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S -depth on Z D -modules and local cohomology

Morteza Lotfi Parsa (2021)

Czechoslovak Mathematical Journal

Let R be a Noetherian ring, and I and J be two ideals of R . Let S be a Serre subcategory of the category of R -modules satisfying the condition C I and M be a Z D -module. As a generalization of the S - depth ( I , M ) and depth ( I , J , M ) , the S - depth of ( I , J ) on M is defined as S - depth ( I , J , M ) = inf { S - depth ( 𝔞 , M ) : 𝔞 W ˜ ( I , J ) } , and some properties of this concept are investigated. The relations between S - depth ( I , J , M ) and H I , J i ( M ) are studied, and it is proved that S - depth ( I , J , M ) = inf { i : H I , J i ( M ) S } , where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with...

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