Cohomologically full rings
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Dao, Hailong
Inspired by a question raised by Eisenbud-Musta\c{t}\u{a}-Stillman regarding the injectivity of maps from Ext modules to local cohomology modules, we introduce a class of rings which we call cohomologically full rings. In positive characteristic, this notion coincides with that of F-full rings studied by Pham and Ma, while in characteristic 0, they include Du Bois singularities. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras. Joint work with Alessandro De Stefani and Linquan Ma.
 I joined work with Linh Chuan Ma and Alexander Ode's Stephanie. I guess I should also thank the neighboring institution for hiring so many great young people. And I have benefited a lot from working with them. So, I should also talk about nearby institution. I also advertised KUMUNU 2018, which is annual community algebra conference. This year will be organized on October 13th and 14th in Lawrence. Last year we had a candlelight dinner and an open bar. Many people told me that is the best algebra conference they've been to. Anyhow, a lot of people will come. So, I wanted to talk about, since I'm literally eating to your dinner time, I want to tell you some stories right away. So, what are these objects? This is a new class of rings that include Kohen-McCauley rings, Stanley Reissner rings, the so-called Dubois Singularity Singularity Zero, and FPU Singularity Incarretera Ristate P. So, they include a lot of good, you know, I also say that life is worth living, Kohen-McCauley rings, and a lot of you live very comfortably in the Stanley Reissner ring. So, this seems to be a good class of rings. Of course, you know, for this class to be interesting, we should also be able to say something, otherwise you can just define any ring to be a combo. So, we should be able to say something interesting about it. And since the theme of this conference is about CG's, I will state one of the consequence, one of the results, and then after this you can go to dinner if you want to. So, that's simple I, you could add some more I is logically full, whatever it means, I'm going to be five later. So, if S is regular local, then the projective dimension of R over S is less than or equal to the number of generator I. So, it satisfies a very strong individual banal sort of stillment, as a very strong stillment question, right? And also, if S is just a polynomial ring and I is gridded, then it also, we also have the regularity of I is less than or equal to the number of generator I times the, so here D of I is just the maximum degree of a generator, of a minimal generator of I. So, again, it shows you that this class of rings are not every ring, they satisfy very, very strong, you know, rather nice property homologically. On the other hand, they include also all of these classes in the variety itself. So, it seems to be a right side, that's what I'm trying to convince you. Okay, so that's the main thing I want to talk about today, but let me actually start with a motivation question, which actually comes from a paper by David Michemustata and stillman. And so, let me just remind you that, okay, so S is a commutative material ring and I and S is an ideal. And so, the local homology of S is, the ideal homology of S is supported in I is by definition and following. So, okay, so the direct limit of X module and here I sub E is any nested, you know, system that go final with, go final with, with the power. So, for example, you can take, you can take any system. So, for example, IE equal to I to the E or IE equal to I to the symbolic power or IE to the, equal to E to the Frobenius power in characteristic Cp and so on. So, this is a direct limit. Local homology module contains a lot of build, you know, each useful information. On the other hand, a big drawback of this module is, I'd say, really finally generated. And so, that motivates the following question, which I, as I said before, is due to, was first raised by, in a paper by David Michemustata and stillman. So, when is, when is this, direct limit actually a union? Okay. Well, the obvious reason why we want this to be true, namely that any question about this module can be now reduced to a question about IE, which is a finally generated module. Right? Of course, this is just mean. It's not hard to see, it's just mean that each of these map is injected. Okay? So, in order for this to be true, you need to, and so it motivates the next question, which is when is this map injected? So, when is this map? And in fact, one of our first observation is that this class of idea, which I haven't defined yet, will sort of answer this question precisely. So, let me, let me now focus on this question. So, here's a quick observation, namely that if you have a map, so if you have any idea in our system, then the map from here to here factors through this. So, for this map to be injected, you need this to be injected for yourself. And if you, let's say that you leave in a, if S is a complete local ring for simplicity, then by local duality, say S is a complete local, I mean the question here is, this is a local question, you can, you know, the kernel here is a finally generated module. So, to be injective just mean that local is, so, so, so you're going to work completely lovely if you want to. So, this is just mean that the J log of homology, so let's say SM here, of S mod IE is so J1 to J i and here J is just equal to N times i and the dimension of S. So, this is a suggestion, this is equivalent to that. Okay, so this motivates the following definition, that