Compatible systems along the boundary
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Zheng, Weizhe
A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field, based on Gabber's theorem on compatible systems. This has applications to the equicharacteristic case of some classical conjectures on l-independence. I will also discuss the relationship with compatible wild ramification.
 I would like to thank the organizers for the invitation. It is a great honor for me to speak in this conference for Alfred Gabber. Over the years I have benefited tremendously from him. Just yesterday I received some corrections from him for a paper by other people that will be using this talk. But fortunately there was nothing serious. Here's another illustration of his tremendous generosity. This talk will be based on applications of a couple of marvelous theorems of Gabber that were written up by other people. Here the keywords are after Gabber. The subject of this talk is about some old questions on L independence. So CER certainly made many conjectures on L independence. The ones that I'm going to talk about are those he made in his definition in his recipe of the Hasselweig data function of smooth productive varieties over global fields. So he made these conjectures C1 through C8, eight of them. Actually a couple of them already appeared earlier in his paper that were stated as problems. So here's the definition of the Riemann's data function. This can be generalized without any problem to a rhythmic data function of a scheme of finite type over spec z. I just have to replace a set of integers by the set of effective zero cycles on x and to replace n by the norm of the cycle. With the Eulerian product you can write it as a product of local data factors. As we all know Grottinger gave a common logical interpretation of these local data factors. So now let x be a variety over a finite field. Grottinger defined this added common logic with compact support. Compared with a continuous action of the Galois group, then the data function can be written as alternating product of characteristic polynomials of Frobenius acting on the compacted supportive added homology as a consequence of the Grottinger trace formula. So a natural question here is whether these polynomials which are a priori with coefficients in ql, actually zl, are actually integer coefficient and independent of l. So in the case where x is proper smooth, very conjectures tell us that the action of Frobenius on the i's homology is actually of weight i. So you can extract these pil from the data function. So as a consequence of this, these polynomials actually has coefficients in z and independent of l. And these were Ceres conjectures C1 and C2. So these are now known. Another consequence of weight conjectures is that by threading out, you get that for proper smooth variety over an arbitrary field, then the better number of the added common logic is actually independent of l. So this spreading out argument will be a major theme of the rest of the talk. So now let's get back to the Hasse-Weisei function. So here's Ceres's recipe. The local factors are defined as characteristic polynomials of Frobenius acting on the inertia invariant part of the added homology. So in order for this formula to make sense, he has to assume that this polynomial is actually independent of l. So that is his conjecture C5. He stated it in a slightly more generality. Now we work over a local field k by which I mean a complete description field of finite residue field fq. So now assume only that x is defined over k. Then Ceres conjecture says that the Frobenius acting on the inertia invariant part of the added homology has characteristic polynomial with coefficients in z and independent of l. Related conjecture. Is it also a conjecture concerning the full structure of the local Galois action on HIL? So is it a more refined conjecture? So you are, this one, this C8? Yeah. So yes, there is. He also made this conjecture C8 that for the 4HI, the action of the wave group actually has traces independent of l. So does it imply the previous one? Yeah, I'm coming to that in a moment. So to explain the relationship between these two conjectures, let me recall the monogamy weight conjecture which says that the eigenvalues of Frobenius liftings on the graded pieces of these homology for the monogamy filtration are pure of weight i plus n. So combined with the conjecture C8, this implies that all these graded pieces are actually independent of l. Because you can extract them, they have different weights. And from here, actually you get that all the primitive parts are independent of l. So it follows that this conjecture C5 on the inertia invariant part holds. So actually, Ser didn't make this monogamy weight conjecture, but he made conjectures C6 and C7 between C5 and C8 that were special cases of the monogamy weight conjecture. So these are Ser's conjectures. Now for over a local field, a finite residue field. Now if we now allow the residue field to be an arbitrary field, then well, there's no weight group, but we can still consider the action of the inertia group on the homology and Ser conjectured, this conjecture C4 says that for any element in the inertia group, the characteristic polynomial on the ice elliptical homologies has coefficients in z and independent of l. So to put this into context, let me also recall the local monogamy theorem of a gronadig, which says that for acts of variety, which I mean a separated scheme of finite type over k, not