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In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in particular, by choosing the Grothendieck topologies appropriately.

Subsequently I've looked into Voevodsky's paper "Motives over simplicial schemes" (Journ. K-theory 2010) and found that he seems to explain, in the introduction, that there are two kinds of motivic cohomology for singular varieties: the effective and the stable one. The effective motivic cohomology of a variety $X$ over a field $F$ with coefficients in an abelian group $A$ is given by $$ H^i_M(X,A(j))=Hom_{DM(F)}(M(X),A(j)[i])=Hom_{DM(X)}(\mathbb Z,A(j)[i]), $$ (see also section 4 of the same paper), while the stable motivic cohomology is $$ H^i_{stable}(X,A(j))=\varinjlim\nolimits_n Hom_{DM(X)}(\mathbb Z(n),A(n+j)[i]). $$ The reason for the two theories being different is that the cancellation theorem (claiming that the Tate twist is fully faithful) does not hold for motives over a singular variety.

As I tried to explain in my previous question, it seems that the Beilinson-Lichtenbaum etale descent rule for motivic cohomology with finite coefficients $$ H_M^i(X,\mathbb Z/m(j)) = H_{Zar}^i(X,\tau_{\le j}R\pi_*\mu_m^{\otimes j}), $$ where $\pi\colon Et\to Zar$, breaks down when $X$ is no longer smooth. Can it be true that an isomorphism like $$ H_{stable}^i(X,\mathbb Z/m(j)) = H_{Zar}^i(X,\tau_{\le j}R\pi_*\mu_m^{\otimes j}), $$ holds for arbitrary singular varieties $X$? Or can it be made to hold by replacing the pair of topologies (Zariski, etale) with a different pair, with one or both of the topologies including resolutions of singularities as covers, perhaps?

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I believe you can get results along these lines if you replace Zar with something called the $\ell$dh topology which is a modification of the cdh topology. This topology depends on $\ell$ which is a prime that is not equal to the characteristic you are working over and is a replacement for resolution of singularities over finite fields. The result is originally in Shane Kelly's PhD thesis:

And this is a consequence of corollary 4.25 of which states that every $M\mathbb{Z}_{(\ell)}$-module spectrum satisfies $\ell$dh descent.

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Thank you, I was not aware of these references. – Leonid Positselski Aug 29 '14 at 16:35
Hope they turn out useful! – Elden Elmanto Aug 29 '14 at 16:37

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