the main definition of the top, so let me introduce, so this is some time called thickening of S mod IE. So, let me just, let me, so to give an intrinsic definition without refer to some embedding and you need to make a formal definition of thickening, so let R and be local. A thickening of R is a suggestion from T to R such that the, such that the induced map on the repo, sorry, this reduce, on the reduce part of T and R is a nice form of esmophism. Well, example is T equal to Kx mod x square and R is Kx mod x. So, you know, this is a picture that you've probably seen in many beginning course into, you teach a beginning course in, in, in, under regulatory right, is a, a fattening of this, this point, a double point, right, so it's a thickening. Okay, so now I give the definition, which is motivated by this observation. So, so definition, so Rm, again I already gave the local definition point, is called homological full beef for any thickening. Well, obviously you have the same skin structure, so your thickening will be a local ring. Then the map, sorry, is injective for pi, so sorry, it's ejective. So, this is the reason why we call it's homologically full, so for all thickening, you don't lose any, it hit all the local modi element in R. And so that, the motivation question and the definition. I should remark that if you have an integrated, so remark, you can define the grid, because in fact, the equivalent to the, so R is standard grid, you can also take this as definition, then R is homologically full, even though if Rm is, so that's the definition. So let me, so the first result we can prove is that this relationship is actually a lot tighter than we just saw, so in fact the, this definition is almost the answer to the question by David. So long after a remark, a definition or a statement? Okay, we have a definition, if you want. So, okay, so a little bit of story here, so, so you know, you can, this, okay, so this is a lazy way to define it, right, so, so, so, formally you should define it as like any grid of thickening or that, but it will say it. So, the G point is defined, that's why I called it remark, I guess. Okay, anyway, so I will, so what is this? Oh, right, yeah, so, so the first theorem, say what I just said, so, okay, so we, we don't have the full strength, but let's say i is S mod i and here S is an unremifed regular local ring. If you don't, well, if you know what it is, you know what it is, if you don't know it, probably doesn't affect your life. This one, right, so, so then, then, then i is co homologically full in our sense, is actually equivalent to this map, injective of i, so, so in fact, it is the same definition. However, they are, you know, so, so we found that it's, it's when working with this, so then you can sort of take one of this as a definition. On the end, this is intrinsic definition and also when you work with this concept, this is useful to think in both ways. We won't see some examples of that if I can get to some proof, but yeah, so let's, yeah, so, so let's look at the examples. Right, so co homicoly, why is that? So co homicoly, well, if you have a, if you have a map like this, right, and you have some kernel here, then just as long as I see what a little homo should tell you that this is a co homicoly. So, hi, okay, hi, hi, and so below the dimension, it is this zero, so it's nothing, but above the dimension is, you know, this is vanish, right, so it's this, sorry, above the dimension, this vanish, so this is, or it shrinks away to look at co homicoly ring, it turns out to be an useful way. Right, what about Stanley Ryan's ring? Well, this is actually, this is true because, because actually Mirscher proved that this is an injective, an injection for square free monomial ideal. So, if you believe this then, you know, although I mean, directly it's not obvious to see that this this is property is false, but, and there are things like FQ and reward singularity, so the reward singularity property was proved by, in a paper by, by Lindscher and, and Carl Schroed and FQ is followed from the work by Lindscher and Fa. Okay, so, so, so this, although they didn't explicitly say it, but it's the same thing, they did prove those things. All right, oh, so, also I want to give some small dimension example, so basic, so basic property here is that if R is co homologically full, then R is unmixed. And so if, so if dimension of R equal to 1, then it's equivalent to the R's primary quality and if dimension of R is equal to 2, then, then R is homologically full, is equivalent to the following. So, let's say, okay, okay, so I will need to assume that I will ask for I when, when S is a complete rate, okay, so maybe, and this guy, this is a little bit of subtlety because when you complete, you lose some irrelusibility thing, so, so it needs to be a little bit careful of I as I, and this is, it's this probability, okay, so, so I is equal to, so basically all the connected component of the puncture spectrum of R are quite equivalent, okay, so, S mod I actually, component, connected component, component of a spectrum R, and each of these has to be equivalent. So, this fact use, the fact that the combative dimension of something that whose puncture spectrum is connected at most at minus 2, so it's actually a non-trivial statement, but anyhow, so this is some example, so you can see that obviously not everything is converging full if you need further evidence than you can classify in, in small dimensional case. Okay, so, right, so there are many properties that we can prove about this, this class, so before talking about