Cohomological Dimensions With Respect To Sum And Intersection Of Ideals

Alireza Vahidi

Abstract: Let $R$ be a commutative Noetherian ring with non-zero identity, $𝔞$ and $𝔟$ proper ideals of $R$, $M$ a finitely generated $R$-module with finite projective dimension, and $X$ a finitely generated $R$-module. We study the cohomological dimensions of $M$ and $X$ with respect to $𝔞+𝔟$ and $𝔞\cap 𝔟$. We show that the inequality ${cd}_{𝔞+𝔟}(M,X)\le {cd}_{𝔞}(M,X)+{cd}_{𝔟}\left(X\right)$ holds true and we find an equivalent condition for it to be equality.

Keywords: cohomological dimensions; generalized local cohomology modules; local cohomology modules

Classification (MSC2000): 13D05; 13D45

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