 Mulkudan from IAT Delhi and his talk is a reduced type in one-dimensional analytics algebra. Are you able to hear me? Okay so thank the organizers for inviting me and also ICTP for conducting this wonderful conference. I get to see quite a lot of people who I wouldn't be able to see if I were in India I'm guessing. So today's talk is going to be defining what reduced type is and how it compares to the usual type and why the word reduced comes into play. I came up as some kind of an offshoot of another problem I was working with. I'll just explain what reduced type is the setting and why it came to be defined so and how does it compare to the usual kind of Macaulay type. So the setup is the following. Of course you have one-dimensional so one I'm going to assume algebraically closed and R is S mod i which is the k joint x1 up xn modulo the ideal i and i is contained inside the square of the maximum. So I don't get to reduce the embedding dimension and it's also a prime idea so this guy is actually a prime idea so I work only with domain though I'm suspecting that you could generalize it to reduce. Of course one dimensional the dimension of the ring is actually equal to one and complete local and domain. So one of the nice consequences is that the integral closure is a nice DVR with unifying parameter T. What one can do is basically write the ring as k joint alpha 1 d bar a1 alpha n d bar a n where alpha i's are units in R bar. You can define it as one-dimensional and becomes a nice ring set. So of course my handwriting is so bad but these are capital X's and if I write small X's these are the images of Xi's in R. So basically this guy is my X1 blah blah blah this guy is my Xn and these guys these numbers are kind of very important so you want to play in and it's kind of you can define valuations on AR. So on R bar I have to follow in the valuation of t of t is summation of ci t par i of course the power series in t par i to be the minimum among all the i's that the ci is not equal to the order value standard order valuation on the pfd. So this induces a valuation on on R also induces a valuation on R so you have a valuation semi-group on R. V of R is the valuation semi-group. I'll be talking a lot more on the valuation semi-group but there's a very strong relationship between the ring and the valuation semi-group also. If the valuation semi-group exhibits some properties then the ring also exhibits some equivalent properties. I'll come back to that in a little bit and also know that the minimal reduction of the maximal ideal one of Xn is the ideal generated by X1 is precisely because if you extend the maximal ideal to the integral closure it's actually princely generated. The minimal reduction of the maximal ideal is nothing but the ideal generated by X1 itself. You can arrange it in such a way that the a's are in increasing order you have the minimal valuations in which are also the multiplicity. Okay so one quick example of that take for example k adjoint t power 4, t power 6 plus t power 7 and t power 11. Now this guy is nothing but t power 4, t power 6 times 1 plus t and t power 11. The valuation semi-group is of R, where this guy is my R, is actually generated by 4, 6, 11 and 13. It's kind of easy to just say that you just take these valuations and valuation semi-group will be generated by a1, a2, an but it's not true. You have to be a little bit more careful. The valuation semi-group may be generated by more elements than the a1, a2, a5. Okay so the next definition I want to make is about the conductor. Conductor is nothing but R adjoint R bar on the quotient field and it's the largest common ideal between the ring and the and the integral closure and there is no way that the conductor is always contained inside a principal ideal. And this guy is very special because one can define this guy to be generated by t power c, t power c plus 1, all the way up to t power c plus a1 minus 1 and this guy is special because it contains every monomial after t power c but not t power c minus 1. c minus 1 is not in the conductor but it contains every monomial after t power c. For example a quick example for that one is for example take K adjoint t power 4, t power 11 and t power 17. You can show that the conductor contains everybody after 19 and t power 18 is not in the conductor. It's not very hard to prove it. You can basically check all the valuations that appear generated by 4, 11, 17 and you can prove that everybody after 19 is present. Okay so for example this one if you want to check the conductor the conductor is going to be 10. Okay so what is this reduced type? Reduced type is basically the K dimension of the conductor plus x1 modulo x1 that's the reduced type. I'm going to call this as s throughout the talk, s of r or s. Either one of these will be called as the reduced type. Okay first I'll explain why this is important then I'll explain why it's called a type or reduced type. So one of the principal reasons why we were studying this is it actually comes up as an attempt to solve cases of burgers. So what we were able to show is that you construct an overring which is the conductor modulo x1. Conduct an overring of the ring r. What this one is essentially doing is that it will take the ring r and adjoint t power b1 up to say bs where bi is not in the valuation semi group of r but it is actually inside the interval c minus a1 comma c minus 1. So essentially what's happening is that the embedding dimension of s is actually n plus s where this guy is a reduced type and this guy is the embedding dimension of the original ring. And we are able to show that if s is quasi homogeneous among other results. This is not the complete set of cases where we can show the burgers but and the other one is also pretty nice. So m power 4 is contained inside the conductor plus x1 and there's also bound on the reduced type. The module of differential omega r has torsion. So the model of differential omega r will have torsion. So that affirming what burgers is true. The first one is nice because it's strictly better than the old result of... Sorry, sorry. Okay, okay, I'll