necessarily proper smooth, the action of the inertia is actually quasi-unipotent on the compactly supported homology. And this has several variants due to the link, Gabber and Uduzi, which says that first of all you can replace the compactly supported homology by a bioro-alleted homology. And this subgroup, which acts unipotently, can be taken to be independent of l. And also the index of unipotency can be taken to be i, which is the homological degree. So this is some other problem of independent of l. Now let me concentrate in the case of equal characteristic for a moment. So let me recall that in equal characteristic, the monogamy weight conjecture was known, it was due to the linear teresoma. So the linear in the case where it is defined over a curve and teresoma maybe in general and veto for a generalization to residue fields of finite type. So here's the statement of our results. So still in the equal characteristic case, let's recall k is a completely discrete variation field of residue field k of characteristic p greater than 0. So both of characteristic p greater than 0. Let x be a proper smooth variety over k. Then conjectures c4 and c8 actually hold. So let me recall that for c4 it says that for every f in the inertia group, the characteristic polynomial of f acting on h i l is in z bracket t and is independent of l. And for c8 it says that if you assume the residue field to be finite then for the lifting of Frobenius above holds. So this was known, this c8 was known in special cases to the linear in teresoma. Now as mentioned before, so this combined, so this c8 combined with the monogamy weight conjecture implies conjecture c5 that the inertia invariant part of the homology is independent of l. Now let me make a number of remarks. So previous results on, well previous results on in general characteristic without equal characteristic assumption which holds for, not for individual conjecture but holds for the alternating sum of the traces. So first of all there were this theorem by Gabber over a finite field already that for f in the weight group, the alternating sum of the traces is a rational number independent of l. And then for local fields, first of all for general completely square fashion field, the action of the inertia is actually in z and independent of l when you take the alternating sum, this was a theorem of V-doll. So these are variants of the conjecture c1 and c4. And finally there's this variant of conjecture c8, so if k is now local field, a finite residue field, fq, then for the action of the elements of the weight group are independent of l, which was a theorem of RCI in the proper case that can be extended to the general case. So now let me explain the strategy of the proof of such theorems. So basically the proof is a spreading out argument. So when never axis defined over a field f of characteristic p, we can spread it out to something of finite type over fp. So there is this chronology on the left hand side is now spread out to least sheaf on the right hand side. And when we vary l, we get a system of least sheafs on the base b. So we are led to study instead of this system of galvanized representations to the system of added sheaves on the base scheme. So now our basic setting is a slight generalization of the previous one. So we work not just over a finite field, but over the ring of integers, okay, of a local field. So here I do not assume that k is equal characteristic. So k is just a Hensselding, so okay is an excellent Hensselding discrete evaluation ring of residue of field fq with no restriction on the characteristic. And we take x to be a scheme of finite type over spec okay. And I denote by k the growth in a group and we should fix a number of primes, a system of primes li. Now given a system of elements of the growth in a group, a system of virtual sheaves li. So each li is in is a qli bar addict sheaf. This system is called compatible if for every point x in x, so here by a point I consider only locally close points, for every locally close point x and every element in the wave group, the local wave group, the trace of f on the stock of li is a rational number independent of i. So basically the local traces are rational numbers independent of i. Let me recall that this being a locally close point here just means that it is either a close point in the generic fiber or a close point in the special fiber. Of course there is a more general notion of compatible systems with traces in a field extension of q when you fix it in baddings. Now fundamental theorem of Gabber says that a compatible systems defined in this way and is over a finite field are preserved by a growth in variability and growth in six operations. And this can be extended over spec OK over the ring of integers of a local field. Now for our purpose it will be important to consider not just points on the curve but also points on the variety but also points off the variety, points on the boundary of the variety. So let's take the simplest case. If c bar is a smooth curve or a rough q and I take a zarsky open dense subset c, we are interested in just having compatible systems on c but we look at points on the boundary, on the complement of c and c bar. And then of course we have these local fields. So we look at these local Galois groups which has this short exact sequence. And this can be defined algebraically by taking the hand solization of this curve at this point. And then removing this point itself. Now this generalizes to this general setting. So now let