Bass-Quillen Conjecture And Lower K-groups Over Coherent Rings

The Bass-Quillen Conjecture combines many branches of mathematics such ascommutative algebra, homological algebra, algebra K-theory, matrix theory andalgebraic geometry. It is involving not only a new theory of algebra, but also thefrontiers of geometry theory and the important achievements on integral theory.The research on it left human precious theoretical knowledge and thinking methodsin mathematical culture. An attention with particular worth, is that the studyon Bass-Quillen Conjecture contributes greatly to the development of the algebraK-theory.In Chapter 1 of this paper, we summarize the history of the zero-divisor graph,the background and main results of this paper. At the same time, we gave the basicnotation of ring theory, graph theory and some new definitions.In Chapter 2, we firstly introduce the origin of Bass-Quillen Conjecture in section1. Nextly, we analysis two classic solutions of Serre’s Conjecture, and obtain someequivalent statements of Bass-Quillen Conjecture in section 2. Finally, we alsosummarize the main results on Bass-Quillen Conjecture in section 3.In section 3.1, three kinds of equivalent definitions of K0R will be introduced,with some useful simple properties. In section 3.2, there is some facts on finitelypresent modules, coherent modules, coherent rings, regular rings, stably coherentrings etc. In section 3.3, we will show an isomorphism of K0 group between a stably coherent regular ring R and the polynomial ring over R, which will greatly promotethe research into Bass-Quillen Conjecture over non-Noetherian ring.In Chapter 4,we will transfer to investigate another class of lower K-groups Insection 4.1, there is the straightaway definition and properties of K1 group. Insection 4.2, as a tool, the category with exact sequences will be introduced. Insection 4.3, we show some properties on K1 group.