# Trivial morphism between local cohomology groups

I have two questions concerning morphism between local cohomology groups which I think are related.

Let $$G$$ be a reductive group with Weyl group $$W$$ and $$B \subset G$$ a Borel. Let $$X=G/B$$ be the flag variety, $$C(w)=BwB/B \subset X$$ the Schubert cell associated to $$w \in W$$, and $$X(w)$$ its closure. Additionally fix an abelian sheaf $$\mathcal{F} \in \mathit{Ab}(X)$$.

1. Let $$v,w \in W$$ with $$l(w)=l(v)+1$$ and we abbreviate $$\operatorname{codim}(C(w))$$ by $$c_w$$. In Kempf's paper "The Grothendieck–Cousin complex of an induced representation" he mentions in a remark after Lemma 12.6 that $$H^{c_w}_{C(w)}(X,\mathcal{F}) \rightarrow H^{c_v}_{C(v)}(X,\mathcal{F})$$ is trivial if $$C(v) \not\subset X(w)$$ (equivalently $$v \nleq w$$) and says that it follows by trivial support related reasons. But unfortunately I don't see the concrete argumentation. So how does it follow in more detail?
1. Let $$v,w \in W$$ with $$l(w)=l(v)$$ and $$v\neq w$$. Is it then also true that $$H^i_{X(w)}(X,\mathcal{F}) \rightarrow H^i_{X(v)}(X,\mathcal{F})$$ is trivial in general or at least for $$i=c_w$$? (Notice that the local cohomology groups vanish for $$i.)
• What you quote in (1) is not quite what the remark says; it says rather that $H^{c_w}_{C(w)}(U_{c_w}, \mathcal F) \to H^{c_v}_{C(v)}(U_{c_v}, \mathcal F)$ is $0$, where $U_i$ is the complement of the union of codimension-($i + 1$) Schubert varieties. (This is important, because individual cells are usually not closed in $X$.) Jan 8, 2021 at 0:20
• Here he is using the material before the lemma, which gives a map $H^{c_w}_{C(w)}(U_{c_w}, \mathcal F) \to \bigoplus_{C'} H^{c_v}_{C'}(U_{c_v}, \mathcal F)$, where $C'$ runs over the Bruhat cells of codimension $c_v$. The map $H^{c_w}_{C(w)}, \mathcal F) \to H^{c_v}_{C(v)}(U_{c_v}, \mathcal F)$ is projection on the $C(v)$ factor (i.e., restricting the section to $C(v)$); but a section with support in $C(w) \subseteq U_{c_w}$ has support in $X(w) \cap U_{c_v} \subseteq U_{c_v}$. In particular, its support does not intersect $C(v)$ unless $C(v) \subseteq X(w)$. Jan 8, 2021 at 0:23
• To your first answer. I think it doesn't matter as we have the excision isomorphism $H^{c_w}_{C(w)}(X,\mathcal{F})\cong H^{c_w}_{C(w)}(U_{c_w},\mathcal{F})$. Local cohomology is defined for locally closed subsets if you look for example at "Local cohomology" by Hartshorne/Grothendieck.
– KKD
Jan 8, 2021 at 17:12