Chow schemes in mixed characteristic

In this thesis we compare Suslin–Voevodsky’s sheaves of proper effective relative cycles with presheaves representable by certain monoid objects. We give two results in this direction; the first describes a higher dimensional analogue of Suslin–Voevodsky’s comparison between relative zero cycles and the graded monoid of symmetric powers (Thm. 6.8 of "Singular homology of abstract algebraic varieties") and the second is a new proof of a direct generalization of loc.cit. The key component of our efforts is a theorem, proved on the way, telling us that after restricting ourselves to seminormal schemes the morphism from the presheaf represented by a commutative-monoid object (satisfying reasonable assumptions) to its sheafification in the h-topology, becomes an isomorphism after appropriate extension of scalars. This thesis was written with the additional purpose of providing a self contained presentation of the theory of relative cycles and the construction of the Chow scheme. To achieve this we recall many of the definitions and results from the literature and occasionally expand on the explanations found there