Stony Brook University Problem Set 3 Jason Starr Fall 2014 MAT 614 Problem Set

(a) Prove that there exists a (left-right) inverse σ(t) in K [[t]] to the image of λ(t) if and only if a0 is invertible in K. In this case, prove that the inverse σ(t) is unique. More generally, for every element a0− a1t+ . . . in K [[t]], prove that the element is invertible if and only if a0 is invertible in K, in which case the inverse is unique. (b) Assume that λ(t) ∈ K[t] has an inverse σ(t) in K [[t]]. Prove that the following sequence of K[t]-modules is well-defined and exact, where the arrows are scaling by the given element in K [[t]].