 Okay, welcome everybody to this workshop on characteristic P methods in algebraic geometry and commutative algebra. This is of course a second week in a larger program here at ICTP and I've heard that things have been going very well last week. For me this is my fourth time here at ICTP and I must say I very much appreciate the mission of ICTP and what ICTP is doing for mathematics worldwide. This is a conference on characteristic P methods and we are very sorry that Merhockster and Craig Unicky are not able to attend in person, but since this is an event in characteristic P methods, such an event can do nothing but celebrate the tremendous contributions of our two colleagues. So to start, our first speaker, I'm very happy to introduce my old friend, David Kalkowski from the University of Missouri and he'll talk about the analytics spread of filtration. Thank you, Berndt and thank you for the organizers for inviting me to be in this wonderful place. One thing, a practical thing is I forgot, I was told that there is some concern about where I write on the blackboards for the camera. Does anybody know if that's an issue or not? I can write anywhere in the camera just, okay, okay that's good, okay maybe it will be today, okay so if I go over here somehow it should shift, does it pick it up? Can I read this? But I don't see really, it's sort of a long angle shot. You can see, okay perfect, okay so now I can start, okay so this is on filtrations, analytic spread, I have a new sort of added, and most of this is with Prangamah Sarkar which I will mention. So is this, this is going to be seen everywhere, okay, okay you're right up front though, so but no, okay, okay good, alright so what is a filtration is equal i sub n on a Northeerian ring R is a sequence, R is equal to i naught contained in i1 and so on, and the critical condition is im times in is contained in im plus n. So this allows one to form a kind of a Reese ring of this which is from this which I'll mention later. Okay so what are some examples of this, there's the, if i is an ideal, there's the iatic filtration of powers of i, and this is the most very classical and most important one, okay and there's another type of filtration I want to talk about. So if we take R to be a domain, or the way this is usually used is if you start out with R and you mod out by P, where P a minimal prime, and if I take V a valuation on the quotient field, which has a property that non-negative on R, and it has the property that the transcendence degree of the quotient field of R mod the maximal ideal of the valuation ring over R, the residue field of the valuation ring over the residue field of the quotient field, this is equal to the height of this prime ideal minus one. So if this happens, this is called a divisorial valuation, okay so this in itself seems like sort of a strange and unmotivated definition maybe, but it has the, in the case for reasonable rings this has a very nice interpretation. So if R is excellent, then V is divisorial if and only if there exists some mat pi from X to speck of R, which is a blow up of an ideal, X is normal and the valuation ring of V is going to be local ring of E, where E, I'll call it a co-dimension one sub-variety of X, so if you will an integral sub-scheme of co-dimension integral closed sub-scheme of X of co-dimension one, okay so from this you can construct, well first off we need to know the concept of valuation ideal and this is, if I take a lambda in R, this is going to be the elements X in R such that the value of X is bigger and equal to lambda, okay so in fact these divisorial valuations are always discrete, rank one discrete, so in fact this is equal to the roundup of lambda, so there's some technical reason we'll see why we want to consider this and divisorial valuations are discrete, okay so now there's a concept of a divisorial filtration which will be I sub n, what does this mean, this means that each I sub n is I, it's the intersection of multiples of valuation or a volt of valuation ideals over different n, we're here v1, vr are divisorial valuations, lambda one, this should have been in R, this is in, this is non-negative real numbers, okay so one can consider the case where these guys are all, where these guys are all rational numbers or integers, so I will give those different names, I need something to, okay erase with this, so in the case where these guys are all rational numbers, call this a rational, the lambdas are all rational then, this is a is a q divisorial filtration and if they're in z then this is z divisorial, so this is the most natural one, the z divisorial one, okay so a couple examples showing that this comes up naturally is let's let I equals I sub n be an attic, an I attic filtration and this is on a local noetherian ring and then we take the filtration J is equal to I n bar, where I n bar is the integral closure of I to the n, okay and then this is in fact a is a z divisorial filtration and the valuations v1 to vr are the so-called Rease valuations, okay this is the theory of Rease, so also another example is if you take p a prime ideal in say a regular local ring r then I can take the the filtration of symbolic powers, powers of p and this is p symbolic n, so this is just if I take p to the n, multiply it times r to the p and then you intersect with r, so this is a this is also a z divisorial filtration, okay so then there's this