try. Over here? No, both of them. Either one of... Okay, yeah, sorry, you're right. So either this is true or this is true then the burgers. This is not the complete set of cases but two illustrative examples where the reduced type comes into play. The first one is nice because it's strictly better than the original result of Shea where he proved that if r is quasi homogeneous then the burgers is true. But this one is strictly better. Second one is nice because the original results involved when the maximal ideal is contained inside a principal ideal. I mean the fourth power of the maximal ideal is contained in the principal ideal then the burgers is true. But this is much better. m power 4 is not contained in the principal ideal but c plus. But you do have some bound in the reduced type. Okay, so this is the reason why we ended up starting the reduced type and I'll try to write a bit more bigger for why characteristic is zero. It's always zero type. It's always zero. Okay, so why the reduced type? So where does the word reduced in the word type comes into play? It's because of the following. So c plus x1 modulo x1 is always contained inside the x1 colon m. You can see that the conductor times the maximal ideal is always contained inside the principal ideal. And that is because m times the conductor is nothing but x1 times the conductor. x1 is the minimal reduction of m. Therefore, m times the conductor is nothing but x1 times. The dimension of this one is your usual coin Macaulay type, whereas the dimension of this one is the reduced type. k dimension is s. This one is the type. k dimension. That's why this is called as the reduced type. So one thing is apparent. One is less than the reduced type is always less than the type. So reduced type can either be one or it can be the type. And that's why we coined the word reduced type. Okay, so what the purpose of this talk is to understand what kind of rings exhibit extremals of these. When is it one? When is it the type? So question so far on the definition? S of r equal to one and S of r is actually equal to the type of r. Okay, when does it exhibit extremal condition? So this we call it the minimal reduced type and this we call it as the maximal reduced. When does the ring exhibit minimal reduced type and when does the ring exhibit maximal reduced type? Okay, so it's true that when r is going to change, the reduced type is actually equal to one because the type is one, then the reduced type is four should be one. But not conversely. But not, a simple example, one of the simplest examples is r is K join C to the 4, T power 11 and T power 17. One can easily check that this guy has reduced type one but it's not going to change. And the way to check going to change is pretty simple. It's not as hard as it sounds because all you have to check is that the map from Z to Z where this takes X to C minus one minus X takes elements to non-elements and non- elements to elements so you can easily check that four is not a member of the semi-group but C minus one minus four is also not a member of semi-group. So take non-element to non-element therefore it cannot be going to change. How did I say that the reduced type is one very quickly here? Ideally you would have to go through the definition but there's a simpler way to do this. All you have to do to compute the reduced type is count the number of integers in C minus A one comma C minus one but not in the valuation semi-group. The cardinality of the set of integers between C minus A one C minus one but not in a valuation semi-group that will give me the reduced type. And why does that happen? Well if you want the length of this quotient C plus X one model that is like computing the length of C model or C intersect with X one. Now if I want to compute the length all I have to do is count the valuation here and then delete the valuation coming from here that the cardinality will give me the length of the quotient. So in this case it will happen that this guy is always contained inside M times the conductor. So the relevant valuation coming from here corresponds to the valuation inside the valuation semi-group also inside the C. So if I'm deleting the valuation coming from C intersect with X one I'm looking at the valuation which are not there in B of R but it is there. So for example if I go back to this one the conductor I think it is nineteen and I just have to look at nineteen minus four and then look at the valuation which appear between nineteen minus four and eighteen and check how many of them are not there. So this is a very quick way to compute what the reduced type is. Okay so the first thing is like I said before we are trying to conclude or make some observations on when the ring is minimal reduced type or maximum reduced type. So there is a very nice containment which is C plus X one contained in X one colon M is contained inside M. And the first thing we had was this very nice containment and we were looking at can you conclude something about the minimal reduced type or the mass reduced type depending on the co-length of the conductor. And the first result we had was the following if the length of the conductor, co-length of the conductor is actually two or the co-length of the conductor is three and R has minimal reduced type, I mean minimal multiplicity then the then R has maximal reduced time. It has R has maximal reduced time. But unfortunately there could be a lot of containment in between this one and this one which kind of did not help us actually improve this result beyond the co-length of four. So we turned our attention to numerical semi-group and we thought that maybe you have a nice ring R you can always construct the numerical semi-group which is related to this one to the ring R. And we were trying to see how the type of R is related to the type of numerical semi-group and reduced type of R is related to a reduced type of numerical semi-group and you see make some conclusions on this side to get some information on this side. And there is obviously a lot of information that you have on the right hand side for numerical semi-group because