Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T15:57:40Z http://mathoverflow.net/feeds/question/82564 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82564/does-artins-vanishing-hold-for-e-2-weight-pieces-for-torsion-cohomology-of Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties? Mikhail Bondarko 2011-12-03T16:16:53Z 2011-12-06T08:31:25Z <p>Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for $i>n$ (and any $l>0$). Now, one can consider the weight spectral sequence $S(H,X)$ converging to $H^\ast(X)$ (using the results of Guillen-Navarro Aznar, Gillet-Soule, or my own 'motivic' version of those; cf. <a href="http://mathoverflow.net/questions/21236/weight-filtration-for-smooth-analytic-manifolds" rel="nofollow">http://mathoverflow.net/questions/21236/weight-filtration-for-smooth-analytic-manifolds</a>); this is canonical and 'very much functorial' starting from $E_2$. My question is: does $E_2^{pq}(S(H,X))$ necessarily vanish for $p+q>n$? Certainly, Artin's vanishing yields that $E_{\infty}^{pq}(S(X))=0$ in this case (and also, that $E_2^{pq}=0$ for rational coefficients); yet the weight spectral sequence for cohomology with torsion (or integral) coefficients (probably) does not have to degenerate at $E_2$.</p> <p>Remarks. </p> <ol> <li><p>Since singular and etale cohomology theories can be extended to Voevodsky's motives, there also exists such a (functorial) extension for (Deligne's) weight spectral sequences; see section 2.4 of <a href="http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.2672v4.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.2672v4.pdf</a>. Then loc.cit. yields: for any fixed $q\in \mathbb{Z}$ the correspondence $H_q^{\ast}:X\mapsto E_2^{\ast q}(S(H,X))$ yields a cohomology theory for motives (this is the so-called virtual t-truncation of $H$ with respect to the Chow weight structure). In particular, we obtain: a positive answer to my current question automatically yields a positive answer to my previous one (http://mathoverflow.net/questions/82432/does-the-torsion-zariski-cohomology-of-a-hyperplane-section-of-a-smooth-project). </p></li> <li><p>It seems that the cohomology theory $H_q^{\ast}$ mentioned above does not satisfy etale descent (since otherwise it would be given by a complex of torsion etale sheaves with transfers, and those are constant ones only).</p></li> </ol>