budget dimension regularity, let me just mention that you can prove some sort of weak or a intervention property for this rings, you can compute Lubesnik numbers and all that, so, so basically a lot of things are true for this sort of statement that's true for this class is true for more, but let me just prove the connected components from the point, or want to be, does this pass from connected components? Yeah, so, yeah, so we prove that it's pass from connected components, so this full if and only pitch connected component of the puncture is full, and so, yeah, so if you have a Gwen Macaulay each component don't create some, so that's another way to get a new example. Right, okay, so let me just remind you of what I, so yeah, so for the last five minutes maybe I will sketch the statement, so once you, once you have this is not so hard, so one is like a positive dimension, so again i equal to s more i, and r is from degree four, this less equal to a number of generator of i, and also the regularity of i is less than equal to this number of generator i times this degree of the maximum generator, maximum degree of generator plus the dimension of r, yeah, so this is a two of the property that's the best most relevant to the theme of this workshop, so how do you prove the, how do you prove the first, so this is really quite easy, the first one, so let's see what do I want to say, so I just want to say is that, oh yeah, so x i of s more i, s inject into h i i s, right, and so this means immediately that if this is non-zero, this is non-zero, and so that means that the positive dimension of s more i is less than equal to the homological dimension of the ideal i in s, and this is less than equal to the number generator, that's very easy, because you can compute low commuji with a with a check complex on the generator i, so so that is easy, the second statement is a bit more involved, and again I know everyone's hungry, so let me just, so for the second statement actually we start by proving it in characteristic p, and basically we, first we prove the following, so step one, so if s more i is commuji full, then s more i to the Frobenius power, okay, oh note that not all monomial ideals are commuji full, so all standard rational ideals are commuji full, but if you look at the thing I just talked about in the dimension two, there are many monomial ideals that are not member-colleges, even though, so we are, so for instance this is a statement is, okay, and this is a Frobenius power, i to the, f to the, p to the e, i to the, p to the e, so this is important, so this is like a base chain statement, right, because you basically base chain using Frobenius, so you prove some base chain statement and it follows that this is also commuji full, but now you, so for hq, so for hq we have some sort of nq, so that i to the nq is the, let's see, what do I want to see, I want to see this containing, it's a thickening of, so right now I, okay, it's inside i to the q, okay, so because of this, right, so then that means that h i of s mod i to the nq is subject on to h i of s i to the, so then the a invariant here is, so the a, so the top degree here, right, have to be at least the top degree, so i, okay, so well Frobenius is flat, so this you can, you know, just multiple, and then on this side you have results that estimate the top, well the a i of power of ideal, you know, using, we know that this thing eventually become, behave like a linear function, right, so if you, and then you do, then you have to understand how this nq related to q, which is just a, bishanhole principle, so if you just, I'm running out of time, but once you have this estimate, and you use the fact that that's a, that's a, you know, the regularity of power of ideal is a linear function, you get the statement, so I think this is just about time. Okay, any questions? Yes? Yeah, so well at least asymptotically it's about for completing sections, right, I mean, this is about that happened to a lot of nice things, you know, this is about, so if you take a bunch of, a complete section then the regularity is basically a sum of dv of generator, that's, so if you take a Athenian completing section, it's the Bao-i-Shah, yes? How about predictive services? Oh, so you're talking about isolated combative pool on a function? Yeah, I mean you have, you understand the curve, you know? Oh, right, right, right, oh you mean like a same statement for dimension three? We have some thought about it, but it's already there, I think it's pretty subtle, so it's some action on the local motion, it's two local motions, so it's, it's not something I can write down, but yeah, that, yeah, that might be, yeah, that's right, yes, yeah, well, Alexandra is here, so, so in fact the reason we want to give this talk here so that you can ask me questions like that, yes, absolutely. So this means that you can write the local homologous unit of X-modules into a test of P for this homological pool? Yes, yes, in fact this is, this shows you that, that, that, that, that for question two, for the first question of Eisenberg, Musselstatt and Stielman, homologous pool is the answer, so if you, so, so the local homologous in characteristic P, this is a union of X, even though if S-module i is, I don't know if it's true in characteristic zero, but because, because this gives you a ready made thickening system too, yeah, so yeah, so this is, so, so in case you're, this answer precisely the question by, by Eisenberg, Musselstatt and Stielman. Any more questions? Thank you. If not, let's thank you all again. Applause. We'll meet tomorrow night.
