 of Jacobson in Yale and perversely wanted to work on Communitive Algebra instead of what Jacobson was doing and so Jacobson I guess I had had some contact with Jacobson mostly through Kaplansky previously when I was a student Kaplansky liked my work and suggested my name to Jacobson at some point and so Jacobson had the idea for which I'm forever grateful of sending Craig to Brandeis to play games with me. I think I didn't know quite what a jewel I had there but we had a good time together I remember sitting on the lawn and talking about Communitive Algebra with Craig and he of course wrote his thesis with D sequences I was very interested in in matrix things and minors of matrices at the time and still am and this was the beginning of a long friendship he too has been kind to me in ways that cost him personally I think in later years and kind to mathematics generally really a very generous nature I always associate that with being Midwestern though I think there are unpleasant people in the Midwest too but Craig certainly not among them and one of the sweetest people I know and one of the best mathematicians so with that said let me go on to the subject I think probably everybody here knows what a resolution is you have some module of a some ring I'll mostly work over local rings Kailey will come into the discussion but doesn't need to be there yet let's see I think one of these is the right one yes and so resolution you have some local ring R with maximum ideal and perhaps as you feel K and the resolution you have some R module finitely generated everything in sight today will be finitely generated and the theory and so M is a finitely generated module you want to understand it so you begin to understand it by producing generators of N and relations or let's call this N zero or N one some relations and then you say well I still don't understand them very well and Hilbert had the idea that you could understand it better by continuing this process and looking at the relations on the relations are in two and so on and Hilbert proved famously that if you have if ours a polynomial ring then the process is finite pre-resolution can be finite and in this minimal in this local case you could say a minimal free resolution and then it just is fine so at a certain point you get to our sub Nk and then the next one was zero so so that's Hilbert's synergy theorem but let me back up a little further I want to say so much not so much what is a free resolution since you all know that but rather why is a free resolution and I think the first person to have a good use for these things is actually predates Hilbert by quite a bit Arthur Cayley in 1848 was interested in one of the primary algebraic problems of the time namely elimination theory given a system of polynomial equations can you tell whether it has a solution or not and Cayley adapted what was not yet called the kazoo complex to answer that question as you all know a system of elements is a regular sequence if and only if the kazoo complex is exact and you can test that exactness in a finite dimensional way to just by looking at a certain element of the co-cranel to see whether it's zero or not in the case when you have a homogeneous regular sequence let's say so can he was interested in the question in projective space so his polynomials were homogeneous and of course the system has a non-trivial solution then if and only if the polynomials don't generate an irrelevant ideal and so if you have n plus one variables and n plus one polynomials you can test whether it is a generates the irrelevant ideal by testing whether it's a regular sequence and the kazoo complex was invented by Cayley for that purpose so I don't know whether Hilbert knew about Cayley's work or not but he gave us as an example first examples of free resolutions he gave the kazoo complex Hilbert of course was interested for a different reason he was interested in counting the number of independent invariance of a group action finite groups are relatively easy in this regard but the problem of the day and really the main problem that afflicted the people in that period of the 19th century around 1890 was to compute and count the invariance of certain actions mostly of SL2 but other groups as well so if you look at the SL2 acting on polynomials in or GL2 acting on polynomials in two variables then nearly then there's an induced action on the polynomials of degree D and if you think of polynomials of degree D in terms of their coefficients then you're looking at the D symmetric power of the standard representation and you can look at polynomials on that which are invariant into the group action and this had been computed with great effort and difficulty in a number of cases basically they could do D up to about 8 at that time and they always found out that the number of invariance number of new invariance that you got in degree N let's say if you made that into a power series that turned out to be a rational function in M which amounts to saying that there's some linear recurrence relation for the number and people had observed that in all these cases and Hilbert sort of wiped up the field by computing nothing but proving that that was always the case and the reason was that if you have a finite resolution over a graded ring then you can compute the what we now call the Hilbert function of M in terms of the resolution so you have a graded ring and you can compute the a graded free resolution and I think this will work so you want the Hilbert function of M you know the Hilbert function of the ring itself that's just some combinatorial thing about the number of monomials of a given degree and so if you take the alternating sum of the Hilbert functions