## Journal of Commutative Algebra

### On formal local cohomology modules with respect to a pair of ideals

#### Abstract

We introduce a generalization of the formal local cohomology module, which we call a formal local cohomology module with respect to a pair of ideals, and study its various properties. We analyze their structure, upper and lower vanishing and non-vanishing properties. There are various exact sequences concerning formal cohomology modules, among them we have a Mayer-Vietoris sequence with respect to pair ideals. Also, we give another proof for a generalized version of the local duality theorems for Gorenstein, Cohen-Macaulay rings, and a generalization of the Grothendieck duality theorem for Gorenstein rings. We discuss the concept of formal grade with respect to a pair of ideals and give some results about this.

#### Article information

Source
J. Commut. Algebra, Volume 8, Number 3 (2016), 337-366.

Dates
First available in Project Euclid: 9 September 2016

https://projecteuclid.org/euclid.jca/1473428564

Digital Object Identifier
doi:10.1216/JCA-2016-8-3-337

Mathematical Reviews number (MathSciNet)
MR3546002

Zentralblatt MATH identifier
1348.13026

Subjects

#### Citation

Freitas, T.H.; Pérez, V.H. Jorge. On formal local cohomology modules with respect to a pair of ideals. J. Commut. Algebra 8 (2016), no. 3, 337--366. doi:10.1216/JCA-2016-8-3-337. https://projecteuclid.org/euclid.jca/1473428564

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