x bar be an enormous scheme of finite type over s. And we take a zarsky open x in x bar. For any point of x bar, we take the hand solization of x bar at x and then we remove the boundary. This gives us an open sub scheme which gives us a rejection. So this pi1 of this gadget is an analog of the local Galois group and the kernel is the inertia which actually appeared in Techesh's talk this morning. So here's the key definition. Now given a system on x, we say that it is compatible on x bar or it is compatible along the boundary of x bar minus x. Yeah, I assume that these LIs are least sheets so that I get maybe a representation of the pi1. So what is the value group here? So just take the inverse image of the value group here. So no, but a normal scheme of finite, what is s? s is s. So I only consider locally closed points. So it's just the euro value group I think. Yeah, I only consider, so this bar means locally closed points. So the residue fields are either finite fields or local fields. So there's a well-defined notion of value group. Well, not quite locally because you're only in the Azzelian. Yeah, yeah, Hanselian. Okay, but it's the same. So everything goes to s and we can take the map from pi1 of this thing to pi1 of s and that gives a way of defining this value group. So the definition is that for every point, every locally closed point on x bar or actually, for example, on the boundary, the action of the local value group has traces in q and independent of i. Now a natural question here is given a system that is compatible on this open subset x, is it automatically compatible on x bar? So is it compatible along the boundary? So I don't know the answer to the question, but it does hold up to stratification or modification. So let me explain what I mean by this. So what we actually proved is the following. So let x be a scheme of finite type of rs and we take a system that is compatible. So I also assume that the index set i is finite. Then there exists a stratification of x into locally closed up schemes such that each stratum x alpha, and it's a normal compactification x alpha bar over s, such that the restriction of the system to each of the stratum is compatible along the boundary. This is what I mean compatible along the boundary up to stratification. A more or less equivalent formulation is that it is compatible up to a modification. So now the setting is slightly different, but the conclusion is that there exists a modification. So I take x bar and I take zaracid dense open x, then there exists a modification of x bar such that the pullback of the system is compatible along the boundary of y bar. So you don't change the good locus. I think I don't know. So in the proof we may have to change the good locus. So of course these are theorems of delin in the case where x is a curve over fq. In the case of a curve you have no need for stratification or modification. It is simply compatible implies compatible along the boundary. Now what is perhaps at least I find it funny that these statements imply a evaluative criterion for a compatible system. So I have this system that I assume, well for the moment I don't assume it's compatible. And we consider this all-valuative criterion commutative squares like this where all l is a hands-on evaluation ring and l is a field of fractions. Which part seems to be discrete evaluation? Not necessarily discrete. Then a system li on x is compatible if and only if the restriction of the system to spec l with l quasi-finite over s is compatible. For such l you can also define the wave group. Of course one side of the implication from right to left is just a definition. So the only content here is from left to right. So the consequence of being compatible is that it satisfies this evaluative criterion. Now also in the strictly local case, so if you just restrict to the inertia then the action is also independent of l. These are the consequence of theorem up to stratification or modification. Now let's come back to the, to Sarah's conjectures. So actually a consequence of this evaluative criterion combined with spreading out argument is the following slight generalization of Sarah's conjectures and eq characteristics. So now all l is just a Hanselian evaluation ring not necessarily discrete. And for x a proper smooth scheme over l, the second point, the second point of this evaluative criterion implies that the action of the inertia is independent of l. Here I have left out the part on the indigradity, which are given by some other arguments. At least the evaluative criterion implies that the characteristic polynomial is in q bracket t and is independent of l. And the first part of the evaluative criterion gives the generalization of Sarah's conjectures c8 in eq characteristic. So for any element of the wave group, the trace is in q and is independent of l. So this evaluative criterion was very much inspired by Gaubert's evaluative criterion for the ramified part of the fundamental group. So let me explain this relation with wild ramification. The ramified part of pi1 was defined in the first paper of VDAL perhaps with the help of Gaubert. So the setting is over any… The definition of the evaluative criterion was described. So this is VDAL's definition. So now we take any excellent Hansel and Desgris evaluation ring of residue characteristic greater than 0. Now VDAL defined these subsets of the fundamental group for a normal integral scheme separated the final type over s. There's a subset of ramified part, this ramified part