of the different free modules you get the Hilbert function of M that was Hilbert's contribution in his case he proved the finiteness finite generation of the invariance that gave him a finitely generated module to work with and then in introduced finite free resolutions in order to compute the Hilbert function so this was a big deal at the time and really made Hilbert famous as a mathematician so this was Hilbert's great great papers on commutative algebra around 1890 then the next use big use of finite free resolutions was perhaps the one by ser so let's jump almost a hundred years not quite and ser use them for a different purpose if you have two varieties meeting in a point then you would like to say that there's a nice nicely defined intersection multiplicity and people had assumed that if they met at a point low represented by this local ring are let's now say that are as regular you to avoid restrooms which will come up later so I'll call my regular ring s s n regular let's say the local ring of a point a smooth variety then you'd like to say if you have two sub varieties of that smooth variety meeting at a point there meeting at that point point p let's say in x then if you have y and z and y intersect z as a set is just p then you'd like to assign an intersection multiplicity to y intersect z and people are initially thought by analogy with plane curves that you could just take the ideal of y and the ideal of z and add them and look at the length so it's the fact then that if you take s modulo the ideal of y plus the ideal of z then the length of that is finite and so people thought perhaps the length would be equal to the multiplicity of intersection that's okay for plane curves where s is the local ring of a point in the plane but there are troubles in higher dimensions for example you have two planes meeting in a point in four space supposed to be planes meeting a point and if you cut that with another plane so in four space typically a plane meets another plane in a point so if you cut it with a general plane so here's use y let's say and z is a plane then y intersect z typically for general z is two different points one for each plane but if you have the temerity to pass your your plane z through that intersection point so the point is there p is there then the length of s mod of the ideal of y plus the ideal of z everybody can write down those ideals easily and make this computation that turns out to be three and not two so this was a problem it's very desirable in algebraic geometry that intersection multiplicities should be preserved when you move the varieties around so this this the wrong Grubner who put forward the definition as the length said that this just showed that that property of of intersections being the same number for every choice of z just was wrong and there was a controversy between him and other mathematicians fundervarden and he had a correspondence about this in which fundervarden said well you know it's very nice to think about lengths but we have to give up these things if they don't agree with our our real desires which is for continuity preservation of number let's go and fundervarden carried the day and there were definitions that then people played with I think Samuel had one and fundervarden proposed something also and there were definitions that made the number come out to be two but the real key was I think then or the most impressive thing for me was a theorem by ser and ser said this length this thing is really just tour one of s mod i s mod sorry towards zero s mod i of y s mod i of z so this is the length of that but if you subtract tour one you get the right answer so he approved famously that the right multiplicity of y and z meeting at a point p is the alternating sum of the lengths of the tour eyes and this turns out to have all the good properties there's still mysteries about when it's zero or positive but this was very satisfactory result however notice it depends on the resolution being finite right we compute this by resolving a small i y let's say and if the resolution is not finite then there'll be infinitely many of these tours and the alternating some might not make any sense so this is very much a theorem about finite resolutions now certainly in the 60s people were interested in the more general case and the first reason I think that was around for a serious interest was probably in homology of groups there the homology let's say of group g with coefficients and some g module m is by definition the tour over the group algebra k of g of what amounts to the residue field I guess I'm working over case I'll say k here with coefficients in m so the resolution of k was the important object resolution of k over the group now it turns out that in a lot of cases the complexity of the representation theory of g and the complexity of g itself doesn't depend on the whole group algebra which is of course very non commutative but it depends mostly on the maximal of billion p subgroups person in characteristic zero this is all rather trivial characteristic zero only towards zero everything is semi-simple but in characteristic p which is where people were interested in they're infinitely many of these things and the complexity somehow depends how big a copy of Z mod p to the end is contained in in g so of course there's a silo p subgroup the silo p subgroup might can is probably not a billion but certainly contains a copy of Z mod p but it might contain a direct sum of copies of Z mod p by the way the case where it contains just one copy those are called thin groups and a huge deal and in finite group theory is made over the classification of thin groups and then the next case is quasi thin groups where