 For giving me this opportunity to present my work here So I'll be talking on bounds for the reduction number of primary ideals in dimension three and In dimension three because These statements are there for high dimensions as well, but it is more interesting in dimension three for dimension less than equal to two Results are already known. So I need my basic setup for that So throughout the talk R is a Noetherian local ring of positive dimension and I is an M primary ideal now a sequence of ideals is Called an eye admissible filtration if It satisfies these three properties so the first one just says that it is Decreasing filtration and the second one is the multiplicative property and the third one is what makes it eye admissible So for a filtration we define reduction Reduction is an ideal J such that J i n is equal to i n plus one for all n large and And it is called minimal reduction if it is minimal with respect to containment among all the reductions So when the residue field is infinite then minimal reductions exist and therefore We assume for all our statements that Residue field is infinite. There is a standard way to reduce the problem to that case Okay now to keep track of these indices to Keep track of these in the incidence and here we define the reduction number so basically This equality ensures that the result of I will be finitely generated over the result of J And the generators will have degree at most N at most The the value of n where after which it becomes equal so that number is preserved in the invariant reduction number So we define the reduction number of I with respect to J as Supremum of All the values n such that I n is not equal to J i n minus 1 Okay, so it is an important problem to give bounds on this reduction number and Desireively computable bounds So one such bound is known in terms of Hilbert coefficients So let me define Hilbert coefficients for any i admissible filtration. We look at this length function Length of our mod I n and this is known as the Hilbert Samuel function of I Then it is well known that there is a polynomial this this function this length function is of polynomial type Which means that there is a there is a polynomial with rational coefficients Such that this length function coincides with this polynomial For all the large values of n and this polynomial will have degree D, which is the dimension of the ring And this is a rational polynomial and in fact We can normalize the coefficients and then we can write it in this fashion Where all the coefficients are? Integers so this e i i's are integers and these coefficients are known as the Hilbert coefficients of I so it looks like that these Hilbert coefficients are Invariance of asymptotic powers of I I mean the asymptotic ions for this filtration But it turns out that these are very important and they contain a lot of information very deep in information about the structural properties of the filtration I or the ring and even the blow-up algebras of I So We restrict ourselves for the stock to the i-addict filtration so we always consider I is the powers of I The I is the filtration defined by the powers of I then Yeah, so now going back to the reduction number So if R is one-dimensional coin Macaulay local ring then R I is less than equal to E naught I minus one This R I is a minimum of all the reduction numbers Rji's which I defined earlier so R I is less than equal to E naught minus one Later Vaskuncelos proved that in any dimension positive dimension This R I is bounded above by D times multiplicity of I by order of I minus 2d plus 1 Where order is defined as the largest positive integer n such that I is contained in m to the power n Okay, and this is the best bound known for reduction number In higher dimension, but we will see that this is a large bound. This is large This is larger than what we have got Okay, and a non-cohen Macaulay version of the above results can also be found in literature so Now Rossi proved that the reduction number is less with respect to a Minimal reduction J is less than equal to the first Hilbert coefficient even minus the multiplicity E naught plus length of R mod I plus one and This is true only when dimension is at most two Okay, and here is this year is not correct. This is not 1988. This is 1999 or 2000 maybe so This this bound was given in terms of the first Hilbert coefficient and the multiplicity. This is a linear bound. It is nice and it is computable But it is known only for dimension at most two For D greater than equal to three. It is believed to be true. No counter examples are known But it is an open problem Okay, so our motivation was to understand the difficulties in extending the proof of Rossi for this bound in dimension larger than two and And we figured out that there are two major difficulties The first one is when the first one is that the reduction number does not behave well with respect to superficial elements Which means that this say I say are I write R prime as R mod X where X is a superficial element Then R of IR prime and R of I These are related in this manner and this inequality is not useful for this for us We would like to know a precise relation between these two if we want to follow