This question is from the proof of Theorem 11.34 in the book: *Twenty-four Hours of Local Cohomology*.

Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite generated $R$-module, $k$ and $l$ are residue fields of $R$ and $S$ respectively. Let $t = dim R - dim S$, and $\omega$ be a canonical module of $R$. Let $I$ be a minimal injective resolution of $\omega$.

Now we have natural isomorphism of l-vector space:

$$Hom_{l}(l,Hom_{R}(S,I) \cong Hom_{R}(l,I) \cong Hom_{k}(l,k) \otimes_{k}Hom_{k}(k,I)$$

Passing to cohomology yields isomorphisms

$$Ext^i_{S}(l,Ext^{t}_{R}(S,\omega)) \cong Hom_{k}(l,k) \otimes_{k}Ext^{i+t}_{k}(k,\omega)$$

Question: Can we change the LHS to $Ext^a_{S}(l,Ext^{b}_{R}(S,\omega))$, with $a+b=i+t$? If not, how to decide the indices $i$ and $t$ of $Ext$ functor in the LHS?

The same question is at here