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I'm absolutely new to this stuff I'm asking about, so I hope this is not nonsense.

If X is a smooth scheme over a perfect field k, I can study its motivic cohomology in the sense of Voevodsky and Morel. More precisely, to $\mathbb{Z}$ there is associated an Eilenberg-MacLane T-Spectrum. I'll write $H^{p,q}(X,\mathbb{Z})$ for the motivic cohomology of $X$ defined by Hom into this Spectrum.

One fact for smooth $X$ now is that $$ H^{2p,p}(X,\mathbb{Z}) = CH^p(X). $$

Now I've got two questions:

1) Can I feed something singular into this Voevodsky-Morel machine?

2)If the answer to 1 is yes, are there some maps from $ H^{2p,p}(X,\mathbb{Z})$ to $CH^p(X)$? Maybe defined by some spectral sequence? I'm really interested just in some maps, no isomorphisms.

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1 Answer 1

up vote 21 down vote accepted

The answer to question 1) is yes. However, Chow groups do not form what we should call a cohomology theory, but are part of a Borel-Moore homology theory. This ambiguity comes from the fact that, by Poincaré duality, motivic cohomology agrees with motivic Borel-Moore homology for smooth schemes (up to some reindexing), while they are quite different for non-regular schemes. You have the same phenomena in K-theory: K-theory of vector bundles (which defines a kind of cohomology) agrees with K-theory of coherent sheaves (which defines the corresponding Borel-Moore homology) for regular schemes. That said, they are plenty of ways to extend motivic cohomology into a cohomology theory for possibly non-singular schemes; they all agree in char. 0, or if you work with rational coefficients. As you seem to be interested by Chow groups, it seems that what you want is motivic Borel-Moore homology for possibly non-singular schemes (which agrees with Bloch's higher Chow groups, whence gives classical Chow groups for the appropriate degrees).

To define motivic Borel-Moore homology of $X$ (separated of finite type over a field $k$ of char. 0, unless you tensor everything by $\mathbf{Q}$, or admit resolution of singularities in char. $p$), you may proceed as follows: there is a motive with compact support $M_c(X)$ in $DM(k)$, and we define

$H^{BM}_i(X,\mathbf{Z}(j))=Hom_{DM(k)}(\mathbf{Z}(j)[i],M_c(X)).$

If you prefer the six operations version, for $f:X\to Spec(k)$ a (separated) morphism of finite type, we have

$H^{BM}_i(X,\mathbf{Z}(j))\simeq Hom_{SH(k)}(f_!f^*\Sigma^\infty(Spec(k)_+)(j)[i],H\mathbf{Z})$

where $H\mathbf{Z}$ denotes the motivic Eilenberg-MacLane spectrum in $SH(k)$. To answer your question 2), the link with Bloch's higher Chow groups is that

$H^{BM}_i(X,\mathbf{Z}(j))\simeq CH^{d-j}(X,i-2j)$

for $X$ equidimensional of dimension $d$, and that, for any separated $k$-scheme $X$,

$H^{BM}_{2j}(X,\mathbf{Z}(j))\simeq A_j(X)$

where $A_j(X)$ stands for the group of $j$-dimensional cycles in $X$, modulo rational equivalence.

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That's amazing, thanks a lot! The last formula is exactly what I was looking for. Mathoverflow rules! –  Timo Schürg Oct 15 '10 at 9:59
    
I added backticks ` to get your formulas to render properly. –  Donu Arapura Oct 15 '10 at 12:43

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