This is a relatively elementary question in commutative algebra. My motivation comes from trying to compare some results in the theory of deformation quantisation. Let $k$ be a field, let $k[h]$ be a polynomial ring in one variable and let $k[[h]]$ be the ring of formal power series. If $M$ is a free $k[h]$-module then the $h$-adic completion of $M$ is flat over $k[[h]]$, and obviously it's $h$-adically complete (see Lemma 15.27.5 https://stacks.math.columbia.edu/tag/06LD for example). Someone once told me that the converse is true but I've never been able to piece together the proof.

Question: If $M$ is a flat, $h$-adically complete $k[[h]]$-module then is it true that $M$ can be constructed as the $h$-adic completion of some free $k[h]$-module?

Any references, proofs or counterexamples would be greatly appreciated. Thanks in advance.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.