What is the relation between $H^i\_I(-)$ and $H^i\_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?



Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$ 
There's a map between them, and these do fit into a long exact sequence together. This is explained in the book on local cohomology by Hartshorne, see Lemma 1.8: You can even download this book **[if your institution has access...](http://link.springer.com/book/10.1007/BFb0073971/page/1)**


I will sketch it briefly.
Let $V(I) = Y \supseteq Z = V(J)$ and set $W = V(I) \setminus V(J) = Y \setminus Z$, this $W$ is only locally closed in $\text{Spec} R$.


We define $H^i\_W(\bullet)$ to be the right derived functors of $\Gamma\_W(\bullet)$. Here $\Gamma\_W(M)$ for any $R$-module $M$ is defined to be the set of elements of $M$ which are zero in $M\_{p}$ for all $p \in (\text{Spec }R) \setminus W$.


With this notation, for any $R$-module $M$ we have a long exact sequence
$$
... \to H^i\_Z(M) \to H^i\_Y(M) \to H^i\_W(M) \to H^{i+1}\_Z(M) \to ...
$$
Note $H^i\_Z(M) = H^i\_J(M)$ and $H^i\_Y(M) = H^i\_I(M)$.
I doubt this gives you much information unless $Y$ and $Z$ have some special relationship. 


There are related things you can do too. If for example $Y = Z \cup X$ where $X$ is some other closed subset of $\text{Spec }R$, there's another long exact sequence but this is a special case. Can you tell me what $Y$ and $Z$ are in your case?


