According to Chapter 4 of Beilinson, Bernstein, and Deligne's "Faisceaux Pervers" (Asterisque 100, 1980) the inverse image $Rf^\*$ with respect to a finite morphism $f$ is right t-exact with respect to the perverse t-structure. Is there an example that shows that it is not necessarily t-exact? Is there a class of finite morphisms other than the etale morphisms where it is t-exact?



Example to show that the inverse image under a finite morphism is not t-exact with respect to the perverse t-structure
By Noether normalisation any affine variety admits a finite map $f$ to affine space. Hence it is enough to find an example of an affine variety for which the constant sheaf is not perverse. (In this case $f^\* \mathbb{Q}\_{\mathbb{A}^n}[n]$ will provide a counter-example.)


Now it is easy to find examples where the constant sheaf is not perverse. For example any affine cone $C\_X$ over a smooth projective variety $X$ provides a counter-example as long as the cohomology of $C\_X \setminus \{ 0 \}$ doesn't look like that of a sphere. So I guess the cone over an elliptic curve would do it.


Edit (...not quite:) see Daniel's comment below.