Rease algebra, the Rease algebra of the filtration I, I will write it in this way, okay and here where you benefit from it being a filtration that makes us into a ring and this this is make the comment that well maybe I'll just say this that for the I attic filtration this is of course noetherian but for divisorial filtrations in general it's not, this thing makes a lot of noise, is this a camera that's doing that or to help maybe this is oh there's this this I have a second okay so maybe I can put this in my pocket or something is it good enough on the desk okay I'm sorry I know I'm sorry yeah anyway I think I think this will work and material can make a comment if someone can let me know if it's you hear me from here okay so this seems to work and is so okay thank you okay okay so attached to this is an integral closure we can take the integral closure of this ring the integral closure our I in r of t and this is a graded ring associated to a filtration where J sub n is the elements x in r such that x to the r is in I n times our integral closure for some r bigger than 0 okay so this is a little more general than the what you get from the I attic filtration so the classical case if I is is an I attic filtration then J n is just equal to the integral closure of I n you don't need the r here okay so now I want to make another comment is that the Riese algebra of a divisorial filtration is always integrally closed okay so now I will define the analytic spread so I will take noetherian local ring and I a filtration or then you can define the analytic spread of I is the analytic spread of I is the dimension of the Riese ring modulo the maximal ideal times the Riese ring so if I is the attic I at if I is an I attic filtration this is exactly the usual definition of analytic spread that after Prangama and I started working with this that we learned that there's another definition of analytic spread of a filtration which is originally due to Bruins and Schrenzel it's also there's work on this by Hoa, Kamura, Terai, and Chung and Dao and Mentanyo and I'm sure there's some other work too that that I've missed but it's a related but there's other questions there which I won't go into today okay but this is our definition okay well the the first statement and the only statement that we have for general filtrations is the following this is with myself and Prangama which is that the dimension of our sorry the l analytic spread of I is always bounded above by the dimension of the ring okay so it might think with all these noetherian rings that might be infinite sometimes but no okay one thing which I will mention is that if we always have this map over here there's a projection from speck of the Riese out sorry proj to speck of R and the L of I then the analytic spread of I is going to be just the dimension of the preimage of the maximal ideal minus one okay so in the classical case of the attic filtration this is just the blow up of an ideal and this is so one can check easily this is a nice birational map so the fiber can has to be at most dimension of the base minus one but it's true in this non noetherian case also there's an interesting statement a nice statement in for the attic filtration which is that or the attic filtration one has another inequality which is height of I is less than or equal to the analytic spread of I which is in our notation analytics filtration of the I attic filtration but this is not true the mention about how this is true that's actually the upper semi continuity of fiber dimension because this is a proper morphism yeah what let's get it right you're right it's a plus one exactly yeah thank you plus one okay but this maybe I'll just write it like this that in fact this fails and it's not difficult to find examples of of just general strange filtration for which this fails but even for naturally occurring filtration this fails and so this is a even for two dimension a height two prime ideal p in a regular local ring r of dimension three so a space curve singularity and then you have the length of the analytic spread of the symbolic filtration is equal to zero which is less than the height of p which is equal to two so you can do this if this occurs this means that the preimage of the maximal ideal under this map is the empty set so somehow this is not a proper morphism it's not dominant okay so now this is the first main theorem I want to talk about now and this is the second theorem so I explain why some of these assumptions here are here where R be an excellent local domain which is either of equi characteristic zero or of dimension less than or equal to three uh and i is equal to i sub n a q divisorial filtration on r and then the following are equivalent the first statement is the analytic spread of i is equal to the dimension of of r so that's the maximal analytic spread the second is that there exists some n not such that the maximal ideal maybe I should have written this is in is an associated prime of r mod i n for all n bigger than equal to n not and the third statement is that there just exists some such that the maximal ideal is in the associated prime of r mod i m not okay so these are the equivalent and uh first off I'll mention just the following the proof is in two halves where the first half is uh is uh one implies two uh which is with uh with uh prangamas our car and this is true much more generally we don't need all