 I joined work with Linh Chuan Ma and Alexander Ode's Stephanie.  I guess I should also thank the neighboring institution for hiring so many great young people.  And I have benefited a lot from working with them.  So, I should also talk about nearby institution.  I also advertised KUMUNU 2018, which is annual community algebra conference.  This year will be organized on October 13th and 14th in Lawrence.  Last year we had a candlelight dinner and an open bar.  Many people told me that is the best algebra conference they've been to.  Anyhow, a lot of people will come.  So, I wanted to talk about, since I'm literally eating to your dinner time, I want to tell you  some stories right away. So, what are these objects? This is a new class of rings that include  Kohen-McCauley rings, Stanley Reissner rings, the so-called Dubois Singularity Singularity  Zero, and FPU Singularity Incarretera Ristate P. So, they include a lot of good, you know,  I also say that life is worth living, Kohen-McCauley rings, and a lot of you live very comfortably  in the Stanley Reissner ring. So, this seems to be a good class of rings.  Of course, you know, for this class to be interesting, we should also be able to say  something, otherwise you can just define any ring to be a combo.  So, we should be able to say something interesting about it.  And since the theme of this conference is about CG's, I will state one of the consequence,  one of the results, and then after this you can go to dinner if you want to.  So, that's simple I, you could add some more I is logically full, whatever it means,  I'm going to be five later. So, if S is regular local, then the projective dimension of R over  S is less than or equal to the number of generator I. So, it satisfies a very strong  individual banal sort of stillment, as a very strong stillment question, right? And also,  if S is just a polynomial ring and I is gridded, then it also, we also have the regularity of I  is less than or equal to the number of generator I times the, so here D of I is just the  maximum degree of a generator, of a minimal generator of I. So, again, it shows you that  this class of rings are not every ring, they satisfy very, very strong, you know,  rather nice property homologically. On the other hand, they include also all of these  classes in the variety itself. So, it seems to be a right side, that's what I'm trying to convince  you. Okay, so that's the main thing I want to talk about today, but let me actually start with  a motivation question, which actually comes from a paper by David Michemustata and stillman.  And so, let me just remind you that, okay, so S is a commutative material ring and I and S is an ideal.  And so, the local homology of S is, the ideal homology of S is supported in I is by definition  and following. So, okay, so the direct limit of X module and here I sub E is any nested,  you know, system that go final with, go final with, with the power. So, for example, you can take,  you can take any system. So, for example, IE equal to I to the E or IE equal to I to the  symbolic power or IE to the, equal to E to the Frobenius power in characteristic Cp and so on.  So, this is a direct limit. Local homology module contains a lot of build, you know,  each useful information. On the other hand, a big drawback of this module is, I'd say,  really finally generated. And so, that motivates the following question, which I, as I said before,  is due to, was first raised by, in a paper by David Michemustata and stillman. So, when is,  when is this, direct limit actually a union? Okay. Well, the obvious reason why we want this  to be true, namely that any question about this module can be now reduced to a question about  IE, which is a finally generated module. Right? Of course, this is just mean. It's not hard to see,  it's just mean that each of these map is injected. Okay? So, in order for this to be true, you need  to, and so it motivates the next question, which is when is this map injected? So, when is this map?  And in fact, one of our first observation is that this class of idea, which I haven't defined yet,  will sort of answer this question precisely. So, let me, let me now focus on this question.  So, here's a quick observation, namely that if you have a map,  so if you have any idea in our system, then the map from here to here factors through this.  So, for this map to be injected, you need this to be injected for yourself.  And if you, let's say that you leave in a, if S is a complete local ring for simplicity,  then by local duality, say S is a complete local, I mean the question here is, this is a  local question, you can, you know, the kernel here is a finally generated module. So, to be  injective just mean that local is, so, so, so you're going to work completely lovely if you want to.  So, this is just mean that the J log of homology, so let's say SM here,  of S mod IE is so J1 to J i and here J is just equal to N times i and the dimension of S.  So, this is a suggestion, this is equivalent to that.  Okay, so this motivates the following definition, that the main definition of the top, so let me  introduce, so this is some time called thickening of S mod IE. So, let me just, let me, so to give  an intrinsic definition without refer to some embedding and you need to make a formal definition  of thickening, so let R and be local. A thickening of R is a suggestion  from T to R such that the, such that the induced map on the repo, sorry, this reduce,  on the reduce part of T and R is a nice form of esmophism. Well, example is T equal to Kx  mod x square and R is Kx mod x. So, you know, this is a picture that you've probably seen  in many beginning course into, you teach a beginning course in, in, in, under regulatory  right, is a, a fattening of this, this point, a double point, right, so it's a thickening. Okay, so now I give the definition, which is motivated by this observation.  