The preparation theorem and the freeness of 𝐴[[𝑋]]/𝐼

Abstract

Listen

Translate article

Let I I be a nonzero ideal of A [ [ X ] ] A[[X]] , the ring of formal power series over a commutative Noetherian ring A A . These are equivalent: (i) I I is generated by a single series f = a 0 + a 1 X + … f = {a_0} + {a_1}X + \ldots such that for some s , a s s,\;{a_s} is a unit, the first s s coefficients a 0 , … , a s − 1 {a_0}, \ldots ,{a_{s - 1}} of f f lie in the Jacobson radical of A A and A A is complete in the adic topology defined by the ideal they generate. (ii) A [ [ X ] ] / I A[[X]]/I is a free A A -module.