n is 2 and then quasi quasi thin groups and is 3 and then things get more uniform for larger n but this is really the the in some sense the main thrust of the classification theory of finite groups is to study these elementary a billion subgroups of the group so this is as I say a big deal in that case p is two the interesting case there is two when I was a student at the University of Chicago years and years ago John Thompson was one of the main figures in the faculty and he was working on groups of odd order so the idea is that odd order groups which have none of these Z mod twos in them are somehow simpler not very simple I have to say in the big paper of Thompson and fight settled that there are no extra simple groups except the obvious ones in odd order for odd order anyhow that's a digression but important in our story because John Tate was interested in group homology in the service of number theory and so he was interested in computing these things and interested in particular in computing the tour over this subgroup so Tate this is about 1960 now computed a free resolution residue field K over the ring K this is a characteristic P ring necessarily to be of interest K of Z mod P to the end now this this ring is rather familiar to commutative algebras in a different form unfortunately we have a noise maker in my roof I hope you can't hear it objectionably they promised me they wouldn't make noise until eight we can we can hear you fine David can you okay good so if you look at the group elements that generate this group Z mod P to the end so G to the P is is zero or G to the P is one I should say let's write that in that form and if I look at G then mod one to the P it's zero in the group algebra so it's generated by these things so this is really the polynomial ring in any elements mod each XI to the P so this is a complete intersection and Tate recognized this of course and and wrote a paper then about resolutions of the residue field over complete intersections so resolution of K over any complete intersection ring I'll abbreviate that as CI for the moment so those are the first infinite resolutions which are important I think and Tate observed that the free modules could be written in a very simple way you have the kazoo complex of the variables inside so let's call this F and then what you have is the kazoo complex first the kazoo complex of course begins K and then the ring S and then we have n variables S to the n and wedge 2 S to the n and so on nice and finite wedge n S to the n and that's the end of it however it's never exact except in a regular ring at this point and what's wrong is that the generators of the ideal I should have put an S mod I here sorry the generators of the ideal go to zero and or r0 and you can express them in terms of the maximum ideal this is the row of variables and so that you can express the generators in terms of that and those are zero so you have another term which is as many as there are generators of the ideal as to the M maybe necessary there and Tate proved that the whole resolution F was a tensor product of the kazoo complex wedge of S mod I and S mod I too tensored with this the dual of the symmetric algebra sim of S mod I to the A the great dual of that so it's very simple structure and that got people interested in infinite resolutions I think this is about 1960 as I say and so Kaplan's key at that time was thinking about projective dimension homological algebra and he posed the problem whether all infinite resolutions might be like this one so what is like this one this is this is almost a polynomial ring right it's mostly a polynomial ring tensored some little bit finite dimensional piece and so it's Hilbert function is rather simple this is the Hilbert function of a graded module whose generators are actually all in one degree so it's really just a polynomial but again if you make a power series out of Hilbert functions of things with generators in many degrees then those are rational functions so kaplan's he either conjectured or posed the problem let's say it's a conjecture I think that's a it's plausible conjecture that kaplan's he conjectured it because kaplan's he had the philosophy which he expounded to his students including me I always listened to him when he talked though I wasn't officially his student he had the philosophy that you should always pose as a conjecture the strongest thing that you think would be possibly true because that would give people the most interesting thing to shoot at and if they proved disproved your conjecture then they would feel very good or if they proved your conjecture they would feel even better perhaps but I would it would get the competitive juices flowing I'm sure Craig and Mel would understand that philosophy very well and and therefore we produce the most mathematics he wasn't interested in being right he was interested in in getting people to produce mathematics very sensibly so let's say he conjectured that the Poincare series every local ring is a rational function and here the Poincare series is defined the following way you take the resume k you take its minimum free resolution smi this is the local ring now I guess I should start using our for a local ring you back up and say our then you get some stuff f1 f2 this goes on probably forever and you take summation well he took the positive sum I'm not quite sure why summation of t to the i times the rank of f sub i as a free module equals zero to infinity okay that's the Poincare series p r of k so that should be a rational function and then people started proving it was a rational function and they proved it in lots and lots of special cases many special cases are known now at this point I want to