EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES

• Published in: Taiwanese journal of mathematics Vol. 19; no. 1; pp. 211 - 220
• Main Authors: Abazari, Nemat, Bahmanpour, Kamal
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.02.2015
• Subjects: Algebra; Functors; Integers; Mathematical rings; Mathematical sets
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm.19.2015.4315

Let (R, m) be a complete Noetherian local ring,Ia proper ideal of R andM, Ntwo finitely generated R-modules such that Supp(N) ⊆V(I). Lett≥ 0 be an integer such that for each 0 ≤i≤t, theR-module H I i ( M ) is in dimension <n. Then we show that each elementLof the set 𝔍, which is defined as: { Ext R j ( N , H I i ( M ) ) : j ≥ 0 and 0 ≤ i ≤ t } ∪ { Hom R ( N , H I t + 1 ( M ) ) , Ext R 1 ( N , H I t + 1 ( M ) ) } is in dimension <n− 2 and so as a consequence, it follows that the set Ass R ( L ) ∩ { 𝔭 ∈ Spec ( R ) : dim ( R / 𝔭 ) ≥ n − 2 } is finite. In particular, the set Ass R ( ⊕ i = 0 t + 1 H I i ( R ) ) ∩ { 𝔭 ∈ Spec ( R ) : dim ( R / 𝔭 ) ≥ n − 2 } is finite. Also, as an immediately consequence of this result it follows that theR-modules Ext R j ( N , H I i ( M ) ) are in dimension <n− 1, for all integersi, j≥ 0, whenever dim(M/IM) =n. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in the case of complete local rings. 2010Mathematics Subject Classification: 13D45, 14B15, 13E05. Key words and phrases: Associated prime ideal, Cofinite module, Complete local ring, Krull dimension, Local cohomology.