Bloch cycle complexes and higher Chow groups

PDF p. 4: Smooth comparison with Bloch cycles

Let $k$ be a field, $X$ smooth of finite type over $k$, and $r\ge 0$. For $n\ge 0$ put ${\mathrm{\Delta }}^n=\mathrm{Spec}k[t_0,\dots ,t_n]\mathrm{/}({\sum }_a{t}_a-1)$. A nonempty coordinate face is obtained by setting a proper subset of the coordinates to zero. An integral closed subscheme $W\subset X\times {\mathrm{\Delta }}^n$ of codimension $r$ is admissible if every component of its intersection with every such face has codimension $r$ in that face; empty intersections are allowed. Codimension is measured on the ambient irreducible component containing the generic point. Define $z^r(X,n)$ as the free abelian group on the admissible subschemes.

For $n\ge 1$, identify $\{t_i=0\}$ with ${\mathrm{\Delta }}^{n-1}$ by retaining the remaining coordinates in increasing order and set $d_i^n[W]={\sum }_T{\mathrm{\ell }}_{{\mathcal{O}}_{W,{\eta }_T}}({\mathcal{O}}_{W,{\eta }_T}\mathrm{/}(t_i))[T],$ where $T$ ranges over the components of $W\cap \{t_i=0\}$ and ${\eta }_T$ is its generic point. The empty sum is zero. Extend by integral linearity and set ${\mathrm{\partial }}_n={\sum }_{i=0}^n(-1{)}^i{d}_i^n$. Put $z^r(X,n)=0$ for $n<0$ and ${\mathrm{\partial }}_n=0:z^r(X,n)\to z^r(X,n-1)$ for every $n\le 0$. The resulting homological complex is Bloch's cycle complex $z^r(X,\bullet )$, and its higher Chow groups are ${\mathrm{C}\mathrm{H}}^r(X,n)=H_n(z^r(X,\bullet ))=\ker {\mathrm{\partial }}_n\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{n+1}\mathrm{.}$ In particular they vanish for $n<0$. This is the unnormalized complex; no quotient by degenerate simplices is taken.

Admissible face restrictions and their cycle identities

Alternating face boundaries square to zero

Homology object of a chain complex

The local rings in the coefficient formula need not be discrete valuation rings and the cycles need not be smooth. The proper single-face calculation in [F1] supplies exactly the finite lengths shown. The cycle-level compatibility in [F1] uses the explicitly imported simplicial-cycle input locally on integral smooth affine opens; no quotient by rational equivalence is used. Thus all face maps are defined, and [F2] gives ${\mathrm{\partial }}^2=0$ and makes the homology quotient well-defined. This definition does not assert the separate comparison with motivic cohomology or the identification of degree zero with the classical Chow group.