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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).

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Yes, this can happen. See for example https://projecteuclid.org/download/pdf_1/euclid.nmj/1118801622

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