Denote $\mathbb k_{[0, \infty)}$ as the sheaf (of $\mathbb k$-module) over $\mathbb R$ only supported in $[0, \infty)$ where $\mathbb k$ is a field. By definition, $\Gamma_{[a, \infty)}$ is a functor such that for any sheaf $\mathcal F$, $\Gamma_{[a, \infty)}(\mathcal F)(U) = \Gamma_{[a,\infty) \cap U}(U; \mathcal F)$ (and again by definition $\Gamma_Z\mathcal F(U): = \ker(\mathcal F(U) \to \mathcal F(U\backslash Z))$ whenever $Z$ is closed in $U$). As we know functor $\Gamma_{[a, \infty)}$ is only left exact, we will be interested in its derived functor.
Question: What is $R\Gamma_{[a, \infty)}(\mathbb k_{[0, \infty)})$?