So, so definition, so Rm, again I already gave the local definition point, is called homological  full beef for any thickening. Well, obviously you have the same skin structure, so your thickening  will be a local ring. Then the map, sorry, is injective for pi, so sorry, it's ejective.  So, this is the reason why we call it's homologically full, so for all thickening,  you don't lose any, it hit all the local modi element in R.  And so that, the motivation question and the definition. I should remark that if you have  an integrated, so remark, you can define the grid, because in fact, the equivalent to the,  so R is standard grid, you can also take this as definition, then R is homologically full,  even though if Rm is,  so that's the definition. So let me, so the first result we can prove is that this  relationship is actually a lot tighter than we just saw, so in fact the, this definition is  almost the answer to the question by David.  So long after a remark, a definition or a statement?  Okay, we have a definition, if you want. So, okay, so a little bit of story here, so,  so you know, you can, this, okay, so this is a lazy way to define it, right, so, so, so,  formally you should define it as like any grid of thickening or that, but it will say it.  So, the G point is defined, that's why I called it remark, I guess.  Okay, anyway, so I will, so what is this?  Oh, right, yeah, so, so the first theorem, say what I just said, so, okay, so we, we don't have  the full strength, but let's say i is S mod i and here S is an unremifed regular local ring.  If you don't, well, if you know what it is, you know what it is, if you don't know it,  probably doesn't affect your life. This one, right, so, so then, then,  then i is co homologically full in our sense, is actually equivalent to this map,  injective of i, so, so in fact, it is the same definition. However, they are, you know, so,  so we found that it's, it's when working with this, so then you can sort of take one of this as a  definition. On the end, this is intrinsic definition and also when you work with this concept,  this is useful to think in both ways. We won't see some examples of that if I can get to some proof,  but yeah, so let's, yeah, so, so let's look at the examples.  Right, so co homicoly, why is that? So co homicoly,  well, if you have a, if you have a map like this, right, and you have some kernel here,  then just as long as I see what a little homo should tell you that this is a co homicoly.  So, hi, okay, hi, hi, and so below the dimension, it is this zero, so it's nothing,  but above the dimension is, you know, this is vanish, right, so it's this, sorry,  above the dimension, this vanish, so this is, or it shrinks away to look at  co homicoly ring, it turns out to be an useful way. Right, what about Stanley Ryan's ring?  Well, this is actually, this is true because, because actually Mirscher  proved that this is an injective, an injection for square free monomial ideal.  So, if you believe this then, you know, although I mean, directly it's not obvious to see that this  this is property is false, but, and there are things like FQ and reward singularity,  so the reward singularity property was proved by, in a paper by, by Lindscher and, and Carl Schroed  and FQ is followed from the work by Lindscher and Fa.  Okay, so, so, so this, although they didn't explicitly say it, but it's the same thing,  they did prove those things. All right, oh, so, also I want to give some small dimension example,  so basic, so basic property here is that if R is co homologically full,  then R is unmixed.  And so if, so if dimension of R equal to 1, then it's equivalent to the R's primary quality  and if dimension of R is equal to 2, then, then R is homologically full,  is equivalent to the following. So, let's say, okay, okay, so I will need to  assume that I will ask for I when, when S is a complete rate, okay, so maybe,  and this guy, this is a little bit of subtlety because when you complete, you lose some  irrelusibility thing, so, so it needs to be a little bit careful of I as I,  and this is, it's this probability, okay, so, so I is equal to,  so basically all the connected component of the puncture spectrum of R are quite  equivalent, okay, so, S mod I actually, component, connected component, component of a spectrum R,  and each of these has to be equivalent. So, this fact use, the fact that  the combative dimension of something that whose puncture spectrum is connected at most  at minus 2, so it's actually a non-trivial statement, but anyhow, so this is some example,  so you can see that obviously not everything is converging full if you need further evidence  than you can classify in, in small dimensional case. Okay, so, right, so there are many  properties that we can prove about this, this class, so before talking about  budget dimension regularity, let me just mention that you can prove some sort of  weak or a intervention property for this rings, you can compute Lubesnik numbers and all that, so,  so basically a lot of things are true for this sort of