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I am looking for a reference for the following result: If $X$ is an algebraic variety and $$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$ is a stratification (edit: filtration) by closed subvarieties, then there is a spectral sequence $$E^{p,q}_1 = H^{p+q}_c(T_p \setminus T_{p-1}) \implies H^{p+q}_c(X).$$ Looking for a reference I have found Dan Petersen using it several times here in MathOverflow, such as here, here and here. So what is the specific result and where can I find a proof of it?

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    $\begingroup$ I don't know a reference (in fact had not heard of this before), but I assume what you do is take the filtration of sheaves $0 = \mathbf Z_{U_n} \subseteq \cdots \subseteq \mathbf Z_{U_{-1}} = \mathbf Z_X$ and take the spectral sequence associated with this filtration Tag [0BKK] (where $U_i = X \setminus T_i$). (I'm a little unhappy about this because I want to use sheaf cohomology with compact support, not usual sheaf cohomology.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 16, 2021 at 16:24
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    $\begingroup$ @R.vanDobbendeBruyn The filtration still makes sense after extending by zero to a compactification, so applying the definition of compactly supported cohomology there is no problem. $\endgroup$
    – Will Sawin
    Commented Apr 16, 2021 at 19:06
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    $\begingroup$ If you just want a reference, you can look at p 575 of my paper "The Leray spectral sequence is motivic", and then specialize, but in terms of understanding, it's probably simpler to work it for yourself along the lines of Will Sawin's first comment. PS I'm pretty sure Hatcher's title is Spect. Seq. in Algebraic Topology, not that it matters. $\endgroup$
    – Donu Arapura
    Commented Apr 17, 2021 at 0:18
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    $\begingroup$ I guess this question is for me. It would be easier to answer if you could say what you're unhappy about with the answer to mathoverflow.net/questions/233067 . That answer contains both the argument in Remy's comment and a citation to Donu's paper... $\endgroup$
    – Dan Petersen
    Commented Apr 17, 2021 at 4:40
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    $\begingroup$ Try Dimca, Sheaves in Topology, Section 2.3 and thereabouts. $\endgroup$
    – Balazs
    Commented Apr 18, 2021 at 19:04

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