Definition 5.1.1.
Let \(A\) be a \(\delta\)-ring. An element \(d \in A\) is distinguished if \((p, d, \delta(d))\) is the unit ideal of \(A\text{.}\) That is, the intersection of the zero loci of \(p, d, \delta(d)\) on \(\Spec A\) is empty.
If \(A\) is \((p, d)\)-local, then \(d \in A\) is distinguished if and only if \(\delta(d)\) is a unit in \(A\text{;}\) in fact this is the definition used in [18] and [25]. The discrepancy will not affect the definition of a prism because the latter already includes a completeness hypothesis (see Definition 5.3.1). One confusing aspect of our definition is that units in \(A\) are always distinguished.
In many arguments that follow, we can reduce to the \((p,d)\)-local case by localizing \(A\) at \((p,d)\text{.}\) By Remark 2.4.10, the result is still a \(\delta\)-ring.