change gears a moment and tell you a political story about mathematics yes mathematics has politics in it young Eric Roos was a postdoc at Berkeley when I was a student there very attractive interesting mathematician very powerful mathematician he was working on category three at the time and he was from Stockholm he was to there at Chicago I think just for a year and after I graduated a few years later I was long-term visit in France I went to visit Roos and Stockholm so we were both very young men at the time and he and young Eric Björk who was also there were plotting to change mathematics in Sweden at that time mathematics in Sweden was essentially dominated by partial differential equations and partial differential operators it was a very powerful group there and there was essentially no algebra and Roos and Björk were both algebraists and wanted to change that and so they plotted and Roos felt that it would be possible to choose a problem in algebra and gather a group of people who would work on that problem and that would be the way to build up algebra in Sweden and this strategy worked brilliantly actually and algebra in Sweden has been going strong ever since the problem he chose was Kiplanskis problem or Kiplanskis conjecture about Poincaré series of a local ring and so he built up the school there Fröberg is a representative of it Mats Boe is a descendant of it honors Björner came in some point in that time Roos died a few years ago but the school lives on and they worked on this problem a lot Lucho Avramov became their collaborator and did a lot of important things in this area and the focus was very much on the ranks of these FIs it was a numerical problem that they were trying to solve in some sense so a great deal of work was done and then the balloon was popped in a way David Annick found a counter example now to understand where that came from you have to know that there's another place where resolutions of the residue field and over a local ring are as important and that was in Sullivan's work on rational homotopy theory something called a minimal model in homotopy theory and it turns out that it has something to do with this question with this resolution so David Annick was really a topologist by training and he pointed out basically that the topologists had known a counter example to this conjecture for a long time already in this other context of rational homotopy and he was able to adapt it to the commutative algebra situation so he found a counter example quite simple actually just beyond the examples that people had been able to do or discuss so well did that stop the field not at all mathematicians are always capable of pivoting to a slightly different problem and then the problem became well when is the Poincare series rational or if not what is it anyway how how fast the Betty numbers grow so the field still revolved around and still does resolve around size of the Betty numbers the Betty numbers are the ranks of these fi but there's more to a resolution than that if you have a resolution minimal resolutions over a local ring are unique up to isomorphism so all those matrices up to equivalence of matrices are also part of the invariance that you get and you can ask much more refined questions about the matrices now in the case of finite resolutions this had been already a big deal so back to finite resolutions for a moment David Buxbaum and I and many other people Mel and Craig among them other people no doubt in this audience we're interested in the structure of finite resolutions and we proved some theorems of course Hilbert already had understood the structure of resolutions of length two of a cyclic module so a resolution of s mod i s is a polynomial ring and i is homogeneous and and projective dimension i is projective dimension one or s mod i is projective dimension two and that was generalized by by Lindsay Birch to the case of local rings same for local regular local rings or any local ring where s mod i has predicted dimension two actually so this and the structure was really a theorem about the matrices in the resolution and the ideals of minors in the matrices so the emphasis was ideals of minors of matrices in the resolution but for some reason this didn't lead over into the study of infinite resolutions and people kept on studying this numerical question well it turns out that there's a lot to say about ideals of minors in a resolution and recently I got interested in this also for the case of infinite resolutions in work with Heilong Dao which I'll tell you a little bit about first let me backtrack for a moment and just mention that the case of finite resolutions is by no means closed and there are plenty of problems there and a typical one is well Birch did the case of length two how about length three and Jerzo Weyman who was a student of bookspelm put forward a very revolutionary kind of idea and back in in 89 that there could be a structure theorem for resolutions of length three two but the base of the of the family of resolutions would not be a finite dimensional base in general but rather an infinite dimensional graded Lie algebra horrible we thought because who among commutative algebraists knows about infinite dimensional Lie algebras but Weyman wasn't afraid of this and he pursued this off and on over the years without making decisive progress until quite recently and now he and a group of his students among them in particular a very bright student of mine named Shanglong Ni has really finished this program in a special case turns out that sometimes these infinite dimensional Lie algebras are only finite dimensional they're sometimes really infinite dimensional and the cases when they're finite dimensional