We give axiomatic conditions in order to calculate the local cohomology of some idempotent kernel functors. These results lie in some new dimension introduced by T. Levasseur for Auslander-Gorenstein rings. Under some hypothesis, we generalize previous results.

Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T:𝒜\to ℬ$ a left exact additive functor. Then one has a comma category $\left(ℬ\downarrow T\right)$. It is shown that if $T:𝒜\to ℬ$ is $𝒳$-exact, then ${(}^{\perp }𝒳,𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and ${(}^{\perp }𝒴,𝒴)$) is a (hereditary) cotorsion pair in $ℬ$ if and only if $\left(\left(\genfrac{}{}{0pt}{}{{}^{\perp }𝒳}{{}^{\perp }𝒴}\right)\right),\langle 𝐡(𝒳,𝒴)\rangle )$ is a (hereditary) cotorsion pair in $\left(ℬ\downarrow T\right)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping classes...