# On motivic cohomology with compact support

For a smooth projective variety $X$ and its closed non-smooth subvariety $Z$ I would like to say that a cone of the morphism between the motivic cohomology of $Z$ and those of $X$ is the motivic cohomology of $X\setminus Z$ with compact support. Is this statement known over a positive characteristic (perfect) base field (at least, when $Z$ is a smooth normal crossing divisor; I am interested in $l$-adic motivic cohomology)? Also, I would like here to relate the motivic cohomology of $Z$ with those of its Henselization in $X$; cf. On (the cohomology of) Hensel pairs

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I'm not sure what you mean by "motivic cohomology", since I've run into different kinds : Nisnevich, Beilinson, Voedvodsky motivic cohomology... I'm also not sure what you mean by "motivic cohomology with compact support". Do you define this using duality ? Anyway, as far as I can see, if I fill in the most likely (for me) definitions, your question should have a formal answer if you have triangulated categories of motives over "nice" schemes of any characteristic with the 4 operations and a good formalism of duality. I think this is known with with Q coefficents (eg by Cisinski-Deglise). – Alex Sep 21 '11 at 19:07
Unfortunately, I am interested in torsion (l-adic) coefficients.:) I wonder whether the methods of Voevodsky (that use resolution of singularities) together with the prime-to-l alterations of Gabber yield the result over a perfect characteristic $p\neq l$ field. – Mikhail Bondarko Sep 21 '11 at 19:38
That seems very likely. Whether this is written anywhere is another problem. You could try asking Cisinski, Deglise or Ayoub directly. – Alex Sep 21 '11 at 19:46