Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:46:19Z http://mathoverflow.net/feeds/question/72287 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72287/does-n-multiplication-maps-of-cohomology-groups-vanish-if-it-vanishes-at-the-0th Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology? Hiro 2011-08-07T14:33:45Z 2011-08-07T15:08:12Z <p>In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\$vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes.</p> <p>However, for a fixed object $F$, we cannot say that $f^{q} (F) = 0$ for every $q$ even if $f^{0} (F) = 0$.</p> <p>Now, in order to make this statement true, what kind of condition we need for the morphism $f \$? </p> <p>In particular, I am interested in the following situation:</p> <p>Let $n$ be an integer, $X$ be a scheme, $F\ \colon \ X _ {et} \to Ab \$ be a $n$-torsion etale sheaf on $X$, and $S=T=H^{\ast}(X,-)$, then can we say that the morphism $H^{q}(X,F)\to H^{q}(X,F)$ induced by the $n$-multiplication map vanishes? Thanks!</p>