of the fundamental group, and there's this wildly ramified part of the fundamental group. These are closed subsets stable and their conjugation. So the definition is a bit technical. So for every normal compactification there was this subset pi1r subscript x bar, which is the closure of the union of the conjugates of this image for x bar running through geometric points of x bar. So basically these are the local inertia groups that appeared in Takeshi's talk this morning. So we take the image of all these, take the union, take the conjugate, and take the closure. So you get a well-defined subset. And then we take the intersection with respect to all x bar with respect to all normal compactifications. Now to define the wildly ramified part is just the intersection of the ramified part with the union of the proprio silos. Now Gabbard gave this evaluative criterion for this pi1r, which says that pi1r is the closure of the union of the conjugates of images of Galois group of L for L appearing in these evaluative diagrams where OL is a strictly hands-on evaluation ring and L is a fraction field of OL. No, it's just a subset. So there's also a evaluative criterion for pi1r for the wildly ramified part where you just restrict to the pro-order of elements. Now one can define using this wildly ramified subset a notion of compatible wild ramification. So given a scheme x over s, a system, now I take a system of F L bar sheaves, virtual F L bar sheaves because now we restrict to the proprio silos, so it suffices to consider torsion sheaves. Now such a system has said to have compatible wild ramification if for every, first of all we have to restrict to pieces where all the sheaves are least and then we can talk about a broader traces for G in the wildly ramified part of the fundamental group. And we ask that these broader traces are rational numbers independent of L. And as Takeshi mentioned this morning, so there's also this notion of same wild ramification which is actually more general, it's a less restrictive notion. And that was studied by Takeshi and his students Yuri Yadagawa and Yadagawa herself. And there are some old work by Deline and not so old work by Vida and some very recent work by Saito Yadagawa, Yadagawa and Kuo. There's also a related work by Hiroki Kato which is not exactly studied like this. Which says that this notion of compatible wild ramification is preserved by all six operations. Well this notion of same wild ramification is preserved by four operations. And the corporations. Upper star, lower star, upper tree. Not preserved by duality. Not preserved by tens. Preserved by duality. Preserved by duality, sorry, yeah. It is. But not preserved by tens or harm. So actually there's just a quick explanation of this non-stabilized by tenser but I didn't put it in the slides. Basically this is what Takeshi explained this morning. So a way of defining the same wild ramification is to ask for the trace to be in. So if you take the trace, so this broader trace the upper aria lens in a bigger field than Q and then you take the trace to Q and you ask that to be independent of L. But by doing so you are actually getting some sub spaces which are not sub-algebra. So that means this won't be compatible. This won't be preserved by tenser product. Okay. Now we have this extension of scalars morphisms, this maps as in modular representation theory. So there's an isomorphism between Kx with coefficient in Z bar with Kx with coefficient in Q bar. And there's also a reduction from Kx with coefficient in Z bar to Kx with coefficient in FL bar. Using these two maps we get the decomposition map from Kx Q bar to Kx FL bar. Then by combining our evaluative criterion with that of Gavir, we get the following consequence that for any compatible system LI. So this decomposition map carries compatible systems of analytic sheaves to systems of mod L sheaves with compatible wild ramification. So this gives some examples of systems with compatible wild ramification where the recent work can be applied to. That will be all of my talk. Thank you. So time for questions. Please, can you give a hint in the question of the application of the application theorem? Well, so at least one ingredient is to use an equivalent form of the Gavir theorem. So we have to work with Dion's alterations in the equivalent form. I have the paper stashed in my backpack. I believe that if you have some alteration, something that makes ramification tame for one KL, Ki, then probably you can prove that the other ones are also tame and then you can prove that the other ones are also tame. So we are speaking about this different notion of compatible and compatible in the boundary. Compatible means that all localy close to each other. So you have compatibility. Let's say for least sheaves. So you have a compatibility with some normal crossing device or maybe I don't know, the generic one was over all of us. So suppose you know that one of them is tame, then it follows that the other is also tame. Is it the case? The others are tame. But in any case, we only have a finite number of Ls. So we can make everything tame. And then you can study the action of the group and look at the, but I believe that probably the decision to find that number is not needed. If you can indeed get the, I mean, if you prove. Yeah. The one tameness implies the other. Yes. And then you can probably put all of, then you can find a comb on the alteration, the alteration, the alteration that works for everything, guys. Okay, so another technical