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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. Weight filtration for smooth analytic manifolds); 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 $E_2$.


  1. 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 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 (Does the (torsion) Zariski cohomology of a (singular) hyperplane section of a smooth projective variety vanish (in small degrees)?).

  2. 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).

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