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For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):

$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots i_{p} }}^{q} (X, F) \Longrightarrow H_{Y}^{q-p}(X, F).$

Here, $X$ is a scheme, $Y\to X$ is a closed subscheme, and $Y_{i}$'s are a closed covering of $Y$.

$Y_{i_{0} \cdots i_{p}}$ denotes $Y_{i_{0}} \cap \cdots \cap Y_{i_{p}}.$

$F$ is an etale sheaf on $X$, and $H_{Z}^{\ast}$ denotes the cohomology with supports in a closed subscheme $Z$ on $X$.

My questions are: is there any spectral sequence of the similar form for motivic cohomology (or higher Chow group)?

If yes, then how one can prove it?

If no, then why is it?

Please give me any advice.

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1 Answer 1

up vote 2 down vote accepted

I believe that cycle groups do not behave nicely for singular varieties. On the other hand, one could define certain reasonable motivic cohomology via Voevodsky's motives. In any case, look at (section 2 of) the paper http://www.math.uiuc.edu/K-theory/0529/glr.pdf

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