these assumptions it's just uh noetherian local domain uh and then uh three implies two this I did a little later and this the hopefully I'll have time to come back and say a little a few words about the proof but that's where these assumptions come in where I use geometric methods and the uh what I use is resolution I need resolution of singularities so in fact something uh many people don't know is that actually in his original paper hirinaka proved resolution of singularities over an excellent local ring which of equa characteristics zero so this follows immediately from this in fact I've seen people referring to this and they seem not to understand that hirinaka actually proved this because the theorem we all know is that he of course is an immediate quarrel area this is the resolution of singularities uh over uh and out over a field over an algebraic variety okay so uh the this other one this dimension less than equal to three that uh abjuncker of course about the same time in the mid 60s proved resolution of singularities for three-dimensional varieties in the geometric context at least for characteristic bigger than five uh but about a decade ago maybe in a very very long paper that uh uh venison cosar and levier pilter finished it up in dimension three for excellent ranks and beyond that it's all a mystery that that we'll see I mean it could be in characteristic p one could make use of some of these uh uh diong type methods to do something in fact that just occurred to me now that might be a possibility so that that's something that might be a way of of getting around this at least in rings essentially a finite type over a field or a perfect field okay so I'll sit hopefully I'll say a little more about this uh this later another uh comment I want to make is that uh this theorem generalizes a classical theorem of uh McAdam and Birch which uh can be found in the excellent book by uh swanson and uniki on uh integral closure where this is one of the theorems that they uh they present there okay so now uh the next topic on epsilon multiplicity uh so we can define the epsilon multiplicity of a of a filtration i equals i sub n d dimensional noetherian local ring is defined like this this is equal to the limb soup of n goes to infinity of the length as an r module of the colon ideal this is a saturation of i to the n you remove the m primary component divided by d factorial uh and of course this is uh defined by Berndt and Javid Berndt Ulrich and Javid Valedashti who developed many of the properties that for the iatic filtrations and for some some so for ideals and for modules okay so it doesn't take tremendous amount of imagination to make the extension to uh to filtrations okay so uh this is the um uh the first theorem so suppose that i is an attic filtration a divisorial filtration then this thing exists as a limit then epsilon i is in fact equal to the limit as n goes to infinity of the lengths of these uh saturations modulo the ideal and this exists as a limit even for an ideal this can be a an irrational number okay so now uh more oh i forgot something thank you yeah yeah this isn't in complete generality like the definition this is the r is and analytically unrammified on analytically unrammified okay so the next theorem has even more assumptions but i'll remember to write them this time okay so so really um what our theorem is is a is a generalization of this characterization of analytics spread which we had uh before and this is again with uh prangama sarkar uh which is that had the same restrictions uh let r be a d uh dimensional excellent local ring which is either of equi characteristic zero or of dimension lesser than equal to three and i uh be a q divisorial filtration uh on on r okay then we have that the epsilon multiplicity of i is positive uh if and only if the uh analytics spread of i is equal to which the dimension of r okay and and in fact uh this theorem is implicit in this paper of uh javid and uh and baron that somehow they write the proof in their papers but they don't actually quite state it so and i think other people have maybe pointed this out okay this is uh this this uh theorem holds uh for um for idea ideals and it's in a very general situation i think it's uh formally equidimensional i think okay so uh there's a uh i think what i'll do is i'll try to say a little bit of the proof and then there's a corollary i might come back to if i if i have time so i will say a little bit about how one proves theorem two and uh and five so say some ideas of the proof of theorem proves of theorems two and four these are the statements uh characterizing maximal uh analytics spread uh and uh the first case is uh and i think i uh i forgot something for this one i need normal somehow i wasn't quite able to get rid of the normal assumption uh and but i don't need the normal for the theorem two uh so and so one can reduce to r being normal for theorem four this is a triviality okay so what i do then is i can uh struct uh pi from x to spec of r oh which is the uh such that x is pi is the blow up of an ideal not not anything directly connected to the filtration of some ideal uh and this is where i use a resolution of singularities x is non-singular uh there exists uh uh divisor d equals a1 e1 plus ar er on x such that uh such that uh oh uh uh if i take the sheaf of this o x and d and i take the uh round up of this and i take the sections of