the proof of Rossi but in general there are no Relation clear relation between these two invariants and the next problem is that the Ratliff rush filtration of I Which I will not define It does not behave well with respect to superficial elements, which means that these two operations do not commute I and tilde R prime is not equal to I and R prime tilde for all and greater than equal to one so if we Get hold of these two difficulties then we can prove Rossi's bound and there are some cases when we can get rid of these two difficulties For example, these are the cases So when depth of so in dimension three now if depth of gi is greater than equal to one or if depth of the associated grid a ring of the Ratliff rush filtration is Greater than equal to two or if the second and the third Hilbert coefficient vanish Or if e2i vanish and I is asymptotically normal or e2i is zero and gi is generalized coin Macaulay so in all these cases we get that Rossi's bound hold in dimension three and Basically, what is happening is those two problems, which I Stated can be handled in these cases So but of course these are very restrictive conditions and we would like to Get we would like to get rid of these conditions or We would like to get Some bound on reduction number without putting a strong hypothesis So so and there is a hope because we noticed that Yeah, if depth of gi is positive then the reduction numbers are preserved going modular superficial element but this relation holds even if depth of gi is zero and Here is an example for that This is taken from a work of Rossi and Walla. So in this case depth of gi is zero But this is dimension two case depth of gi is zero, but reduction number Modular superficial element is the same So basically we have to look at the question that when does it happen that? Reduction numbers are preserved Okay All right, so we tried to find relation between the reduction numbers and we got this useful lemma So in a Noetherian local ring of dimension D Positive dimension D and also positive depth We have that if if reduction number of I mod X is strictly less than reduction number of I Then in this interval Which is closed from left side and open from the right side I until I is not equal to I in Okay, and using this lemma now if I define this Invariant rho i which is minimum of i greater than equal to 1 such that iron is equal to i and tilde for all n greater than equal to i Then we get that if rho i is less than equal to rji minus 1 then Rossi's bound holds Okay, and again, this is a restriction here For rho i and this is not in general true. So we have examples when this is not true So And also examples where this is true. So again, there is some restriction on the ideal So this is one more case in that list which I displayed earlier When Rossi's bound hold in dimension 3 Okay, so then moving on in our effort to find the relation between the reduction numbers We could show that if depth of g i t is positive for some t greater than equal to 1 then Reduction numbers are related in this manner Okay, and in particular we can always put this t to be equal to rho i and therefore Rji is always less than equal to r i mod X plus rho i minus 1 So then using this we could generalize the the first condition on the list I displayed To the following theorem that if depth of g i t is positive for some t greater than equal to 1 which always happens Then rji is less than equal to even minus e naught plus length of r mod i plus t in dimension 3 So this is a linear bound Which we have got moreover we can further reduce it looking at the value of the reduction number I mean suppose if we know that reduction number is k mod t for some k between 1 and t minus 1 then this t can be replaced by k here and particularly useful corollary is that if depth of gi square is positive and The reduction number is odd then again we get Rossi's bound here. All right So here is an example which shows that the bound So in this case this depth of gm cube is greater than equal to 1 and therefore by our result This is the upper bound with plus 3 here For reduction number which turns out to be 7. However, if we calculate Vaskan's loss is bound. This is 19 So this new bound is better in this case All right Okay, so but ideally we want to remove all the hypothesis and get a bound So here is the result for a Cohen Macaulay local ring of dimension larger than 2 strictly larger than 2 Suppose depth of gi is greater than equal to d minus 3 then rji is less than equal to this Rossi's bound plus some new terms Which is e2 minus 1 times e2i minus e3 of i Okay, so so the second and the third Hilbert coefficient Here come into picture when we move to dimension 3 Okay, and in particular for small values of e2 here we get linear bound say when e2 is 0 or e2 is 1 Then this is a linear bound This so ideally we want a linear bound but here in our result we have got a quadratic term here of e2 squared Okay, so in dimension 3 this is an interesting result without any hypothesis on i Okay, and then Yeah, so as I said when e2 is 0 or 1 and 2 so for these small values we are getting linear bounds here Okay, now one would also Observe that this this additional term present here e2 minus 1 times e2 