Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:13:15Z http://mathoverflow.net/feeds/question/108819 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108819/is-there-a-mayer-vietoris-spectral-sequence-of-motivic-cohomology-for-closed-cove Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings? Hiro 2012-10-04T13:59:10Z 2012-10-04T19:12:06Z <p>For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):</p> <p>$E_{1}^{p,q}=\oplus_{i_{0}&lt; \cdots &lt; i_{p}} H_{ Y_{i_{0} \cdots i_{p} }}^{q} (X, F) \Longrightarrow H_{Y}^{q-p}(X, F).$</p> <p>Here, $X$ is a scheme, $Y\to X$ is a closed subscheme, and $Y_{i}$'s are a closed covering of $Y$.</p> <p>$Y_{i_{0} \cdots i_{p}}$ denotes $Y_{i_{0}} \cap \cdots \cap Y_{i_{p}}.$</p> <p>$F$ is an etale sheaf on $X$, and $H_{Z}^{\ast}$ denotes the cohomology with supports in a closed subscheme $Z$ on $X$.</p> <p>My questions are: is there any spectral sequence of the similar form for motivic cohomology (or higher Chow group)?</p> <p>If yes, then how one can prove it?</p> <p>If no, then why is it?</p> <p>Please give me any advice.</p> http://mathoverflow.net/questions/108819/is-there-a-mayer-vietoris-spectral-sequence-of-motivic-cohomology-for-closed-cove/108847#108847 Answer by Mikhail Bondarko for Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings? Mikhail Bondarko 2012-10-04T19:12:06Z 2012-10-04T19:12:06Z <p>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 <a href="http://www.math.uiuc.edu/K-theory/0529/glr.pdf" rel="nofollow">http://www.math.uiuc.edu/K-theory/0529/glr.pdf</a></p>