On the uniform bound of Frobenius test exponents

Notes on the Frobenius test exponents

“…It leads the following question. We definite the Frobenius test exponent for parameter ideals of R, F te(R), the smallest integer e satisfying the above condition and F te(R) = ∞ if we have no such e. Question 1 has affirmative answers when R is either generalized Cohen-Macaulay by [7] or F -nilpotent by [14], and is open in general. The Frobenius test exponent for parameter ideals is closely related with an invariant defined in terms of Frobenius action on local cohomology.…”

“…Furthermore, the authors of [7] confirmed the question for generalized Cohen-Macaulay local rings. Recently the second author gave a positive answer for the question for F -nilpotent rings [14]. The main idea in [7,8] Our main technique is to analyze the local cohomology modules by using the Nagel-Schenzel isomorphism (cf.…”

Upper bound of multiplicity in prime characteristic

“…(2) Huneke, Katzman, Sharp and Yao [5] gave an affirmative answer for Question 1 for generalized Cohen-Macaulay rings. (3) Recently, the second author [14] provided a simple proof for the theorem of Huneke, Katzman, Sharp and Yao. By the same method he also proved that F te(R) < ∞ if R is F -nilpotent.…”

“…[8] F te(R) = HSL(R). Moreover the question of Katzman and Sharp has affirmative answers when R is either generalized Cohen-Macaulay by [5] or F -nilpotent by [14] (see the next section for the details). The main result of the present paper is as follows.…”

Frobenius test exponent for ideals generated by filter regular sequences

“…Beyond the Cohen-Macaulay case, Huneke, Katzman, Sharp, and Yao [8] showed that Fte(R) < ∞ for generalized Cohen-Macaulay local rings by using several concepts and techniques from commutative algebra, namely unconditioned strong d-sequences, cohomological annihilators, and modules of generalized fractions. In 2019, the second author [16] not only simplified the proof for generalized Cohen-Macaulay rings but also proved Fte(R) < ∞ for weakly F-nilpotent rings, i.e. [13] extended this result for generalized weakly F-nilpotent rings, i.e.…”