Let $A$ be an integral domain, $B$ is its fraction field. Can the projective dimension of the $A$-module $B$ be greater than $1$? This surely cannot happen if the spectrum of $A$ is countable (since then the presentation of $B$ as a countable direct limit of $A$-modules gives a length $1$ projective resolution for it).



What are the projective dimensions of big fraction fields?
Yes, this can happen. See for example <https://projecteuclid.org/download/pdf_1/euclid.nmj/1118801622>


