 The speaker is Claire DeCruz from Chennai Institute of Mathematics and you find the title there, please. I have to correct, hold it in the right direction. So okay, so I would like to thank the organizers for organizing this conference and for giving me an opportunity to talk here. So this is joint work with Mosme Mandel and J.K. Varma. So since this is a conference where we remember the work of Hoxton Uniki, I would like to say that our main result is actually the generalization of a result of Uniki. It says that if you take a height 2 prime ideal in a three-dimensional regular local ring, then the symbolic resalgebra of p is netherian if and only if there exists an element x, an element x which is a non-zero device in R mod p and you can find two elements f in the kth symbolic power and g in the lth symbolic power so that this length function on the left is equal to the one on the right. Okay, so we will go into a little bit of history of this problem and how it is connected with satirical complete intersection. So we say that an ideal i is defined satirically by S elements. If up to a radical i is the radical of i is radical of f1, f2, fs and if s is the height of the ideal then we say it is a satirical complete intersection and this goes back to the work of Kronecker where he showed that goes back to the work of Kronecker where he showed that if you have a affine close scheme defined by i sitting in an affine d space then it can be satirically defined by d plus 1 elements and there was a question whether if you were wondering whether one could improve and there were several conjectures which came by Forrester, Eisenbody ones and several others but then and there were several partial results but I would just like to state that the work of Storch in 1972 and Eisenbody ones in 1973 says that you could improve Kronecker result from d plus 1 to d. So there were several other results and in the 60s, 70s there were many interesting results that came up. So then Kaushik and Nori in 1978 proved this wonderful result where they showed that if i is a radical ideal of id minus 1 then i is a satirical complete intersection provided the characteristic is p. So this result is not true in we know in positive characteristic and since this is a confidence of characteristic p I would also like to mention the result of Hartzschol in 1979 where he proved that this McCollis curve I think I just was saying that the McCollis curve is actually a satirical complete intersection for all d become into 4 provided the characteristic is p and in characteristic 0 for d become into 4 this is still an open problem and people do not know the answer to it ok. So we have already seen symbolic powers in the we have already seen symbolic powers in the morning and we say that if i is an ideal then the n symbolic power is the intersection of i raise to n r local area p intersection r where p we can take it to be the associated primes or the minimal primes of r mod i but in our case we will be in our case we will be looking at radical ideal so in this case both the definitions are the same and then the symbolic re-salgebra of i is nothing but the direct sum of all these symbolic powers. Now Kaushik proved an interesting result which gives a relation between the symbolic re-salgebra and satirical complete intersection and he showed that if you have a prime ideal of i d minus 1 ok in a nodarian local ring then if the symbolic re-salgebra is nodarian then it is satirical complete intersection and this proof is actually characteristic free and there were several results on several satirical complete intersection but an exact generalization of Kaushik's result was not known. And so in a joint work with Mandel and Varma we give a generalization not just two so Kaushik's result was for height p equal to d minus 1 so even for d minus 2 d minus 3 or lower heights there was no result known. So we generalize it to an ideal i but the only thing is we need the condition that r mod a symbolic n is going to call it for n large so under that condition we can generalize this result and we show that if the symbolic re-salgebra is nodarian the first thing is the analytic spread of a symbolic k is same as the height of a symbolic k for some k bigger than 1 and in this case the ideal a is a satirical complete intersection. Now we know that there are examples where if we drop our assumption that r mod i symbolic n is not going to call it then the result is not true and these examples for study first one by Eisenwood and Evans and the second was Schenzel and Vogel and the first one actually uses it is not a very easy proof it uses Hartzmann's connectedness theorem. The second example Schenzel and Vogel came across a whole bunch of infinitely many examples of varieties in a projective space which are not satirical complete intersection and this was the simplest example on that and even there we see that r mod i symbolic n is not going to call it for all n equal to 1 and so we cannot drop that assumption in our result. So in some sense this is probably the best result we can get. So then to prove that result one of the important results that we need to prove the satirical complete intersection as well as our later result is this lemma which states that if you have an ideal i the dimension of r mod a is s and r mod i n is going to call it for all n become to 1. So this is a very important thing and this in some sense was an obstruction to generalizing the results and once we have this result it seems to be we cannot really drop these assumptions and so it says that if we can find x1, x2, xs elements so that they form a regular sequence in the associated graded ring. Now so suppose r mod i is going to call it then we can find x1, x2, xs which is a regular sequence in the associated ring and x1, x2, xs is r regular. So this is a very important crucial step and then the second part we get is dimension of r mod a is dimension