Local cohomology modules and their properties

UDC 512.5 Let be a complete Noetherian local ring and let be a generalized Cohen-Macaulay -module of dimension We show thatwhere and is the ideal transform functor. Also, assuming that is a proper ideal of a local ring , we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to

Partimenti and the Power of Improvisation

Like a lead sheet in popular music, a partimento gives a performer a single line of music to aid in the performance of a multivoice composition. Lead sheets provide a simplified melody and symbols for chords. Partimenti provide a bass and sometimes figured-bass numbers to indicate specific intervals. In both cases the reconstruction works well if the performer has a good knowledge of the style of music involved and a memory for the kinds of musical patterns needed. In Naples, children played the written partimento with their left hands at a small harpsichord. With their right hands they improvised the types of melodies, chords, and counterpoints implied by the bass. Beginners may have improvised mostly simple chords. Intermediate-level students improvised melodies and counterpoints. And advanced students developed highly contrapuntal realizations that included partimento fugues.

Bass numbers of derived local cohomology with respect to a specialization closed subset

Let [Formula: see text] be a specialization closed subset of Spec [Formula: see text] and [Formula: see text] a left homologically bounded complex with finitely generated homologies. We provide some inequalities between the Bass numbers of [Formula: see text] and its local cohomology modules with respect to [Formula: see text]. As an application of these inequalities, we provide a comparison between the injective dimensions of [Formula: see text] and its nonzero local cohomology module [Formula: see text]. Our versions contain variations of some results already known in these areas.

Chains of semidualizing modules

Let [Formula: see text] be a commutative Noetherian local ring. We study the suitable chains of semidualizing [Formula: see text]-modules. We prove that when [Formula: see text] is Artinian, the existence of a suitable chain of semidualizing modules of length [Formula: see text] implies that the Poincar[Formula: see text] series of [Formula: see text] and the Bass series of [Formula: see text] have very specific forms. Also, in this case, we show that the Bass numbers of [Formula: see text] are strictly increasing. This gives an insight into the question of Huneke about the Bass numbers of [Formula: see text].

Bass numbers of generalized local cohomology modules with respect to a pair of ideals

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] and [Formula: see text] are ideals of [Formula: see text], and [Formula: see text] and [Formula: see text] are [Formula: see text]-modules. We study the relationship between the Bass numbers of [Formula: see text] and [Formula: see text]. As a consequence, it follows that if one of the following holds: (a) [Formula: see text] is a principal ideal of [Formula: see text], (b) [Formula: see text], (c) [Formula: see text] (when [Formula: see text] is local and [Formula: see text] is finitely generated), (d) [Formula: see text] (when [Formula: see text] is local), (e) [Formula: see text] (when [Formula: see text] is local), then [Formula: see text] is finite for all [Formula: see text] and [Formula: see text], whenever [Formula: see text] is finitely generated and flat, [Formula: see text] is minimax, and [Formula: see text].

On the generalized local cohomology of minimax modules

Let [Formula: see text] be a commutative Noetherian ring with identity and [Formula: see text] be an ideal of [Formula: see text]. Assume that [Formula: see text] is a finite [Formula: see text]-module and [Formula: see text] and [Formula: see text] are minimax [Formula: see text]-modules such that [Formula: see text]. In this paper, among other things, we show that [Formula: see text] is minimax for all [Formula: see text] and [Formula: see text] when one of the following conditions holds: [Formula: see text](i) [Formula: see text]; [Formula: see text] (ii) [Formula: see text]; or (iii) [Formula: see text]. As a consequence, we obtain that the Bass numbers and Betti numbers of [Formula: see text] are finite for all [Formula: see text] when one of the above conditions holds.

Local cohomology modules of invariant rings

LetKbe a field and letRbe a regular domain containingK. LetGbe a finite subgroup of the group of automorphisms ofR. We assume that |G| is invertible inK. LetRGbe the ring of invariants ofG. LetIbe an ideal inRG. Fixi⩾ 0. IfRGis Gorenstein then:(i)injdimRGHiI(RG) ⩽ dim SuppHiI(RG);(ii)$H^j_{\mathfrak{m}}$(HiI(RG)) is injective, where$\mathfrak{m}$is any maximal ideal ofRG;(iii)μj(P, HiI(RG)) =μj(P′,HiIR(R)) whereP′ is any prime inRlying aboveP.We also prove that ifPis a prime ideal inRGwithRGPnot Gorensteinthen either the bass numbersμj(P, HiI(RG)) is zero for alljor there existscsuch thatμj(P, HiI(RG)) = 0 forj<candμj(P, HiI(RG)) > 0 for allj⩾c.