I really searched for this quite a while and didn't find an answer - I hope I didn't miss anything obvious. Let $X$ be a complex manifold ($\mathscr O_X$ the sheaf of holomorphic functions on $X$) and $Z\subset X$ a closed analytic subset, $I_Z$ the defining sheaf of ideals of $Z$. I learned that the functor $$\Gamma_{[Z]}\colon F\mapsto \varinjlim_k\mathscr{H}om_{\mathscr O_X}(\mathscr O/I_Z^k,F),$$ for $F$ some $\mathscr O_X$-module, is left exact, and its right derived functor $R\Gamma_{[Z]}$ is called the algebraic cohomology functor (with support in $Z$), and that if $F\in D^b(\mathscr D_X)$ is a (left) $\mathscr D$-module, then $R\Gamma_{[Z]}(F)\in D^b(\mathscr D_X)$ is a (left) $\mathscr D$-module on $X$ as well.
But recently I came across a situation where this functor $R\Gamma_{[Z]}(\cdot)$ was used for some only locally closed complex analytic set $Z\subset X$. I will describe the actual situation below.
My question:
Referring to the above definition, there obviously is no defining sheaf of ideals on the whole of $X$ in the locally closed case, so there must be some generalization to this - do you know a reference to an appropriate definition? (Just by the way: Is there any intuition on why this is called "algebraic cohomology"?)
The locally closed situation mentioned above:
The situation I found was the following: Let $X,Y$ be complex manifolds, $U\subset X$, $V\subset Y$ open submanifolds and $f\colon U\rightarrow V$ a complex analytic map such that the graph $\Gamma_f$ is a complex analytic set in $X\times Y$. But now, $\Gamma_f\subset X\times Y$ is only locally closed, right (closed in $U\times V$, so locally closed in $X\times Y$)? So my questions comes from the construction of $R\Gamma_{[\Gamma_f]}(\mathscr O_{X\times Y})$ which was done there.
What I thought of so far: Say $Z\subset X$ is a locally closed analytic subset in a complex manifold $X$ and set $F=\mathscr O_X$ (in view of the situation which I referred to in the last paragraph). First, one could certainly use the "usual" algebraic cohomology by choosing some open some open $U\subset X$ such that $Z$ is closed analytic in $U$ an building $Rj_!R\Gamma_{[Z]}(\mathscr O_{U})$, where $j\colon U\rightarrow X$ is the open embedding and $Rj_!$ refers to the $\mathscr D$-module direct image with proper supports. Still one would have to ensure that the result doesn't depend on a choice of $U$ and so on...
Second, in Moderate and formal cohomology associated with constructible sheaves of M. Kashiwara and P. Schapira, I found (Theorem 5.12) that, if $Z\subset X$ is closed analytic in the above situation, then $$R\Gamma_{[Z]}(\mathscr O_X)\simeq \mathscr T hom(\mathbb{C}_Z,\mathscr O_X).$$ This seems to give a natural generalization to the locally closed case by using the righthand side in the very same way, for some locally closed $Z$. I encountered the functor $\mathscr T hom$ for the first time here, but if I tracked down the definitions correctly, this actually corresponds to my first guess of a generalization above.
Still I have no clue if these thoughts are reasonable or not, and didn't find any references as well. Any help is highly appreciated!
Thanks in advance, and kind regards!