring R and let M, N be finite R-modules. The concept of an a-filter grade of b on M is introduced and several characterizations and properties of this notion are given. Then, using the above characterizations, we obtain some results on generalized local cohomology modules H i a (M, N ). In particular, first we determine the least integer i for which H i a (M, N ) is not Artinian. Then we prove that H i a (M, N ) is Artinian for all i ∈ N 0 if and only if dim R/(a + Ann M + Ann N ) = 0. Also, we establish the Nagel-Schenzel formula for generalized local cohomology modules. Finally, in a certain case, the set of attached primes of H i a (M, N ) is determined and a comparison between this set and the set of attached primes of H i a (N ) is given.

Let [Formula: see text] be ideals of a commutative Noetherian ring [Formula: see text] and [Formula: see text] be a finitely generated [Formula: see text]-module. By using filter regular sequences, we show that the infimum of integers [Formula: see text] such that the local cohomology modules [Formula: see text] and [Formula: see text] are not isomorphic is equal to the infimum of the depths of [Formula: see text]-modules [Formula: see text], where [Formula: see text] runs over all prime ideals of [Formula: see text] containing only one of the ideals [Formula: see text]. In particular, these local cohomology modules are isomorphic for all integers [Formula: see text] if and only if [Formula: see text]. As an application of this result, we prove that for a positive integer [Formula: see text], [Formula: see text] is Artinian for all [Formula: see text] if and only if, it can be represented as a finite direct sum of [Formula: see text] local cohomology modules of [Formula: see text] with respect to some maximal ideals in [Formula: see text] for any [Formula: see text]. These representations are unique when they are minimal with respect to inclusion.

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a proper ideal of [Formula: see text] and [Formula: see text] a nonzero finitely generated [Formula: see text]-module with [Formula: see text]. Let [Formula: see text] (respectively, [Formula: see text]) be the smallest (respectively, greatest) non-negative integer [Formula: see text] such that the local cohomology [Formula: see text] is nonzero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules [Formula: see text], [Formula: see text] and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch’s conjecture.