Previous |  Up |  Next


formal local cohomology; local cohomology
Let ${\frak{a}}$ be an ideal of Noetherian local ring $(R,{\frak{m}})$ and $M$ a finitely generated $R$-module of dimension $d$. In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to ${\frak{m}}$. Also we prove that for an arbitrary local ring $(R,{\frak{m}})$ (not necessarily complete), we have ${\rm Att}_R(\mathfrak{F}_{\frak{a}}^d(M)) ={\rm Min} {\rm V}({\rm Ann}_R \mathfrak{F}_{\frak{a}}^d(M)).$
[1] Asgharzadeh M., Divaani-Aazar K.: Finiteness properties of formal local cohomology modules and Cohen-Macaulayness. Comm. Algebra 39 (2011), no. 3, 1082–1103. DOI 10.1080/00927871003610312 | MR 2810597
[2] Bijan-Zadeh M. H., Rezaei S.: Artinianness and attached primes of formal local cohomology modules. Algebra Colloq. 21 (2014), no. 2, 307–316. MR 3192350
[3] Brodmann M., Sharp R. Y.: Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, 1998. MR 1613627
[4] Eghbali M.: On Artinianness of formal local cohomology, colocalization and coassociated primes. Math. Scand. 113 (2013), no. 1, 5–19. DOI 10.7146/math.scand.a-15478 | MR 3105540
[5] MacDonald I. G.: Secondary representations of modules over a commutative ring. Symposia Mathematica, Vol. XI, Convegno di Algebra Commutativa, INDAM, Rome, 1971, Academic Press, London, 1973, pages 23–43. MR 0342506
[6] Macdonald I. G., Sharp R. Y.: An elementary proof of the non-vanishing of certain local cohomology modules. Quart. J. Math. Oxford Ser. (2) 23 (1972), 197–204. DOI 10.1093/qmath/23.2.197 | MR 0299598
[7] Melkersson L., Schenzel P.: The co-localization of an Artinian module. Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 1, 121–131. MR 1317331
[8] Peskine C., Szpiro L.: Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. Inst. Hautes Études Sci. Publ. Math. No. 42 (1972), 47–119 (French). DOI 10.1007/BF02685877 | MR 0374130
[9] Rezaei S.: Minimaxness and finiteness properties of formal local cohomology modules. Kodai Math. J. 38 (2015), no. 2, 430–436. DOI 10.2996/kmj/1436403898 | MR 3368073
[10] Rezaei S.: Some results on top local cohomolgy and top formal local cohomology modules. Comm. Algebra 45 (2017), no. 5, 1935–1940. DOI 10.1080/00927872.2016.1226867 | MR 3582837
[11] Schenzel P.: On formal local cohomology and connectedness. J. Algebra 315 (2007), no. 2, 894–923. DOI 10.1016/j.jalgebra.2007.06.015 | MR 2351900
[12] Sharp R. Y.: Some results on the vanishing of local cohomology modules. Proc. London Math. Soc. (3) 30 (1975), 177–195. MR 0379474
Partner of
EuDML logo