Suslin Homology, Chow Varieties, and \(h\)-topology
Incapability of Zariski/étale topology on modelling Chow sheaves
(This part is essentially nonsense)
For a given scheme \(X\) together with a Zariski cover \(\{ U_i \to X \}\), any sheaf \(\mathscr{F}\) on \(X\) should satisfy \[ \mathscr{F}(X)\rightarrow\coprod\mathscr{F}(U_i)\substack{\rightarrow \\ \rightarrow}\coprod\mathscr{F}(U_i\times_X U_j) \] where the diagram represents an equalizer.
Now take \(C_n(X)\) for \(X\) (large site, any scheme over \(\mathbb{Z}\)) to be the set of points of height \(n\) on \(X\). Such a presheaf has no chance for becoming a Zariski sheaf or an étale sheaf, since there is no correct restriction \[C_n(X)\rightarrow C_n(U)\] for point \(p\in X\) when \(U\) avoids \(p\). To mitigate such a problem, a trivial measure is to take not the sheaf of points, but the sheaf of free abelian groups on points of \(X\) (or to say, cycles), in which case \(p\in X\) restricts to \(0\) in \(C_n(U)\) when \(U\) avoids \(p\). However, the sheaf condition still fails. Take \(X=\mathbb{P}^1\) and \(U=X\backslash\{0\}\), \(V=X\backslash\{\infty\}\), we hope the sequence \[ 0\rightarrow C_0(X)\rightarrow C_0(U)\times C_0(V)\xrightarrow{u} C_0(U\cap V) \] could be exact. However, just note that the
\(qfh\)-topology
A morphism of schemes \(p: X \to Y\) is called a if the underlying Zariski topological space of \(Y\) is a quotient space of the underlying Zariski topological space of \(X\) (i.e., \(p\) is surjective and a subset \(A\) of \(Y\) is open if and only if \(p^{-1}(A)\) is open in \(X\)). \(p\) is called a if for any \(Z \to Y\) the morphism \[p_Z : X \times_Y Z \to Z\] is a topological epimorphism.
An h-covering of a scheme \(X\) is a of morphisms of finite type \(\{ p_i : X_i \to X \}\) such that \[\coprod p_i : \coprod X_i \to X\] is a universal topological epimorphism.
A qfh-covering of a scheme \(X\) is an h-covering \(\{ p_i \}\) such that all the morphisms \(p_i\) are quasi-finite.
h-coverings (resp. qfh-coverings) define a pretopology on the category of schemes, h-topology (resp. qfh-topology) is the associated topology.