Voevodsky's 'split standard triple' argument: an explanation; does it work with \$Z/nZ\$-coefficients? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:13:34Z http://mathoverflow.net/feeds/question/114193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114193/voevodskys-split-standard-triple-argument-an-explanation-does-it-work-with Voevodsky's 'split standard triple' argument: an explanation; does it work with \$Z/nZ\$-coefficients? Mikhail Bondarko 2012-11-22T22:42:24Z 2012-11-26T09:14:43Z <p>For varieties over a perfect field (of some characteristic \$p\$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see <a href="http://books.google.ru/books?id=TzUmk87bN9cC&amp;pg=PA85&amp;lpg=PA85&amp;dq=Voevodsky+standard+triple&amp;source=bl&amp;ots=lqFZojfUU-&amp;sig=Jtt57xtmlQwX7XvShXHPyaKVP68&amp;hl=ru&amp;sa=X&amp;ei=SaeuUIzEGZH24QSEwIG4DA&amp;redir_esc=y#v=onepage&amp;q=Voevodsky%20standard%20triple&amp;f=false" rel="nofollow">http://books.google.ru/books?id=TzUmk87bN9cC&amp;pg=PA85&amp;lpg=PA85&amp;dq=Voevodsky+standard+triple&amp;source=bl&amp;ots=lqFZojfUU-&amp;sig=Jtt57xtmlQwX7XvShXHPyaKVP68&amp;hl=ru&amp;sa=X&amp;ei=SaeuUIzEGZH24QSEwIG4DA&amp;redir_esc=y#v=onepage&amp;q=Voevodsky%20standard%20triple&amp;f=false</a>). One says that a triple is split over \$U\$ if a certain line bundle is trivial (see Definition 11.11); this has certain consequences for cohomology of varieties with coefficients in a homotopy invariant presheaf with transfers \$F\$ (see Proposition 11.15).</p> <p>My question is: if \$nF=0\$, is it sufficient to consider triviality modulo \$n\$ instead, i.e. could one replace all the Picard groups considered in this section by their \$\mathbb{Z}/n\mathbb{Z}\$-analogues? I looked at the proofs, and it seems that the answer is positive; yet possibly I miss something.</p> <p>Alternatively, one can find Voevodsky's (Mazza's-Weibel's) book here <a href="http://www.claymath.org/library/monographs/cmim02c.pdf" rel="nofollow">http://www.claymath.org/library/monographs/cmim02c.pdf</a> an earlier exposition of this argument can be found in section 4 of <a href="http://www.math.illinois.edu/K-theory/0368/s3.pdf" rel="nofollow">http://www.math.illinois.edu/K-theory/0368/s3.pdf</a></p> <p>Upd. Possibly, a more clear reference to Voevodsky's argument is <a href="http://www.math.uiuc.edu/K-theory/0832/motvo.pdf" rel="nofollow">http://www.math.uiuc.edu/K-theory/0832/motvo.pdf</a>, section 5.1.1; yet I would be deeply grateful for any 'explanation' of this reasoning. </p>