r minus height a. So we just quickly look at the proof once we have these assumptions then using the previous lemma we get dimension of r mod i is dimension of r minus height i. Then we use the Hilbert-Berge theorem and because of which says that an active spread is bounded over by d minus infimum of depth of r mod i n and then our assumptions give that it is equal to height. But then we know that this proves that equality holds and as an active spread of a symbol k similar height of i k and this gives us the fact from this we get that ideal a is a theoretic complete intersection. So it is well known that the symbolic resalgebra is not Notherian and we have these examples. Last week Supriya gave an example of Rhys where the symbolic resalgebra was not Notherian and we also have the example of Nagata and the example of Roberts which says that the symbolic resalgebra in general is not Notherian. So I would like to mention that how the characteristic plays a very important role and so if you have this monomial curve affine monomial curve of degree 7 n minus 3 phi n minus 2 times n and 8 n minus 3 then if the characteristic of the field is p then the symbolic resalgebra is Notherian but not coin Macaulay and if the characteristic is 0 then the symbolic resalgebra is not Notherian. So it says that it is so much very much different on the character and this was proved by Goto, Nishida and Watanabe in 1994. And the result of Bresinski and Arzog also proved it is an unpublished work says that all affine curves in A3 are a theoretic complete intersection. So this says that the converse of Kaushik's theorem is not true. The symbolic resalgebra is Notherian then it is a theoretic complete intersection but the yeah because Bresinski in his paper writes that it Arzog knew the proof but he has not published it. The same result yes but he has not published it. So then there was no result known how to really show that the symbolic resalgebra is Notherian and so in 1987 Uniki came up with this interesting way of telling how do you show that the symbolic resalgebra is Notherian because if you know that the symbolic resalgebra is Notherian then the ideal is a saturated complete intersection. However this does not ease the problem even finally SO Uniki showed that if you are in a regular local ring and eyes are height ideal of p is a prime ideal of height 2 then the symbolic resalgebra of p is Notherian if and only if you can find this elements f and p symbolic k g and p symbolic l and x in m minus p such that this equality holds true but in general finding these elements f and g itself is not very easy. I mean it is quite a difficult problem and nevertheless many people worked on this afterwards this gave sort of an incentive for people to work on the symbolic resalgebra and not only to work to show the Notherian property but also the Coen Macaulay and the Gordon Steele property. So we had several people working on it but before that I would like to state that more or less in 1991 generalize this result to any ring of dimension d bigger than or equal to 3 but under the assumptions that the ring is analytically unamified formally equidimensional. So this is a generalization of Uniki's result and in 1994 Goto gave a further generalization where for the necessary part he drops the assumptions of Morales but for the converse he needs the fact that R is a unmixed no coloring. So this was the best generalization of Uniki's result that was known for many years and then we so I just like to mention this before we state our result I just like to mention though there were several results known Uniki in the same paper in which he gave the criteria for the Notherianness also showed he considered the Mohr curve and so these Mohr curves are very interesting because there are still some of Mohr curves for which the problem is open we do not know whether the symbolic resalgebra is Notherian and Uniki showed that if you have the Mohr curve which is parameterized in this way then the symbolic resalgebra is Notherian and it p is a security complete intersection so but then there are other Mohr curves which the problem is still open and so then yeah so I just like to mention that there are not many examples known in higher dimension where the symbolic resalgebra is Notherian so in Goto in his paper in 1994 in his memoirs show that if you take the monomial curve which is given by an arithmetic progression then the symbolic resalgebra is Notherian and with a joint work with Sridevi we actually could find these generators of the symbolic powers that is j2, jd minus 1 are the extra generators you need for the symbolic powers and then for d equal to 3, 4 Goto showed that the symbolic resalgebra is Coen Macaulay and in a joint work with Sridevi we showed that for all d we are going to 2 we can show that so we this was open for d we are going to 5 Goto raised this question and we showed that for d we are to 4 the symbolic resalgebra is always Coen Macaulay. How do you choose? These are ideals these are ideals so each i2, i3, i1 are ideals so we know exactly what those generators are okay so then we have a generalization of Unique's Morales and Goto's result and this is joint work with Mosfi Mandala and Jekte Verma and here we show that if you have an ideal i which is a positive i and at most d minus 1 and such that r modulo i symbolic n is Coen Macaulay then and the symbolic resalgebra is Notherian then we can find elements x1, x2, xd minus h which is a non-zero device in r mod i and in fact is a regular sequence in r and then we can find this elements f1, f2, fh which are in certain symbolic powers so that this equation on the last line holds true. This is an exact generalization you put the conditions that r is a regular local ring and height 2 ideal then we get the exact version of Unique's theorem. Now using that previous lemma one can actually choose the assignment so I will not go into this and