correspond as usual to the Dinkin diagrams the A D E Dinkin diagrams and that corresponds to looking at free resolutions over a local ring with just certain bedding numbers certain sequences of bedding numbers so among the sequences they like are one N N one and they're interested mostly in in Cohen-McCauley s-mod eyes and many of you know that when you have a Cohen-McCauley ring s-mod eye and its bedding numbers are it has n generators and and the scissorgies are one of ranks one N N one then booksmen I prove that it's generated by the faffians that's the square roots of the maximum minors of the middle matrix and that's the Gorenstein case so that's one of the cases that turns out to be one of the Dinkin cases and it looks as though the other cases will be not as simple but as manageable in the end as that one so there aren't so many different sequences of of bedding numbers that work famously one six eight three is not one of them that's going to be an infinite dimensional family but the Dinkin case looks like it's going to be completely settled and a famous conjecture of of uh Yuniki and Ulrich I believe is that in in what turns out to be those Dinkin cases the ring s-mod eye will always be Lichi if you know what Lichi means that's too big a digression to go down that road but that's one of the things that Shanglong Ni and Yersha and their collaborators have proven now I think not quite um well written down yet so take it with a grain of salt but I'm convinced that this is going to be written down soon and proved anyway that's the progress in the finite case but what about the infinite case so Long and I began playing with this question and turned out that there was one case which was particularly simple and I'm going to describe it to you and uh also uh tell you some of the experiments we've made which look like they might be good theorems but we can't prove them except in special cases so so what's the easy case turns out that Lindsay Birch the student of David Reese which is who's famous for among us for the Hilbert Birch theorem did some other pretty nice things too and she defined a class of rings which uh Long has christened Birch rings with his collaborators so what's a Birch ring or in fact I'll define something more the Birch index of any ring so suppose you have a local a regular local ring s again ideal in s and I'm going to be interested in r which is s mod i and the first case will be when r is depth zero but we'll reduce to that case by fact we have regular sequences so we say that the uh Birch index so if is zero and I'll say that the Birch index of r is the dimension of the maximum ideal mod a certain Birch ideal and notation isn't of i um let's just call it b of i for this talk b of i what's b of i b of i is the following peculiar looking thing let's look at the maximum ideal so I didn't say what the maximum ideal was so I'll write n for the maximum ideal again and k for the residue field of s and b of i is then n times i colon with the suckle i colon n okay so how should you think about that I think of it this way here's here's i so here's s and it kind of looks like this right here's the unit element here are the variables x one up takes n so i is some funny thing down here let's think about the artinian case and then the suckle of i the suckle of s mod i which is i colon n mod i that's this this thin layer up here right here's the suckle and n times i those are those are the things that are not generators so n times i is the stuff down here so we're interested in what throws the suckle into n times i and from this picture already it's clear that the birch ideal contains the square of the maximal ideal since um i colon n is not contained in i this is not doesn't contain the unit element so it's contained in the maximal ideal so this little sandwich m mod m squared is find a dimensional vector space dimension equal dimension of the ring and some of the variables are in i colon n mod n i and maybe all of them or maybe not so this this birch index is essentially the number of variables which are not in n i colon i colon n so just as an example if you take uh so if you take i equal to a power of n n to the p let's say then i colon n is i is n to the p minus one and n i obviously is n to the p plus one so the birch ideal in this case is n to the p plus one colon n to the p minus one and that's n squared so in that case i wrote m here for the maximal ideal because i usually do but this should have been and so in this case the birch ideal is n squared and the birch index is whatever the dimension of s is so birch index which index is equal to the dimension s simply so that's a typical case rather trivial case and the birch index can be any number it wants up to the dimension of s including zero i'll tell you some computations of the birch index in a minute but here's a characterization of things of positive birch index so a birch ideal i is birch it has birch index at least one and here's a characterization of birch and a birch ring is a birch ideal is s mod a birch ideal uh i wanted to find this outside the depth zero case two if is bigger than zero reduce mod a regular sequence regular sequence linear regular sequence and then compute the birch index so birch s mod i in general is equal to the birch i plus this regular sequence okay so that gives us a definition in general you might ask whether it's independent of all the choices i made and the answer in the zero dimensional case for the depth zero case is yes i don't know whether it's independent of this regular sequence probably is and i'm not so concerned i'm mostly concerned with the case of depth zero okay so suppose for s mod i always depth r is zero then r is birch if and only