So I am trying to understand a statement in the proof of Corollary 5.2.3 in `A proof of Jantzen conjectures' (a copy of the paper can be found at http://www.math.harvard.edu/~gaitsgde/grad_2009/).

The starting assumptions of the Corollary are:

`Let $M_1, M_2$ be pure perverse sheaves of weights $w_1, w_2$ that are both $*$- and $!$-pointwise pure. Suppose that $Ext^1_{mixed}(M_1, M_2)\neq 0$. Then ...'

The first line of the proof says:

`Clearly either $Y_1 \subset Y_2$ or $Y_2 \subset Y_1$ (otherwise $Ext^1 = 0$)'

Here $Y_i = Supp(M_i)$. This statement confuses me. The ordinary ( = non-mixed) $Ext^1$ group should be the extensions between the restrictions of $M_1$ and $M_2$ to the intersection $Y_1 \cap Y_2$. I don't see why this should vanish if either $Y_1$ isn't contained in $Y_2$ or vice versa. Anyway, even if the unmixed group does vanish, without vanishing of the unmixed $Hom$ group I don't see how I would get $Ext^1_{mixed}$ vanishes.

Presumably I am making a stupid error here and both of the unmixed groups above do vanish? Any comments would be appreciated.