Modules Possessing Projective Resolutions of Finite Type

Recently the notions of sfli , the supremum of the flat lengths of injective -modules, and silf , the supremum of the injective lengths of flat -modules have been studied by some authors. These homological invariants are based on spli and silp invariants of Gedrich and Gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups. In this paper we will study these invariants. It turns out that, for any group , the finiteness of silf implies the finiteness of sfli , but the converse is not known. We investigate the situation in which sfli < ∞ implies silf < ∞. The statement holds for example, for groups with the property that flat -modules have finite projective dimension. Moreover, we show that the Gorenstein flat dimension of the trivial Z -module Z, that will be called Gorenstein homological dimension of , denoted Ghd , is completely related to these invariants.

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“…J.Asadollahi et al / Journal of Algebra 335 (2011) [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] …”

mentioning

confidence: 99%

On certain homological invariants of groups

et al. 2011

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“…(i) ⇒ (ii): This follows from Lemma 2.4. (ii) ⇒ (i): An easy generalization of [12,Proposition 6.8] shows that if N is a ZG-module which is completely flat as a ZK-module for all finite subgroups K of G, then N ⊗ ZH ZG is completely flat as a ZG-module for all LHF-subgroups H of G. Then, as we are assuming that G belongs to LHF, the result follows.…”

Section: Proof Of Theorem A: (I) ⇒ (Ii)mentioning

confidence: 89%

Finitary group cohomology and Eilenberg-MacLane spaces

Hamilton

2009

Bulletin of the London Mathematical Society

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We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n. In this paper, we show that if G is a group in Kropholler's class lh픉 with cohomology almost everywhere finitary, then G has an Eilenberg–MacLane space K(G, 1) that is dominated by a CW‐complex with finitely many n‐cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G. We also remark that the converse holds for any group G.

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“…Further homological properties of such modules were studied by Hummel [47]. They have also appeared in group representation theory; see [8,45]. Thinking of modules of type F P ∞ as the correct notion of 'finite' modules over general rings, we are then led to make some more natural definitions.…”

Section: The Absolutely Clean Modulesmentioning

confidence: 99%