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 $A$?
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1$\begingroup$ If $A$ is local, $A[[T]]$ is local. $\endgroup$– abxCommented Feb 13, 2021 at 17:13
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$\begingroup$ Ah! you are right. So it is not quite what I was looking for. I'll remove it. $\endgroup$– user127776Commented Feb 13, 2021 at 17:15
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1 Answer
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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.
answered Feb 14, 2021 at 11:26