Generalized <i>F</i> -depth and graded nilpotent singularities (Q6600742)

Many singularity types of commutative rings have been defined through properties held by the local cohomology modules at their (homogeneous) maximal ideals. In this nice paper, the authors examine singularities through the nilpotence or generalized nilpotence of the Frobenius action on their local cohomology modules. One of their goals is to exhibit several classes of graded singularities which are weakly \(F\)-nilpotent; these correspond to rings \(R\) whose Frobenius action on the local cohomology modules of \(R\) supported at \(\mathfrak{m}\) in all but the dimension of \(R\) are nilpotent or in other words, the orbit closure of 0 via the Frobenius action on \(H^j_{\mathfrak{m}}(R)\) is \(H^j_{\mathfrak{m}}(R)\) for all \(j <\text{dim}(R)\).\N\NIn a previous paper \textit{K. Maddox} [Proc. Am. Math. Soc. 147, 5083--5092 (2019; Zbl 1423.13050)], defined generalized weakly \(F\)-nilpotent singularites \(R\) to be rings \(R\) such that the quotient of \(H^j_{\mathfrak{m}}(R)\) by the orbit closure of \(0\) are finite length for all \(j <\text{dim}(R)\). Another goal in the current paper is to define and examine properties of the generalized F-depth of a ring:\N\[\N\text{gF-depth}(R):=\text{inf}\{ j \mid H^j_{\mathfrak{m}}(R) \text{ is not generalized weakly \(F\)-nilpotent}\}\N\]\Ncomparing it to F-depth and its relation to weakly \(F\)-nilpotent singularities. Some of the constructions that they use to determine classes of weakly \(F\)-nilpotent singularities can also be used to produce examples of generalized weakly \(F\)-nilpotent singularities.