minus e3 of i This must be a non-negative term. Otherwise otherwise Rossi's bound is proved in dimension 3, which is Not believable. So we suspect that this is greater than equal to 0 for any m primary ideal But we could prove it only for integrally closed ideals So this gives a bound on e3 of i an upper bound on e3 of i It is e3 of i is less than is less than equal to e2 into e2 minus 1 for integrally closed ideals Okay, so okay, so the the difficulty as I mentioned is that the Ratley-Fresch does not behave well with respect to superficial elements and Yeah, so and then some if we get rid of that then again we get Rossi's bound So one such condition is when e2 and e3 vanishes and this was proved by Tony that in dimension 3 If e2 and e3 Becomes zero then Ratley-Fresch filtration behaves well modular superficial element But if i is integrally closed then e2 is zero Implies that gi is Cohen Macaulay. So this is a strong condition and it does not differentiate between Cohen Macaulay Associated graded rings and I mean non Cohen Macaulay cases. So This is not desirable. This is too strong Okay, so in fact for integrally closed ideals We we get some necessary and sufficient conditions for the Ratley-Fresch filtration of I Behaving well modular superficial element and one such result is the following Let R be a Cohen Macaulay local ring of dimension greater than equal to 3 and I is an M primary ideal then The Ratley-Fresch filtration behaves well If the Ratley-Fresch filtration behaves well modular superficial sequence, then we have this inequality This inequality was earlier known when with the condition that depth of gi is greater than equal to D minus 1 So this provides a necessary condition for checking where the Ratley-Fresch behaves well Okay, and Here is an example where we can apply this. So in this example This e3 is minus 1 and e2 minus even plus e not minus length of our mod maximal ideal is 0 So the inequality is not satisfied and therefore the Ratley-Fresch of the maximal ideal does not behave well Modular superficial element Okay, but for our problem we need sufficient conditions So we prove this these interesting bounds on e3 of I For integrally closed ideals so if I is an integrally closed M primary ideal and J is a minimal reduction of I then we have these three bounds on e3 of I and Here if you notice that Four so they are not related to each other directly I mean there are some indirect relations because from four we can get to five if we know that Rossi's bound holds in dimension three But we do not know that even then we can prove this bound okay which is a computable upper bound for e3 and Then for integrally closed ideals we know that e2 is larger than even minus is greater than equal to even minus e not plus length of our mod I and so we can bound this number by e2 of I but in fact We can actually bound e3 of I by e2 minus 1 by 2 times this Okay, and so these are interesting bounds and what is more interesting is that if In dimension three if equality holds in any one of four five or six then the Ratley-Fresh filtration of I behaves well modulo a superficial element and Consequently Rossi's bound will hold Okay, so these are some computable at least five and six are computable conditions When we can be sure that Rossi's bound holds Okay, and we have the converse statements as well suppose the Ratley-Fresh filtration behaves well modulo a superficial sequence x1 x2 xd minus 2 Then equality holds in four provided RGI is less than equal to three equality holds in five provided Even minus e not plus length of our mod is less than equal to two and equality holds in six provided e2 of I is less than equal to three okay, so This is for integrally closed M primary ideals and in fact we could actually Improve the bound for integrally closed ideals So in that case the hypothesis is same I is integrally but I is integrally closed and Depth of GI is greater than equal to D minus 3 then now RGI is less than equal to the Rossi's bound plus this term and This term is a smaller than e2 into e2 minus 1 minus e3 of I the earlier bound Okay, so this is the best we can have and Rossi's bound still remains open and mystery for us so That's all and then we have some Generalizations in Noetherian for Noetherian local rings So this bound of Rossi in even in dimension has been equal to two is not known for filtration and for Noetherian local rings If we assume that the ring is book bomb then we can have some bounds so in one So, okay, so this is last slide. I'll just continue if R is one-dimensional book bomb local ring Then we have Rossi's bound With some additional terms here. So even I minus even J Minus enotype plus length of arm or die plus two so in place of plus one we have plus two and even J will appear here this is a standard thing it appears everywhere and Then for two-dimensional book bomb local ring Again if we assume that depth of GIT is positive for some T greater than equal to 1 then RGI is less than equal to even minus even J minus enotype plus length of arm or I plus T plus 1 so this T will appear here Okay, and That's all so these are some references and All these results were joint work with the Mosme Wandel and Anud Kumar Yadav Thank you very much