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It is well known, that the intermediate extension functor $j_{!*}$ preserves injections and surjections. However it seems that it is not exact in general!

1) What would be an example which shows that $j_{!*}$ is not exact?

2) It is easy to see, that $j_{!*}$ is exact if $j$ is the inclusion of a cell into a cell stratified variety and we are considering say $\mathbb Q_l$-sheaves. Is $j_{!*}$ still exact if we replace $\mathbb Q_l$ by $\mathbb Z_l$ in this situation?

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For your first question, see Example 2.7.1. in de Cataldo and Migliorini's "The decomposition theorem, perverse sheaves and the topology of algebraic maps". –  Dan Petersen Oct 2 '12 at 10:06
Thanks! $ $ –  Jan Weidner Oct 2 '12 at 12:19

1 Answer 1

up vote 2 down vote accepted

I think that the answer for 2) is no.

Consider the affine Grassmannian for $SL_2$ and let $X$ denote the Schubert variety corresponding to the three dimensional representation of $PSL_2$. Then $X$ is stratified by Iwahori orbits into three affine strata of dimensions 2, 1 and 0. Let $U$ denote the open stratum.

Consider the exact sequence of local systems

$0 \to \mathbb{Z}_2 \to \mathbb{Z}_2 \to \mathbb{F_2} \to 0$

on $U$. Shifting by $[2]$ we get an exact sequence of perverse sheaves on $U$ and if we apply $j_{!*}$ we get a sequence

$0 \to IC(X,\mathbb{Z}_2) \stackrel{2}{\to} IC(X,\mathbb{Z}_2) \to IC(X,\mathbb{F}_2) \to 0$

(Here one needs to be careful: because $\mathbb{Z}_2$ is not a field there are two possible $t$-structures (denoted $p$ and $p^+$) which are interchanged by duality. See Daniel Juteau's paper "Decomposition numbers for perverse sheaves" arXiv:0803.2326.) Here I am considering the perversity $p$.

I claim that this sequence cannot be exact. Indeed, the stalks of $IC(X,\mathbb{Z}_2)$ are $\mathbb{Z}_2$ in degree -2, and 0 elsewhere. Whereas the stalks of $IC(X,\mathbb{F}_2)$ are $\mathbb{F}_2$ in degrees $-2$, $-1$ and zero elsewhere.

(These stalks calculations are explained in detail in arXiv:0901.3322.)

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