question about this kind of problem with ser and so on. So we work with the Bragg variety, let's say proper smooth or the field of fraction, but we can instead work with digital analytics spaces or formal scheme. So of course you can put the same conjecture for proper smooth digital analytics spaces. So of course some of the results on the work monodromes thing is used as a bright nature because you are, so you don't have this, but the other part of the kind of independence of early results. So is it, how much does it extend to this case? Did you, so? There were some results by Mieda, I think. But we're seeing. But otherwise I don't know about these particular questions there. Are there places where the trace, the greater trace is rational but not integer? Integrality is another problem because you can naturally have, for example, when you undertake dual it will not preserve the integers, when you take duals or it will just take twists. So they naturally occur. No, you just take twist. You can have q to some negative power as eigenvalues. But there are estimates on how the integrality is preserved both under the usual operation for both the. Yeah, for both the integrality and for inverse integrality. So for single file, it's a problem. So if we localize that, so if you invert p then both integrality and inverse integrality is preserved by all the operations, sorry, except that some of them exchanges the two. And for the p-divisibility you have some bounds on all the operations. So if not, it's time to thank the organizers of this beautiful conference. So first I met who I think is the main actor of organizing.
 I would like to thank the organizers for the invitation.  It is a great honor for me to speak in this conference for Alfred Gabber.  Over the years I have benefited tremendously from him.  Just yesterday I received some corrections from him for a paper by other people that will  be using this talk.  But fortunately there was nothing serious.  Here's another illustration of his tremendous generosity.  This talk will be based on applications of a couple of marvelous theorems of Gabber that  were written up by other people.  Here the keywords are after Gabber.  The subject of this talk is about some old questions on L independence.  So CER certainly made many conjectures on L independence.  The ones that I'm going to talk about are those he made in his definition in his recipe  of the Hasselweig data function of smooth productive varieties over global fields.  So he made these conjectures C1 through C8, eight of them.  Actually a couple of them already appeared earlier in his paper that were stated as problems.  So here's the definition of the Riemann's data function.  This can be generalized without any problem to a rhythmic data function of a scheme of  finite type over spec z.  I just have to replace a set of integers by the set of effective zero cycles on x and  to replace n by the norm of the cycle.  With the Eulerian product you can write it as a product of local data factors.  As we all know Grottinger gave a common logical interpretation of these local data factors.  So now let x be a variety over a finite field.  Grottinger defined this added common logic with compact support.  Compared with a continuous action of the Galois group, then the data function can be written  as alternating product of characteristic polynomials of Frobenius acting on the compacted supportive  added homology as a consequence of the Grottinger trace formula.  So a natural question here is whether these polynomials which are a priori with coefficients  in ql, actually zl, are actually integer coefficient and independent of l.  So in the case where x is proper smooth, very conjectures tell us that the action of Frobenius  on the i's homology is actually of weight i.  So you can extract these pil from the data function.  So as a consequence of this, these polynomials actually has coefficients in z and independent  of l.  And these were Ceres conjectures C1 and C2.  So these are now known.  Another consequence of weight conjectures is that by threading out, you get that for proper  smooth variety over an arbitrary field, then the better number of the added common logic  is actually independent of l.  So this spreading out argument will be a major theme of the rest of the talk.  So now let's get back to the Hasse-Weisei function.  So here's Ceres's recipe.  The local factors are defined as characteristic polynomials of Frobenius acting on the inertia  invariant part of the added homology.  So in order for this formula to make sense, he has to assume that this polynomial is actually  independent of l.  So that is his conjecture C5.  He stated it in a slightly more generality.  Now we work over a local field k by which I mean a complete description field of finite  residue field fq.  So now assume only that x is defined over k.  Then Ceres conjecture says that the Frobenius acting on the inertia invariant part of the  added homology has characteristic polynomial with coefficients in z and independent of  l.  Related conjecture.  Is it also a conjecture concerning the full structure of the local Galois action on HIL?  So is it a more refined conjecture?  So you are, this one, this C8?  Yeah.  So yes, there is.  He also made this conjecture C8 that for the 4HI, the action of the wave group actually  has traces independent of l.  So does it imply the previous one?  Yeah, I'm coming to that in a moment.  