ABSTRACT. This paper at first concerns some criteria on Artinianness and vanishing of formal local cohomology modules. Then we consider the cosupport and the set of coassociated primes of these modules more precisely.

Let I be an ideal of a local ring (R, m) with d = dim R. For the local cohomology module H i I (R) it is a well-known fact that it vanishes for i > d and is an Artinian R-module for i = d. In the case that the Hartshorne-Lichtenbaum Vanishing Theorem fails, that is H d I (R) = 0, we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case R is complete we prove -as a technical tool -that H d I (R) ≃ H d m (R/J) for a certain ideal J ⊂ R. Thus, properties of H d I (R) and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.2000 Mathematics Subject Classification. 13D45, 13C14.

Abstract. Let (R, m) be a local ring and a be an ideal of R. The inequalitiesare known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.2010 Mathematics Subject Classification. Primary 13D45; Secondary 13A35, 13N10.

In this paper, we examine the Lyubeznik tables of two linked ideals [Formula: see text] and [Formula: see text] of a complete regular local ring [Formula: see text] containing a field. More precisely, we prove that the Lyubeznik tables of two evenly linked ideals [Formula: see text] and [Formula: see text] are the same when [Formula: see text] and [Formula: see text] both satisfy one of the following properties: (1) canonically Cohen–Macaulay, (2) generalized Cohen–Macaulay and (3) Buchsbaum. Furthermore, we give some conditions for equality of Lyubeznik tables of two linked ideals of dimension 2.