On Segre products, \(F\)-regularity, and finite Frobenius representation type (Q6563083)

Let \(R=\oplus_{n \geq 0} R_n\) and \(S=\oplus_{n \geq 0} S_n\) be finitely generated \(\mathbb{N}\)-graded rings over a field \(R_0=\mathbb{F}=S_0.\) The Segre product of \(R\) and \(S\) is the \(\mathbb{N}\)-graded ring\N\[\NR \# S:=\bigoplus_{n \geq 0} R_n \otimes_\mathbb{F} S_n.\N\]\NIn this paper, the authors study the properties of \(R \# S\), with a special focus on properties in positive prime characteristic defined using the Frobenius endomorphism. Since\N\[\NR \# S \hookrightarrow R \otimes_\mathbb{F} S\N\]\Nis pure, it follows that if \(\mathbb{F}\) is a field of positive characteristic, and \(R\) and \(S\) are F-pure or F-regular, then the same is also true for \(R\#S.\) The authors prove the converse of this statement in the paper, provided that the \(\mathbb{N}\)-grading on each of the rings \(R\) and \(S\) is irredudunt. \N\NNext, they consider the property of F-rationality. Unlike the properties of F-pure and F-regular, the property of being F-rational is not inherited by pure subrings. Nonetheless, they show that if \(R\) and \(S\) are F-rational rings of positive prime characteristic, then \(R \# S\) is also F-rational. The example is given to show that the converse of this is not true.\N\NLastly, they consider the property of finite Frobenius representation type (FFRT) introduced by \textit{K. E. Smith} and \textit{M. Van den Bergh} [Proc. Lond. Math. Soc. (3) 75, No. 1, 32--62 (1997; Zbl 0948.16019)]. They observe that if \(R\) and \(S\) are \(\mathbb{N}\)-graded reduced rings, finitely generated over a perfect field \(R_0 = \mathbb{F} = S_0\) of positive characteristic, then \(R \# S\) has FFRT. They use this to construct normal graded rings that are not Cohen-Macaulay, but have the FFRT property.