Is there an analogue of Quillen-Suslin theorem for power series? Let $A$ be a regular noetherian ring over a field. Consider the power series ring $A[[T]]$. Are projective modules on $A[[T]]$ extended from $A$?



Is there a version of Quillen-Suslin-Lindel for power series?
This is true for any $A$ (assuming projective means "projective of finite type", otherwise I don't know). Let $P$ be projective of finite type over $A[[T]]$. Put $Q:=(P/TP)\otimes\_{A} A[[T]]$. Then $P\cong Q$.


More generally, let $R$ be a ring and $I$ an ideal contained in the Jacobson radical of $R$. If $P$ and $Q$ are projective of finite type over $R$, and $P/IP\cong Q/IQ$, then $P\cong Q$. This is easy but a reference is Lam's book *Serre's Problem on Projective Modules* (Springer), Corollary 1.6 on p. 10.


