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Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...

Let $X$ be an algebraic variety over a field $k$. Bloch defines the "algebraic singular complex" using the algebraic simplices $$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...

Intro. I would be deeply grateful if someone could please clarify the following to me. The question. (the main point is (4)) Let $X$ be a smooth projective variety over $\mathbf{C}$, and $\mathbf{Z}(...