if the residue field is a sum end of the second synergy of k so something as funny as happening in this resolution of k right i have k here's s or r i guess then r to the n here's the second synergy says and we continue r to the n this was r to the n one maybe r to the n two but in the minimal resolution has to be as a sum end in this case another copy of the minimal resolution of k shifted back to so at least every second synergy will have a copy of k in the in it right so this obviously implies that ciz to n of k contains a uh sum end i'm writing k divides that to mean that k is a direct sum end of that not just a sub module very strange behavior and not too hard a theorem is that in fact k divides every synergy starting with the second one k is a sum end if if if the ring is birch so so that's kind of surprising phenomenon that had never been observed as far as i know in uh in the study of infinite resolutions right it's this self-reproducing phenomenon very peculiar well how common is that after all what about other modules for example so here's a theorem that dow and i proved this is all work with dow if r is birch is any finally generated module let's say zero dimensional or depth zero is really what i mean i keep getting this funny stylus ciz m well let's say ciz p of m contains the sum end k for all p at least seven somehow if you think about things like this you might expect p at least dimension embedding dimension of r or some other measure of r but seven is a bit of a surprise there might be true for all p at least five we can't prove that um but this show this is very peculiar behavior indeed for example this is true uh oh i'm sorry birch of index at least two it's not true for all birch rings index at least two uh so but it's true for example for the rings of the form s mod of power the maximum ideal even that was i as far as i know completely unnoticed in the past that some somehow ciz g's cropped up with k's in them k has nothing to do with m a priori right just appears out of nowhere there it's really not a sum end of m um in general so this is for me a very surprising result it brings up the question of what is this condition birch index at least two how many rings have this property and i wanted since i do want to prove something in in a talk i thought i would um try to prove something about this but first let me just mention that we've seen lots of other strange regularities sorry about that is my screen visible again can you see my screen yes you're quite visible as is your tablet um my my ipad turned itself off for a moment so another regularity we've seen is the following if you look at any any local ring r and any finitely generated module and you make the minimal resolution of m minimal resolution phi sub i is the ith matrix in that resolution then in analogy with the finite resolution case you could you could think that maybe the ideals of minors of these fees would be interesting now you can't talk about the ranks of the fees very well because too many things are zero in r maybe r is zero dimensional um so it's not going to be interesting to talk about the ranks of the fees but you could still talk about an ideal of minors of a given size so for example you could take about you could talk about the ideal of one by one minors of phi i that's an interesting sequence of ideals um and what we have observed in a number of examples all the examples we've looked at is that this sequence of ideals stabilizes or becomes periodic of period two that can happen it happens quite quickly in most examples we've looked at we tried to guess how far back you had to go but we were unable to do that well our conjectures turned out to be false after a while but nevertheless if you write down some simple ring where you can really compute this for a number of steps you'll never compute infinitely many of course you'll see that this is true and for golden rings you can just about see why it's true so i think that i haven't dotted the eyes and the proof but i think this is going to be true and relatively easy for golden rings where you can write down an explicit resolution of modules but um in general i don't know so that's uh that's a completely open question you can ask about larger order minors too that seems to be true too but again i can't prove any such thing um so i'll leave that just open and instead let me tell you a computation of oh i see i have one minute left so maybe i won't prove it after all but it's interesting to know some class of rings where you can actually compute the perch index so the of course the simplest class is always perfect ideals of co-dimension two so theorem in s s is low as regular remember perfect of co-dimension two so then i is the ideal of minors of some matrix i is equal to some i n of a matrix m call it something else a of size n by n plus one then the birch index is essentially the number of linear forms in the matrix a dimension of the space of those so birch index of i is the minimum of say this correctly the minimum of two and the um the number of variables that appear so the dimension of this matrix the ideal of one by one minors of a plus the square of the maximal ideal mod the square of the maximal ideal turns out that this is the birch ideal in that case so if there are some linear forms independent linear forms appearing as the leading forms of entries of a then you get significant birch index if there's at least one that's a birch ideal if there are two then you get this very strong theorem about the resolution of every module over s mod i and with that i think i'll stop and thank you for your attention are there any questions there must be someone no okay well thank you david and let us thank you again