statement that's true for this  class is true for more, but let me just prove the connected components from the  point, or want to be, does this pass from connected components? Yeah, so, yeah, so we  prove that it's pass from connected components, so this full if and only pitch connected component  of the puncture is full, and so, yeah, so if you have a Gwen Macaulay each component don't  create some, so that's another way to get a new example. Right, okay, so let me just remind you  of what I, so yeah, so for the last five minutes maybe I will sketch the statement, so once you,  once you have this is not so hard, so one is like a positive dimension, so again i equal to s  more i, and r is from degree four, this less equal to a number of generator of i, and also  the regularity of i is less than equal to this number of generator i times this degree of the  maximum generator, maximum degree of generator plus the dimension of r, yeah, so this is a two of the  property that's the best most relevant to the theme of this workshop, so how do you prove the,  how do you prove the first, so this is really quite easy, the first one, so let's see what do I  want to say, so I just want to say is that, oh yeah, so x i of s more i, s inject into h i i s,  right, and so this means immediately that if this is non-zero, this is non-zero, and so that means  that the positive dimension of s more i is less than equal to the homological dimension of the  ideal i in s, and this is less than equal to the number generator, that's very easy,  because you can compute low commuji with a with a check complex on the generator i, so  so that is easy, the second statement is a bit more involved, and again I know everyone's hungry,  so let me just, so for the second statement actually we start by proving it in characteristic p,  and basically we, first we prove the following, so step one, so if s more i is commuji full,  then s more i to the Frobenius power, okay, oh note that not all monomial ideals are commuji  full, so all standard rational ideals are commuji full, but if you look at the thing I just talked  about in the dimension two, there are many monomial ideals that are not member-colleges, even though,  so we are, so for instance this is a statement is, okay, and this is a Frobenius power,  i to the, f to the, p to the e,  i to the, p to the e, so this is important, so this is like a base chain statement,  right, because you basically base chain using Frobenius, so you prove some base chain statement  and it follows that this is also commuji full, but now you, so for hq, so for hq  we have some sort of nq, so that i to the nq is the, let's see, what do I want to see, I want to  see this containing, it's a thickening of, so right now I, okay, it's inside i to the q,  okay, so because of this, right, so then that means that h i of s mod i to the nq is subject  on to h i of s i to the, so then the a invariant here is, so the a, so the top degree here,  right, have to be at least the top degree, so i,  okay, so well Frobenius is flat, so this you can, you know, just multiple,  and then on this side you have results that estimate the top, well the a i of power of ideal,  you know, using, we know that this thing eventually become, behave like a linear function,  right, so if you, and then you do, then you have to understand how this nq related to q, which is just  a, bishanhole principle, so if you just, I'm running out of time, but once you have this estimate,  and you use the fact that that's a, that's a, you know, the regularity of power of ideal is a  linear function, you get the statement, so I think this is just about time.  Okay, any questions? Yes?  Yeah, so well at least asymptotically it's about for completing sections, right, I mean,  this is about that happened to a lot of nice things, you know, this is about, so if you take a  bunch of, a complete section then the regularity is basically a sum of dv of generator, that's,  so if you take a Athenian completing section, it's the Bao-i-Shah, yes?  How about predictive services?  Oh, so you're talking about isolated combative pool on a function?  Yeah, I mean you have, you understand the curve, you know?  Oh, right, right, right, oh you mean like a same statement for dimension three?  We have some thought about it, but it's already there, I think it's pretty subtle, so it's some  action on the local motion, it's two local motions, so it's, it's not something I can write down,  but yeah, that, yeah, that might be, yeah, that's right, yes, yeah, well,  Alexandra is here, so, so in fact the reason we want to give this talk here so that you can  ask me questions like that, yes, absolutely.  So this means that you can write the local homologous unit of X-modules into a test of  P for this homological pool?  Yes, yes, in fact this is, this shows you that, that, that, that, that for question two, for the  first question of Eisenberg, Musselstatt and Stielman, homologous pool is the answer, so if you, so,  so the local homologous in characteristic P, this is a union of X, even though if S-module  i is, I don't know if it's true in characteristic zero, but because, because this gives you a ready  made thickening system too, yeah, so yeah, so this is, so, so in case you're, this answer  precisely the question by, by Eisenberg, Musselstatt and Stielman.  Any more questions?  Thank you.  If not, let's thank you all again.  Applause.  We'll meet tomorrow night.
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