then once you plug it you just put in the multiplicity formula and get the result. So the difficulty was actually finding the right way of the right way of getting these elements x1, x2, xd minus h and f1, f2, fh in choosing those elements very carefully in the right way that was the most difficult part. Once you do that just plug in and you get the multiplicity formula. So now for the converse for the converse we does not like in the version of go to we need to assume that r is a costly unmixed local ring and i is an ideal of height h and suppose you can find elements x1, x2, xd minus h which is the system of parameters r mod i, a and if you can find this elements f, i in the symbolic powers so that the conditions 1, 2 and 3 of the previous 3 are mod 2 and the symbolic resalgebra is netherian and for this part we need to use two important results. One is the result of Goto, Hermann, Schittmer and Willemard which is that it is a symbolic resalgebra of i is netherian if and only if the analytic split of i is symbolic k is in a height of i is symbolic k for some k greater than 1. So the way we choose our elements actually the proof is a little long I cannot spell the proof but we get this condition that nk square of ik is height of ik and the next thing we need to use is the result of Boyger which says that j is a reduction of i if and only if multiplicity of i is symbolic p is j, multiplicity of j is symbolic p for every prime ideal p for which radical of i is radical of j. Now this proof is quite involved and needs some work so I will not state I will just say that we need these two results to get our results and I mean this example and the next example say that we just cannot drop the assumption we drop either the assumption on the height or if you drop the assumption that a ring is cosy and mixed then you can take it off. If you drop the assumptions, if you drop the assumptions that either the height i is on the height of the ideal then we have an example where the conditions are satisfied but the symbolic resalgebra is not Noetherian and the next example says that if we drop the condition that r is a cosy and mixed local ring and even though we have this example this element x1 which is a non-zero divisive normal i and this element f1 so the x1, x1 is system parameters this equation is satisfied but the symbolic resalgebra is not Noetherian. So this is the best possible result that we can get. In the many time I would like to go through the examples because the examples are the more difficult thing to get especially when you go from prime ideal to non-prime ideal. So first example we take is the, we take a graph which is a complete graph and we looked at the edge ideal of a complete graph. So since this is the edge ideal is defined by monomials it is well E. B. Chung in 2007 showed that if you have monomial ideas and the symbolic resalgebra is Noetherian. So they use different techniques and using our result we can show that if a is the edge ideal then the r mod a symbolic k is coenmecal a for all k become equal and then the symbolic resalgebra is Noetherian and a is a satiristic complete intersection. So here we use the elementary symmetric functions. So the first elementary symmetric function which is sum of all the variables is actually a non-zero device on r mod a symbolic k that works as a element and for the remaining elements f1, f2, fn minus y they are in the right symbolic powers and so this sigma 1, sigma 2, sigma n forms a system of parameters in our ring r and this gives us both that our symbolic resalgebra is Noetherian as well as it is a satiristic complete intersection. A second example we took is more from geometry and this is what is called the Fermat ideal and this came in connection with the containment problem. People were interested in some results, some questions on the containment problem which were raised by Uniki, Hardbone, Bucci and several others and this gave a counter example to the containment problem, one of the questions and the first one for d equal to 3 was done by Duniki, Schoenberg and Gasinka and this is for n equal to 3 and for n bigger than 2, 3 it was studied by Nagel and Celestino and Nagel and Alexandria showed that the symbolic resalgebra is Noetherian. They use a very involved methods to show that symbolic resalgebra is Noetherian. We show that using our method we can very quickly show that the symbolic resalgebra locally is Noetherian. I am sure it will give us the graded case also but we all only need to find the right elements. So, if you choose x1 to be x plus 5 plus z that is a non-zero divisor in r mod i symbolic n for all n and then if you choose f1 to be r Sn minus 2t minus 2 times r Sn minus 2 and f2 to be this equation then x1 f1 f2 forms a system of parameters. So, we have the right elements and this shows that the symbolic resalgebra is Noetherian as well as the fact that a is a septioretic complete intersection and the third example again comes from geometry. So, we take this curve not the curve we take this hypersurface and we look at the Jacobian of it. So, ideal i is a Jacobian of f and this is one interesting example because it is height d minus 2. The previous examples were height d minus 1 this is height d minus 2 and so it is not very easy to get examples of height d minus 2 and so a is a height 2 ideal in a four-dimensional polynomial ring and which r mod a symbolic n is coenmecole for all n beginning to work and then if we choose our elements now we need to choose two elements which are non-zero devices in r mod a and so x and z work as non-zero devices g1 g2 g3 are the generators of our ideal if we take f1 to be g1 plus g3 and f2 to be f then f1 is in the ideal itself f2 is in the second symbolic power the ideal and so one can check all those equations are satisfied and so we get the symbolic resalgebra of a is Noetherian and a is a septioretic complete intersection so this completes the proof