 I would like to thank the organizers for giving me a chance to present my work today. And my title is the Frog and the Test Exponent in Bright Characteristics. It's based on the work with my supervisor, Phap Hung Quy. My talk is divided into two parts, Introduction and Question and Redraws. Throughout my talk, I am via a natural commutative, locating a dimension D and a bright characteristic P. R0 is a set of elements of R such as not contained in any minimal bright ideal. I is generated equal to 80. It's an idea of R. Q is parameter ideal of R. And F is full radius map of R. If I generate this equal to 80, then the local cohomology module with support in I may be compute at a cohomology of the site complex. And the full radius map and its localization induce a natural flow banyard action on local cohomology. So concept of the relative flow banyard action was introduced by P and Pocha when they work on an important ring. Let's keep containing in I be a idea of the ring. The full banyard map F from I over K to itself send a blood K to E2P blood K can factor as a composition of two marks. FI and P. P is a natural flow segment and FI have formula in here. And FI induce a relative flow banyard action on local cohomology by site complex. Let I recall about full banyard closure and tight closure. Full banyard closure of I is ideal IF. Thus consists of on element I certify the condition. The condition in here for some E greater than or equal to result. And the tight closure of I is ideal I star is consisting of on element I certify the condition for some C in I0 and for all light enough E. Similarly, we can define the full banyard closure of 0 submodule of local cohomology module. It's an important part of the local cohomology module by full banyard action. It's easy from definition we have it is contained in IF and both of them is contained in I star for own I. As you know on local cohomology module with the support in Massimo idea at Athenian then by Hudson-Spicer-Libendic, there is a non-negative integral E such that we have this here. The flow banyard closure of 0 submodule of local cohomology is kernel of FE. And the Hudson-Spicer-Libendic number of local cohomology module is mean I that certifies the condition here. The Hudson-Spicer-Libendic number of the ring is mean I that certifies the condition here for own I from 0 to D. And we have the Hudson-Spicer-Libendic number of the ring is finite. By the ring is natural. So for all I there is an integer E of course depending on I such that we have the equality here. The smallest number E that certifies the condition is called the flow banyard test exponent of I. And we define the flow banyard test exponent for parameter idea is the mean I that certifies this condition here for own parameter Q. And FTE of I is infinity if we have no suck integer. We move to the second part of my talk, questions and results. There is very no uniform bound of the flow banyard test exponent for own idea by Reynolds in 2006. So first question we consider is studied in the existence of a uniform bound of the flow banyard test exponent for some classes of idea, parameter idea, idea generator, filter, regular sequence. So question on point one is read by Cartman and staff. And she wants to know does the axis a uniform bound the flow banyard test exponent for parameter idea such E FTE I is finite. Some cases have positive answer for question on point one. The first case is Cohen-McGray-Ring. Moreover, in Cohen-McGray-Ring, the flow banyard test exponent for parameter idea is equal to the Hassan-Spicer-Libernick-Oppler-Ring by Cartman and staff in 2006. In the same year, Hoonicky, Cartman, and staff extend this result for general like Cohen-McGray-Ring. In 2019, we use the relative flow banyard action to prove when the ring is quickly apnipotent. And McDo extended this result for general like with the apnipotent. In the article of Cartman and staff in 2006, they said that if it is a fixed regular sequence, so then there is an integer C depending on the sequence such that we have the flow banyard test exponent of ideal generator, the positive power of the sequence is less than or equal to C H. Note that the regular sequence is a fixed and has this only the power it run from one. And we raise the question is that does there is a positive, does there is an uniform power of the flow banyard test exponent for ideal generator by regular sequence? And we answer question 1.2 in a more general form. Is that if T is non-negative integral less than or equal to D, such that we have more than have finite length for own Z less than T. So there is this uniform power of the flow banyard test exponent for ideal generator by a fixed regular sequence of length at multi. As you know for all parameter idea that there is a system parameter that also a fixed regular sequence generated in Q. So let me take T equal to D and we have a form mentioned positive answer for question 1.2. 1.1, yeah. The second question we consider is less ambiornatory and local ring. A prior characteristic and a dimension D, and a big dimension V supports us the flow banyard test exponent for parameter idea is finite. So we want to find an upper bound of Himbos sub-way multiplicity of the ring in terms of D, V and flow banyard test exponent for parameter idea. In 2015, Hoonigki and Guattana Bay gave an upper bound for multiplicity when the ring is a few and better bound when the ring is a rational issue. As you can see from the slide the bound is depending on dimension D and ambuting dimension V. In 2019, Katman and Zhang removed the appeal condition and give the Hatsun specific number to give an upper bound for multiplicity of the ring. I mentioned before when the ring is cohenmaculate the flow banyard test exponent for parameter idea is equal to the Hatsun specific number of the ring and the bound in this case is depending on dimension D, ambuting dimension V and flow banyard test exponent for parameter idea. And as you know, when the ring is appeal or rational the flow banyard test exponent for parameter idea is equal to zero. And try to extend the resource by naturally we obtain the resource let R M be an interior locating a prior characteristic P and dimension D, ambuting dimension V. If the flow banyard test exponent of the ring is finite and we have upper bound for multiplicity of the ring is here and better bound when the ring is omnipotent the upper bound for multiplicity in here. And the upper bound is depending on D, V and flow banyard test exponent of the ring. That's all from my talk. Thank you for your listening. Thank you.