Mixed motivic sheaves, category theory, and study of cycle complexes

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Quasi DG category, a generalization of DG category given by the principal researcher, is a notion that appears in the theory of mixed motivic sheaves. We developed the basic theory of quasi DG categories. The mail features are basic axioms of a quasi DG category, the notion of C-diagrams with values in a quasi DG category, additivity of the function complexes, the construction of the quasi DG category of C-diagrams, and the proof that the homotopy category of the quasi DG category of C-diagrams has the structure of a triangulated category.In particular, when the function complex (depending on n objects) that constitutes part of the notion, is additive in each variable, the same property was proven to hold for the function complex for C-diagrams.We described the Chow cohomology of an algebraic surface using its resolution of singularities. In particular, we gave a condition for Chow cohomology and homology to coincide.We showed the blow-up formula for the motives and higher Chow groups of a quasi-projective variety.