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I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.

  1. For the algebraic cobordism theory $MGL$ and a smooth variety $X$ over a (perfect?) field is it true that $MGL^{2n+i,n}(X)=0$ for any $n\in \mathbb{Z},i>0$? More generally, are there any reasonable restrictions on a (oriented?) ring spectrum $E$ in $SH$ that ensure the vanishing of
    $MGL^{2n+i,n}(E)$. In particular, is this question related with some sort of effectivity for spectra?

  2. It is well known that 'shifts and twists' of the spectra $\Sigma(X_+)$ generate $SH$, where $X$ runs through all smooth $k$-varieties. If the characteristic of $k$ is $0$, resolution of singularities yields that it suffices to consider only smooth projective varieties here. Now, what statements of this sort are known for $k$ of characteristic $p>0$? I suspect that that one can deduce a similar result for $SH\otimes \mathbb{Z}_{(l)}$ for any prime $l\neq p$, ffrom the Gabber's l'-alterations theorem. Is this true? If this is too difficult, can one prove a similar statement for the triangulated category of $MGL$-modules?

What are the best references for these questions?

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up vote 8 down vote accepted

(1) is true if $char(k)=0$. This follows from a combination of results. First of all, it is true over any field that the spectrum $MGL$ is connective, which means that

$$MGL^{p,q}(X)=0$$

if $p>q+dim(X)$, $X\in Sm/k$ [1, Cor. 2.9]. (Slightly more is true: for any $p\geq q+dim(X)$, the orientation map $MGL\to H\mathbb{Z}$ induces an isomorphism $MGL^{p,q}(X)\cong H\mathbb Z^{p,q}(X)$ [1, Lem 6.4].)

Second, we have the Hopkins-Morel equivalence [1, Thm. 6.11]

$$MGL/(x_1,x_2,\dots) \simeq H\mathbb Z.$$

Assuming this, Spitzweck has shown in [2] that that the slices of MGL are given by

$$s_rMGL\simeq \Sigma^{2r,r}H(MU_{2r}),$$

Moreover, he gives an explicit description of the $r$-effective cover $f_rMGL$ as a homotopy colimit of spectra of the form $\Sigma^{2i,i}MGL$ for $i\geq r$, which shows that $f_rMGL$ is also $r$-connective (being a homotopy colimit of $r$-connective spectra). Because the homotopy $t$-structure is right complete [1, Cor. 1.4], this implies that

$$\mathrm{holim}_{r\to\infty}f_rMGL=0.$$

Now, let $E=\Sigma^{-p,-q}\Sigma^\infty X_+$ with $p>2q$. Take any map $E\to MGL$. Since $H^{p+2r,q+r}(X,A)=0$ for any abelian group $A$ and $r\in\mathbb Z$, this map lifts through all stages of the slice filtration, hence comes from a map $E\to \mathrm{holim}_{r\to\infty}f_rMGL=0$. QED.

(Incidentally, this shows that there is a strongly convergent spectral sequence $H^{\ast\ast}(X,MU_{2\ast})\Rightarrow MGL^{\ast\ast}(X)$.)

If $char(k)>0$ ($k$ need not be perfect), the Hopkins-Morel equivalence is also known if $char(k)$ is inverted [1], so we can at least deduce that $MGL^{p,q}(X)$ is $char(k)$-torsion for all $p>2q$, $X\in Sm/k$ (and it is zero if $p>q+dim(X)$ by connectivity, so for fixed $X$ and $q$ at most finitely many of these groups can be nonzero).

Some comments about (2): if you go through the proof of the characteristic zero case in [3] and try to use Gabber's theorem instead of resolution of singularities, at some point in the proof an isomorphism is replaced by a finite flap map $f: Y\to X$ of degree prime to a given prime $l\neq p$, and the proof will work if that map has a section. Even if you work $\mathbb Z_{(l)}$-locally, you still need a map $g: X\to Y$ such that $fg=deg(f)\cdot\mathrm{id}$, and I don't see why you'd have such a map in $SH\otimes\mathbb Z_{(l)}$. But for $MGL_{(l)}$-modules I guess that the Gysin map [4] should work.

[1] M. Hoyois, From algebraic cobordism to motivic cohomology (pdf)

[2] M. Spitzweck, Relations between slices and quotients of the algebraic cobordism spectrum (pdf)

[3] O. Röndigs, P. Østvær, Modules over motivic cohomology (pdf)

[4] F. Déglise, Around the Gysin triangle II (pdf)

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Thank you very much for such a detailed answer!! –  Mikhail Bondarko Nov 11 '12 at 8:34
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