this uh then this is just going to be equal to i n so this converts it into a problem in geometry uh something which i didn't mention here which i should have uh is uh that this is this fails uh if you if you change this to real divisorial valuations if it's real divisorial valuations it fails okay so maybe i should add this here our false even for our divisorial filtrations okay so that's kind of a subtlety uh in this so this is this is acute divisorial filtration okay so uh now uh what i'm going to do is this was the motivating uh idea for this where there's a a case where everything works out uh very easily uh and one can uh prove all kinds of even more so i'll mention this because this is the motivating case so let's uh first suppose that uh the dimension of r is equal to 2 uh then there exists uh thanks to uh local form of zariski decomposition but you can write uh you can find q divisors delta and b such that first off you can write delta's d plus b uh these are effective effective such that uh delta is equal to d plus b where one has the property that minus d is nef in other words uh every uh curve which contracts to the maximal ideal has uh um non has a non negative intersection number with this and b is uh exceptional for uh for pi in other words every component here is contracted to the maximal ideal okay so and the critical point is that in fact we can compute these sections here from which is we know i this is ultimately i n this can be computed okay so especially in dimension two as uh zariski explained to us uh and it's since been developed in higher dimensions that there's really uh the numerically effective divisors behave much much better than uh than ordinary divisors even if their sections rings are not finitely generated and the reason is uh is that uh because of the uh fugita vanishing theorem uh so we can reduce to the case minus d is uh is uh nef okay so the the reason why this is so uh nice is that uh because uh good i'll call them almost vanishing theorems so basically if you add a little bit of something extra you can get vanishing hold for the powers of this uh and this is by fugita and in his uh in his original paper he uh proved this for um in characteristic p so in fact when we use it we reduce to the case where we're looking over the exceptional fiber which is a and then if you look at an integral component of it that's a projective variety over some field the residue field of the of the variety okay so uh and then you can use this to prove theorem two and four in dimension two okay so this was the this was a motivating idea and this was the first thing i thought of but then there's a big problem in that this fails in uh higher dimension so uh this uh does not generalize to higher dimension sadly it means bigger and equal to three and these are some old examples by myself and nakayama and in fact uh send some really out of town i'll just say a few things that nakayama has some analog of this a very weak forked version of the risky decomposition he calls sectional decomposition uh and uh part of this theory is there's a there's an invariant where i guess i can i can write down what the invariant is but you have about five more minutes five six more minutes okay so maybe um okay so maybe i can even write a little bit bit down then okay so uh okay so maybe that's why i was starting right in dimension r bigger and equal to three uh we use a local version of nakayamas this is not the nakayamas lemma nakayama what do you call this sectional decomposition but you need uh this is where we need the uh non-singularity uh condition uh for it for it which is a weak form i'll just say a weak form of the risky decomposition which holds in higher in dimension bigger and equal to three uh which uh uh to prove two and four and the the critical thing that we need in it is a proof requires a delicate analysis of this function of the function gamma f minus d or maybe other divisors in fact for f in in the case where f is an integral divisor which which contracts uh to mr the maximal ideal uh and uh the way you define this is uh gamma f minus uh d this is a limit it so that's a a small theorem that it's a limit not just a limb soup okay so now we've pushed the definition uh back one what is this tau f md minus md this is uh the infimum of the order of g such that uh g is an effective uh q divisor on x with g uh linearly equivalent uh to uh minus md on x okay so linearly equivalent means that if you take g the divisor g and you add md to it's the divisor of a function on the quotient field of r so even if this is some i mean in this definition you can take uh divisors with real coefficients okay so one of one of the subtle things in this is that it's true for the theorem's q for two q divisors it collapses for r divisors but when you take the sectional decomposition this thing could be a real number okay but the the thing is if it's a if it is then you can add just a little more and still stay in your original divisor because the coefficients are integral and then that puts you in a situation where you can do something uh something you can do something with it okay well thank you thank you very much Dale are there any questions no that occurred to me that that i would hope that could be done i haven't tried it but i would i would think it probably could be done but yeah that's a good that's a good question another question then thanks again