Let $X$ be a complex variety with a good stratification $S$ and consider the category $Perv_S(X)$ of sheaves perverse with respect to the given stratification (with middle perversity) lying in $D^b_S(X,k)$, where $k$ is a field. By good I mean that the derived pushforward from any stratum of a complex with constructible cohomology is constructible with respect to the same stratification.

I want to compute the Yoneda-$\text{Ext}$ groups for arbitrary perverse sheaves $F,G\in Perv_S(X)$. It is true that $\text{Ext}^i_{Perv_S(X)}(F,G)=\text{Hom}^i_{D^b_S(X,k)}(F,G)$ for $i=0,1$, since this is true for the heart of any $t$-structure. I know that this holds for $i>1$ if we do not fix a stratification, since the bounded derived category of $Perv(X)$ is equivalent to $D^b_c(X,k).$ Is the same result true for a fixed good stratification, or maybe with some additional hypotheses?

There is a paper on this topic by Beilinson: "On the derived category of perverse sheaves", where he idendtifies the bounded derived category of $Perv(X)$ with $D^b_c(X,k)$ (and also in a lot of other settings), and then writes:

a) The corresponding statement for the category of sheaves lisse along a fixed stratification is usually false,

but I am not sure whether this refers only to the $l$-adic case.


1 Answer 1


No, this is not a purely $\ell$-adic phenomenon.

Let $X = \mathbb P^1$, $S$ the stratification with one stratum so sheaves constructible with respect to this stratification are lisse and complexes constructible with respect to this stratification complexes with lisse cohomology.

Let $F = G = \mathbb Q[1]$, clearly constructible with respect to $S$ and perverse.

Then $Hom^2_{D^b_S(X,k) } (F,G) =Hom^2_{D^b(X,k) } (F,G) = H^2(X,\mathbb Q) = \mathbb Q$ but $\operatorname{Perv}(X)$ is just the category of lisse sheaves on $\mathbb P^1$, i.e. the category of vector spaces, and has all higher Ext groups vanish.


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