So to explain the relationship between these two conjectures, let me recall the monogamy  weight conjecture which says that the eigenvalues of Frobenius liftings on the graded pieces  of these homology for the monogamy filtration are pure of weight i plus n.  So combined with the conjecture C8, this implies that all these graded pieces are actually  independent of l.  Because you can extract them, they have different weights.  And from here, actually you get that all the primitive parts are independent of l.  So it follows that this conjecture C5 on the inertia invariant part holds.  So actually, Ser didn't make this monogamy weight conjecture, but he made conjectures  C6 and C7 between C5 and C8 that were special cases of the monogamy weight conjecture.  So these are Ser's conjectures.  Now for over a local field, a finite residue field.  Now if we now allow the residue field to be an arbitrary field, then well, there's no  weight group, but we can still consider the action of the inertia group on the homology  and Ser conjectured, this conjecture C4 says that for any element in the inertia group,  the characteristic polynomial on the ice elliptical homologies has coefficients in z and independent  of l.  So to put this into context, let me also recall the local monogamy theorem of a gronadig,  which says that for acts of variety, which I mean a separated scheme of finite type over  k, not necessarily proper smooth, the action of the inertia is actually quasi-unipotent  on the compactly supported homology.  And this has several variants due to the link, Gabber and Uduzi, which says that first of  all you can replace the compactly supported homology by a bioro-alleted homology.  And this subgroup, which acts unipotently, can be taken to be independent of l.  And also the index of unipotency can be taken to be i, which is the homological degree.  So this is some other problem of independent of l.  Now let me concentrate in the case of equal characteristic for a moment.  So let me recall that in equal characteristic, the monogamy weight conjecture was known,  it was due to the linear teresoma.  So the linear in the case where it is defined over a curve and teresoma maybe in general  and veto for a generalization to residue fields of finite type.  So here's the statement of our results.  So still in the equal characteristic case, let's recall k is a completely discrete variation  field of residue field k of characteristic p greater than 0.  So both of characteristic p greater than 0.  Let x be a proper smooth variety over k.  Then conjectures c4 and c8 actually hold.  So let me recall that for c4 it says that for every f in the inertia group, the characteristic  polynomial of f acting on h i l is in z bracket t and is independent of l.  And for c8 it says that if you assume the residue field to be finite then for the lifting  of Frobenius above holds.  So this was known, this c8 was known in special cases to the linear in teresoma.  Now as mentioned before, so this combined, so this c8 combined with the monogamy weight  conjecture implies conjecture c5 that the inertia invariant part of the homology is  independent of l.  Now let me make a number of remarks.  So previous results on, well previous results on in general characteristic without equal  characteristic assumption which holds for, not for individual conjecture but holds for  the alternating sum of the traces.  So first of all there were this theorem by Gabber over a finite field already that for  f in the weight group, the alternating sum of the traces is a rational number independent  of l.  And then for local fields, first of all for general completely square fashion field, the  action of the inertia is actually in z and independent of l when you take the alternating  sum, this was a theorem of V-doll.  So these are variants of the conjecture c1 and c4.  And finally there's this variant of conjecture c8, so if k is now local field, a finite residue  field, fq, then for the action of the elements of the weight group are independent of l,  which was a theorem of RCI in the proper case that can be extended to the general case.  So now let me explain the strategy of the proof of such theorems.  So basically the proof is a spreading out argument.  So when never axis defined over a field f of characteristic p, we can spread it out to  something of finite type over fp.  So there is this chronology on the left hand side is now spread out to least sheaf on the  right hand side.  And when we vary l, we get a system of least sheafs on the base b.  So we are led to study instead of this system of galvanized representations to the system  of added sheaves on the base scheme.  So now our basic setting is a slight generalization of the previous one.  So we work not just over a finite field, but over the ring of integers, okay, of a local  field.  So here I do not assume that k is equal characteristic.  So k is just a Hensselding, so okay is an excellent Hensselding discrete evaluation ring of residue  of field fq with no restriction on the characteristic.  And we take x to be a scheme of finite type over spec okay.  And I denote by k the growth in a group and we should fix a number of primes, a system  of primes li.  Now given a system of elements of the growth in a group, a system of virtual sheaves li.  So each li is in is a qli bar addict sheaf.  This system is called compatible if for every point x in x, so here by a point I consider  only locally close points, for every locally close point x and every element in the wave  group, the local wave group, the trace of f on the stock of li is a rational number independent  of i.  