I am reading Kashiwara-Schapira's Sheaves on manifolds, and I am having trouble understanding their proof of Proposition 2.7.2 which says the following :
Let $X$ be a Hausdorff space, $F \in D^+(\mathbb{Z}_X)$, and let $\{U_t\}_{t\in \mathbb{R}}$ be a family of open subsets of $X$. We assume the following conditions:
(i) $U_t = \bigcup_{s<t}U_s$ for all $t\in \mathbb{R}$.
(ii) For all pairs $(s, t)$ with $s \le t$, the set $\overline{U_t\setminus U_s}\cap supp(F)$ is compact.
(iii) Setting $Z_s = \bigcap_{t>s}(U_t\setminus U_s)$ we have for all pairs $(s,t)$ with $s \le t$, and all $x\in Z_s\setminus U_t$ :
$$(R\Gamma_{X\setminus U_t}(F))_x = 0$$ Then we have for all $t \in \mathbb{R}$ the isomorphism:
$$H^*\left(\bigcup_s U_s, F\right)\cong H^*(U_t, F)$$
In the book they say $R\Gamma\left(\bigcup_s U_s, F\right)\cong R\Gamma(U_t, F)$ but sheaf cohomology is the derived functor of global sections functor, so this is the same. Now assuming the statements $a_k^s, b_k^t$ I understand we get the result we wanted.
- Now in the proof of $a_k^s$ they claim $$H^k(Z_s, R\Gamma_{U_t\setminus U_s}(F))\cong \lim_{\to}H^k(U\cap U_t, R\Gamma_{X\setminus U_t}(F))$$ The limit is over all open subsets $U$ containing $Z_s$. It is not clear to me why this is true.
- Also, they don't prove $b_k^t$ in there, but as far as I understand for the inductive reasoning given in the beginning to hold, we need to prove $b_k^t$ for some $k<k_0$. How to see this?
Sorry if these questions are dumb, I am only a beginner in these topics. Thank you for your help.