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I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful.

I want to construct an example of an intersection complex not concentrated in a single (natural) cohomology degree. Reading BBD, it seems the definition of intermediate extension needs to be made in the derived category, even to discuss intermediate extension of constant sheaves. So I think I can find an example of an open inclusion $j: U_0 \hookrightarrow X_0$ such that $j_{!*} \bar{\mathbb{Q}}_{\ell} [d]$ is not concentrated in degree $d$. I'm looking for a simplest example, but I'm having trouble verifying my work so far. So I would also appreciate if someone could point out any glaring errors in my reasoning (and lack thereof).

The first few attempts I made all seem to have $R^1 j_! \bar{\mathbb{Q}}_{\ell} = 0$, and so $Rj_! \bar{\mathbb{Q}}_{\ell} [d] = {}^p j_!\bar{\mathbb{Q}}_{\ell}[d]$; so ${}^pj_! \bar{\mathbb{Q}}_{\ell} [d] \hookrightarrow {}^p j_* \bar{\mathbb{Q}}_{\ell} [d]$; hence $j_{!*} (\bar{\mathbb{Q}}_{\ell} [d]) = j_!(\bar{\mathbb{Q}}_{\ell}) [d]$.

In particular, the above seems that to hold whenever $X_0$ is smooth and $j: U_0 \hookrightarrow X_0$ is the inclusion of dense open. So this is not the right direction.

Looking now at singular varieties, the first two examples that come to mind are $C_0 = \mathrm{Proj} (\mathbb{F}_q[S,T,U]/(T^2U-S^3))$ (projective cubic curve with a cusp) and $C'_0 = \mathrm{Proj} (\mathbb{F}_q[S,T,U]/(T^2U - S^3 - S^2U))$ (projective cubic curve with a node). Note the nonsingular loci $C_{ns, 0} \cong \mathbb{A}^1_0$ and $C'_{ns, 0} \cong \mathbb{G}_{m, 0}$. (Assume $\mathrm{char}(\mathbb{F}_q) > 2$ for $C'_0$.)

But in the case of $C_0$, taking $j: C_{ns,0} \hookrightarrow C_0$ to be the inclusion of the nonsingular locus, it appears to me that $Rj_!$ is exact. In particular, the stalk at a geometric point ${\bar{x}}$ lying over the node $x \in C_0$ $$ (R^1 j_! \bar{\mathbb{Q}}_{\ell})_{\bar{x}} = \lim_{\to} H^1 (U, j_! \bar{\mathbb{Q}}_{\ell}) \overset{(a)}{=} \lim_{\to} H^1_c (U \times_{C_{0}} C_{ns, 0}, \bar{\mathbb{Q}}_{\ell}) \overset{(b)}{\cong} H^1_c (\mathbb{A}^1, \bar{\mathbb{Q}}_{\ell}), $$ where the limit is taken over étale $U \to C_0$ over $\bar{x}$. Then we have $H^1_c(\mathbb{A}^1, \bar{\mathbb{Q}}_{\ell})$ vanishes by Poincaré dualtiy as $H^1 (\mathbb{A}^1, \bar{\mathbb{Q}}_{\ell}) = 0$. (I think $(a)$ holds by definition of $H^*_c$, and $(b)$ I can't justify.) So, assuming every link in this chain holds, we have $j_! = j_{!*}$, and I have not found my example.

But I believe—if my reasoning is at all accurate for $C_0$—that I have found an example in $j': C'_{ns, 0} \hookrightarrow C'_0$. Repeating the argument above, with $x' \in C'_0$ the self-intersection point, $$ (R^1 j'_! \bar{\mathbb{Q}}_{\ell})_{\bar{x}'} = \lim_{\to} H^1 (U, j'_! \bar{\mathbb{Q}}_{\ell}) = \lim_{\to} H^1_c (U \times_{C'_0} C'_{ns, 0}, \bar{\mathbb{Q}}_{\ell}) \cong H^1_c (\mathbb{G}_{m}, \bar{\mathbb{Q}}_{\ell}). $$ In this case, we have $H^1 (\mathbb{G}_{m}, \bar{\mathbb{Q}}_{\ell}) = \bar{\mathbb{Q}}_{\ell}(-1)$ (this is my understanding after reading Milne's and de Jong's notes on étale cohomology), and so $(R^1 j'_! \bar{\mathbb{Q}}_{\ell})_{\bar{x}'} = \bar{\mathbb{Q}}_{\ell}(1) \ne 0$. Since we have determined now that $j'_!$ is not exact, we need to calculate ${}^p j'_!$, ${}^p j'_*$, and finally calculate $j'_{!*}$. Should I keep going? Am I on the right track? Have I made glaring errors? Is there a reason $(b)$ should hold? What can I read to speed up my progress on these questions? I've read BBD and Kiehl-Weissauer, and a couple of less formal notes on perverse sheaves, and I've seen precious few examples in any detail. I recognize I haven't read the entire literature, so does anyone know where I should look next?

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    $\begingroup$ Tiny comment that isn't really that helpful: the singularity on $C_0$ is a cusp, not a node. $\endgroup$ Jul 22 '20 at 23:41
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    $\begingroup$ The most beautiful exposition of perverse sheaves is (arxiv.org/abs/0712.0349). There are examples there of intersection cohomology complexes supported in multiple degrees. $\endgroup$ Jul 23 '20 at 13:37
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Sorry I haven't read your entire question, which is a bit long. This is really just an extended comment to address the "where I should look next?" part. Suppose $X$ has an isolated singularity $x$, and $j:U\to X$ is the smooth complement. Then the formula on top of page 60 of BBD would simplify to $$j_{!*}\overline{\mathbb{Q}}_\ell[n]= (\tau_{\le n-1}\mathbb{R} j_* \overline{\mathbb{Q}}_\ell)[n]$$ where $n=\dim X$ and I'm using middle perversity. Now let $X$ be a sufficiently complicated singularity, a cone over an elliptic curve will do. Then this won't be a translate of a sheaf. Look at the stalk at $x$, it will have cohomology in 2 degrees.

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    $\begingroup$ Should the index of the truncation be $n-1$ rather than $n$? $\endgroup$
    – Anonymous
    Jul 23 '20 at 3:23
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    $\begingroup$ Thanks. Fixed.. $\endgroup$ Jul 23 '20 at 11:35

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