在收斂冪級數環上的有限生成投影模

Abstract

Serre’s conjecture 所探討的是在多項式環上的有限生成投影模。在這篇論文中我們要先探討在一般冪級數環上的有限生成投影模，再探討在收斂冪級數環上的行為，把兩者做個比較。

'Serre's Conjecture', referred to the famous statement made by J.-P. Serre in 1955, to the e ect that one did not know if nitely generated modules were free over a polynomial ring k[t1; : : : ; td], where k is a eld. Serre made some progress towards a solution in 1957 when he proved that every nitely generated projective module over a polynomial ring over a eld was stably free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin independently proved that the answer was a rmative. Kiran S. Kedlaya have proved the case in Tn, the Tate algebra. Lindel-Lutkebohmert and Mohan Kumar did the case of k[[X]][T], polynomial ring over formal power series ring. In this paper, we try to use the similar method to solve the case that the polynomial ring is replaced by Tn[T], polynomial ring over Tate algebra.