that seems to be more well studied. Okay so numerical semi-group. So like I said you have a ring you construct a corresponding numerical semi-group. How do you do that? Just go ahead and take the if the numerical semi-group is actually numerical semi-group is actually generated by d1 dm just takes the corresponding numerical semi-group to be here join d power d1 d power dm. Okay so what the information that you have why is this why is this kind of important why is the numerical semi-group important? Well like I was doing there I was able to conclude that the ring was actually not Gorenstein because of a result of Kunz who says that the ring is Gorenstein if and only if the valuation semi-group is symmetric if and only if V of R is symmetric. So symmetric basically means that there's a function from Z to Z and it takes X to C minus 1 minus X where C is your conductor for the ring. It should take the elements of here to the non-element and non-element to element. So that's the kind of symmetry that you're looking for. But it has been improved by Barussi and Froberg to the R is almost Gorenstein if and only if V of R is almost symmetric. Okay this definition is a little bit more tricky I'm going to avoid the definition but remember that there is something there. Okay almost symmetric ring almost symmetric semi-group definition there's something happening there so just keep that in mind. Okay what about the type? What the type of this ring? Well it's been proved by a lot of people. Steve Froberg got lead and quiz. I'm pretty sure this word is wrong and you can correct it by yourself. But the type of the numerical semi-group is the cardinality of the set Z belongs to Z mod low the valuation semi-group. So that X plus H belongs to V of R for every H belongs to V. This is the type. The type of the numerical semi-group is a collection of all integers which is not in the valuation semi-group but when you add the numerical semi-group element to it it falls into the numerical symbol. And unfortunately the type is not but you see the type of the ring is always less than or equal to the type of the corresponding numerical semi-group. It's always less than or equal and it can be strictly less than the type of the numerical. But funny enough it doesn't happen for reduced type. The reduced type of R is always equal to the reduced type of the numerical. There's no distinction between the reduced type of the ring and the reduced type of the numerical semi-group. Okay so what the story is that the how do you translate properties now? See on how to go forth from the ring to the numerical semi-group is the following. So R is of minimal reduced type if and only if the numerical semi-group is of you can combine all these equalities up to get the one. It basically says that S of R is actually equal to the S of K join the valuation semi-group which is less than or equal to the type of R less than or equal to the type of the valuation. We have this long string of inequalities here and you should be able to prove that it's of minimal reduced type if and only if the numerical semi-group. So if you want to study numerical I mean minimal reduced type is kind of much easier because it just follows through to the new just take the corresponding numerical semi-group to be able to prove that. But the on the other hand maximal reduced type is weird. So R is of max reduced type implies that the numerical semi-group of V of R is of max reduced and the converse happens if you know that the types are equal. The converse is true if the type of the ring is actually equal to the so max reduced type of much harder to play with than the min reduced type. But either way this kind of says that if you want to know a lot of information on the max or min reduced type of the ring you better go to the numerical semi-group study when is it max or min reduced type there and put it back to the ring R. Okay so you won't it's much better when you have a minimal multiplicity is that R is a numerical semi-group is of min reduced type if and only if the maximum of all the excess that X is in the pseudo Frobenian set. I'll explain what the pseudo Frobenian set minus C minus 1 is strictly less than C minus A1 minus 1. What's the pseudo Frobenian set is this guy is the pseudo Frobenian set. So if it's a numerical semi-group you can easily check out when is the minimal reduced type just check that the maximum among the pseudo Frobenian number except for C minus 1 is strictly less than this number and when you have R is of max reduced type if and only if the minimum among all the excess in the pseudo Frobenian set of the numerical semi-group is less than equal to C minus 1. Now that you bought this study from the ring to numerical semi-group then you can do it for a lot of cases when is it guaranteeing when it's almost guaranteeing when it is far flung guaranteeing almost guaranteeing nearly guaranteeing all these cases can be brought into and you can study the I think I'll stop here thank you. So any question or thank you. My question is when you say that is of maximum reduced time that type then can you deduce something about the some properties of the associated ring in this case? Some property of the? Of the associated grid of ring. I haven't thought about it. I don't know much about it. Yeah but yeah. The minimal I don't think so. Right. The maximal could be. Thank you. Another question? The minimal or maximal reduced type will act as a sufficient condition for good properties of I mean other homological or properties of blow-up algebras. I don't know anything about the blow-up algebras but the story originated from Bergus so I was more concentrated on studying these minimal maximal radius type to solve Bergus or get more information to cases of Bergus but yeah I don't know much about the blow-up algebras. Just something related to what you said at the end we could make some connections with nearly Goronstein almost Goronstein. Do you have some ideas in your head? Sorry? Do you have some ideas to make those connections? I know the connection. I haven't shown it yet. Oh okay okay thanks. Not with a speaker again.