On tame ramification and centers of \(F\)-purity (Q6641557)

In elementary school [\textit{R. Hartshorne}, Algebraic geometry. New York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0367.14001), p.~299), we learn what it means for a finite morphism between algebraic curves (over an algebraically closed field) to have tame ramification at a given point. However, this is insufficient for more advanced purposes.\N\NIn the present paper, the authors consider a finite ring extension \(R \subset S\) and an \(R\)-linear map \(\mathfrak T \colon S \to R\) (Def.~3.2), and they define what it means for \(S/R\) to be tamely \(\mathfrak T\)-ramified over a prime ideal \(\mathfrak p \subset R\) (Def.~3.19). If \(R\) is normal and the corresponding field extension is separable, then the trace map gives a canonical choice for \(\mathfrak T\).\N\NThey show that this new notion extends the classical notion of tameness for Dedekind domains (e.g.~for curves as above) (Prop.~3.33). If instead we consider \(S = F^e_* R\) (where \(F\) denotes the Frobenius map of \(R\)), then tame ramification yields the notion of center of \(F\)-purity [\textit{K. Schwede}, Math. Z. 265, No. 3, 687--714 (2010; Zbl 1213.13014)]. Many more technical properties of tame ramification are also established.