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Let $f:R\to S$ be a flat homomorphism of commutative Noetherian rings. "Flat Base Change Theorem", compares the local cohomology modules $H^i_a(M) \otimes_R S$ and $H^i_{aS} (M\otimes_R S)$ for $i ∈ N_0$ and an arbitrary $R$-module $M$.
I have two questions:
Question.1.
Can one generalize this? such as: for a (flat) $S$-module instead of $S$; ($H^i_{aS} (M\otimes_R N)$)?
Question.2. (which is more important for me)

Is there a known fact about $H^i_{aS} (Hom_R (M,N)$?
In this case any (non-trivial) condition can be posed on modules and rings; (for example $R=S$)

Improve this question
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  • $\begingroup$ If $R=S$, $M$ is finitely generated, and you replace Hom by RHom then $R\Gamma_a RHom_R(M,N) \cong RHom_R(M,R\Gamma_a N)$. See Lemma 3.12 of arxiv.org/abs/1208.4064 $\endgroup$
    – the L
    Commented Apr 30, 2015 at 11:53
  • $\begingroup$ is $a$ suppose to be an ideal $\mathfrak a$? or do you mean the principal ideal generated by $a$? $\endgroup$
    – Floresza
    Commented Apr 30, 2015 at 19:53
  • $\begingroup$ it is ideal. but as in the Q. any (non-trivial) condition can be posed $\endgroup$
    – user 1
    Commented May 1, 2015 at 6:27

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