ag.algebraic geometry - How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K)

Background/Motivation: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note).

Let $R$ be a Noetherian normal domain and $K$ its quotient field. Then there is a natural map $f: Br(R) to Br(K)$. In case $R$ is regular, a well known result by Auslander-Goldman says that $f$ is injective. The natural question is when can we drop the regularity condition? But anything in the $Cl(R)/Pic(R)$ will be in the kernel of $f$, so we need that quotient to be $0$ to make it interesting.

Also, it is known that even assuming said quotient to be $0$, one has example like $R=mathbb R[x,y,z]_{(x,y,z)}/(x^2+y^2+z^2)$ which is UFD, but the kernel contains (I think) the quaternion algebra over $R$. The trouble in this case is that $mathbb R$ is not algebraically closed. In fact, if $R$ is local, then $ker(f)$ injects into $Cl(R^{sh})/Cl(R)$, here $R^{sh}$ is the strict henselization. Most of the examples with non-trivial kernel I know seem a little ad-hoc, so:

Question: Are there more general methods to generate examples of $R$ such that $f$ is not injective?

I would love to see answers with more geometric/arithmetic flavors. I am also very interested in the positive and mixed charateristics case. Thanks in advance.

• Next computer science - does the "convolution theorem" apply to weaker algebraic structures?
• Previous pr.probability - Estimate rate of real correct/wrong from 4 answers quiz.