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 $i<c_w=c_v$.)
  • $\begingroup$ 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 $X$.) $\endgroup$
    – LSpice
    Jan 8, 2021 at 0:20
  • $\begingroup$ 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 $C(v) \subseteq X(w)$. $\endgroup$
    – LSpice
    Jan 8, 2021 at 0:23
  • 1
    $\begingroup$ 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. $\endgroup$
    – KKD
    Jan 8, 2021 at 17:12


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