So basically the local traces are rational numbers independent of i.  Let me recall that this being a locally close point here just means that it is either a  close point in the generic fiber or a close point in the special fiber.  Of course there is a more general notion of compatible systems with traces in a field  extension of q when you fix it in baddings.  Now fundamental theorem of Gabber says that a compatible systems defined in this way and  is over a finite field are preserved by a growth in variability and growth in six operations.  And this can be extended over spec OK over the ring of integers of a local field.  Now for our purpose it will be important to consider not just points on the curve but  also points on the variety but also points off the variety, points on the boundary of  the variety.  So let's take the simplest case.  If c bar is a smooth curve or a rough q and I take a zarsky open dense subset c, we are  interested in just having compatible systems on c but we look at points on the boundary,  on the complement of c and c bar.  And then of course we have these local fields.  So we look at these local Galois groups which has this short exact sequence.  And this can be defined algebraically by taking the hand solization of this curve at this  point.  And then removing this point itself.  Now this generalizes to this general setting.  So now let x bar be an enormous scheme of finite type over s.  And we take a zarsky open x in x bar.  For any point of x bar, we take the hand solization of x bar at x and then we remove the boundary.  This gives us an open sub scheme which gives us a rejection.  So this pi1 of this gadget is an analog of the local Galois group and the kernel is the  inertia which actually appeared in Techesh's talk this morning.  So here's the key definition.  Now given a system on x, we say that it is compatible on x bar or it is compatible along  the boundary of x bar minus x.  Yeah, I assume that these LIs are least sheets so that I get maybe a representation of the  pi1.  So what is the value group here?  So just take the inverse image of the value group here.  So no, but a normal scheme of finite, what is s?  s is s.  So I only consider locally closed points.  So it's just the euro value group I think.  Yeah, I only consider, so this bar means locally closed points.  So the residue fields are either finite fields or local fields.  So there's a well-defined notion of value group.  Well, not quite locally because you're only in the Azzelian.  Yeah, yeah, Hanselian.  Okay, but it's the same.  So everything goes to s and we can take the map from pi1 of this thing to pi1 of s and  that gives a way of defining this value group.  So the definition is that for every point, every locally closed point on x bar or actually,  for example, on the boundary, the action of the local value group has traces in q and  independent of i.  Now a natural question here is given a system that is compatible on this open subset x,  is it automatically compatible on x bar?  So is it compatible along the boundary?  So I don't know the answer to the question, but it does hold up to stratification or modification.  So let me explain what I mean by this.  So what we actually proved is the following.  So let x be a scheme of finite type of rs and we take a system that is compatible.  So I also assume that the index set i is finite.  Then there exists a stratification of x into locally closed up schemes such that each stratum  x alpha, and it's a normal compactification x alpha bar over s, such that the restriction  of the system to each of the stratum is compatible along the boundary.  This is what I mean compatible along the boundary up to stratification.  A more or less equivalent formulation is that it is compatible up to a modification.  So now the setting is slightly different, but the conclusion is that there exists a modification.  So I take x bar and I take zaracid dense open x, then there exists a modification of x bar  such that the pullback of the system is compatible along the boundary of y bar.  So you don't change the good locus.  I think I don't know.  So in the proof we may have to change the good locus.  So of course these are theorems of delin in the case where x is a curve over fq.  In the case of a curve you have no need for stratification or modification.  It is simply compatible implies compatible along the boundary.  Now what is perhaps at least I find it funny that these statements imply a evaluative criterion  for a compatible system.  So I have this system that I assume, well for the moment I don't assume it's compatible.  And we consider this all-valuative criterion commutative squares like this where all l  is a hands-on evaluation ring and l is a field of fractions.  Which part seems to be discrete evaluation?  Not necessarily discrete.  Then a system li on x is compatible if and only if the restriction of the system to  spec l with l quasi-finite over s is compatible.  For such l you can also define the wave group.  Of course one side of the implication from right to left is just a definition.  So the only content here is from left to right.  So the consequence of being compatible is that it satisfies this evaluative criterion.  Now also in the strictly local case, so if you just restrict to the inertia then the  action is also independent of l.  These are the consequence of theorem up to stratification or modification.  Now let's come back to the, to Sarah's conjectures.  