Cycle complexes with fixed support

PDF p. 13: Supported Chow group; PDF p. 16: Curve-supported motivic groups

Let $k$ be a field, $U$ a smooth finite-type $k$-scheme, $T\subset U$ a closed subset, $V=U\setminus T$, and $r\ge 0$. In the unnormalized Bloch complex $z^r(U,\bullet )$, define $z_T^r(U,n)$ to be the free subgroup generated by admissible integral cycles contained in $T\times {\mathrm{\Delta }}^n$, with differential induced from $z^r(U,\bullet )$. Put ${\mathrm{C}\mathrm{H}}_T^r(U,n)=H_n(z_T^r(U,\bullet ))$ and ${\mathrm{C}\mathrm{H}}_T^r(U)={\mathrm{C}\mathrm{H}}_T^r(U,0)$. Its cohomological indexing is $H_T^{p,r}(U;z)={\mathrm{C}\mathrm{H}}_T^r(U,2r-p)$.

The restriction $j^{\ast }:z^r(U,\bullet )\to z^r(V,\bullet )$ has homotopy-fiber model $F_n=z^r(U,n)\oplus z^r(V,n+1)$, with differential $(a,b)\mapsto (\mathrm{\partial }a,j^{\ast }a-\mathrm{\partial }b)$. The inclusion $z_T^r(U,\bullet )\to F$ is a quasi-isomorphism, so the displayed supported groups are also the homology groups of this derived restriction fiber, with the same indexing. A quasi-isomorphism is a chain map inducing isomorphisms on all homology groups.

Bloch cycle complexes and higher Chow groups

Face boundaries preserve closed support

Open restriction of admissible cycles

The kernel of restriction is the supported subcomplex

Restriction modulo its kernel

Kernel models the homotopy fiber when the cokernel is acyclic

Higher-cycle localization including nonseparated schemes

By [F1] and [F2], the displayed groups form a subcomplex, so its homology quotient is defined. The hypotheses of [F4] hold for the stated field, smooth finite-type scheme and open complement; [F2] identifies the kernel, and [F3] records the quotient used in the comparison. Thus [F4] gives the stated quasi-isomorphism. The cohomological notation is a reindexing of these homology groups. This construction depends only on the closed subset $T$ and does not identify the whole support fiber with the image of ordinary Chow groups of $T$ in those of $U$. A comparison with a separately defined motivic cohomology theory requires its own compatible comparison theorem.