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?



Intermediate extension functor exact?
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.)