So actually a consequence of this evaluative criterion combined with spreading out argument  is the following slight generalization of Sarah's conjectures and eq characteristics.  So now all l is just a Hanselian evaluation ring not necessarily discrete.  And for x a proper smooth scheme over l, the second point, the second point of this evaluative  criterion implies that the action of the inertia is independent of l.  Here I have left out the part on the indigradity, which are given by some other arguments.  At least the evaluative criterion implies that the characteristic polynomial is in  q bracket t and is independent of l.  And the first part of the evaluative criterion gives the generalization of Sarah's conjectures  c8 in eq characteristic.  So for any element of the wave group, the trace is in q and is independent of l.  So this evaluative criterion was very much inspired by Gaubert's evaluative criterion  for the ramified part of the fundamental group.  So let me explain this relation with wild ramification.  The ramified part of pi1 was defined in the first paper of VDAL perhaps with the help  of Gaubert.  So the setting is over any…  The definition of the evaluative criterion was described.  So this is VDAL's definition.  So now we take any excellent Hansel and Desgris evaluation ring of residue characteristic  greater than 0.  Now VDAL defined these subsets of the fundamental group for a normal integral scheme separated  the final type over s.  There's a subset of ramified part, this ramified part of the fundamental group, and there's  this wildly ramified part of the fundamental group.  These are closed subsets stable and their conjugation.  So the definition is a bit technical.  So for every normal compactification there was this subset pi1r subscript x bar, which  is the closure of the union of the conjugates of this image for x bar running through geometric  points of x bar.  So basically these are the local inertia groups that appeared in Takeshi's talk this morning.  So we take the image of all these, take the union, take the conjugate, and take the closure.  So you get a well-defined subset.  And then we take the intersection with respect to all x bar with respect to all normal compactifications.  Now to define the wildly ramified part is just the intersection of the ramified part  with the union of the proprio silos.  Now Gabbard gave this evaluative criterion for this pi1r, which says that pi1r is the  closure of the union of the conjugates of images of Galois group of L for L appearing  in these evaluative diagrams where OL is a strictly hands-on evaluation ring and L is  a fraction field of OL.  No, it's just a subset.  So there's also a evaluative criterion for pi1r for the wildly ramified part where you  just restrict to the pro-order of elements.  Now one can define using this wildly ramified subset a notion of compatible wild ramification.  So given a scheme x over s, a system, now I take a system of F L bar sheaves, virtual  F L bar sheaves because now we restrict to the proprio silos, so it suffices to consider  torsion sheaves.  Now such a system has said to have compatible wild ramification if for every, first of all  we have to restrict to pieces where all the sheaves are least and then we can talk about  a broader traces for G in the wildly ramified part of the fundamental group.  And we ask that these broader traces are rational numbers independent of L. And as Takeshi  mentioned this morning, so there's also this notion of same wild ramification which is  actually more general, it's a less restrictive notion.  And that was studied by Takeshi and his students Yuri Yadagawa and Yadagawa herself.  And there are some old work by Deline and not so old work by Vida and some very recent  work by Saito Yadagawa, Yadagawa and Kuo.  There's also a related work by Hiroki Kato which is not exactly studied like this.  Which says that this notion of compatible wild ramification is preserved by all six operations.  Well this notion of same wild ramification is preserved by four operations.  And the corporations.  Upper star, lower star, upper tree.  Not preserved by duality.  Not preserved by tens.  Preserved by duality.  Preserved by duality, sorry, yeah.  It is.  But not preserved by tens or harm.  So actually there's just a quick explanation of this non-stabilized by tenser but I didn't  put it in the slides.  Basically this is what Takeshi explained this morning.  So a way of defining the same wild ramification is to ask for the trace to be in.  So if you take the trace, so this broader trace the upper aria lens in a bigger field  than Q and then you take the trace to Q and you ask that to be independent of L. But by  doing so you are actually getting some sub spaces which are not sub-algebra.  So that means this won't be compatible.  This won't be preserved by tenser product.  Okay.  Now we have this extension of scalars morphisms, this maps as in modular representation theory.  So there's an isomorphism between Kx with coefficient in Z bar with Kx with coefficient  in Q bar.  And there's also a reduction from Kx with coefficient in Z bar to Kx with coefficient  in FL bar.  Using these two maps we get the decomposition map from Kx Q bar to Kx FL bar.  Then by combining our evaluative criterion with that of Gavir, we get the following consequence  that for any compatible system LI.  So this decomposition map carries compatible systems of analytic sheaves to systems of  mod L sheaves with compatible wild ramification.  So this gives some examples of systems with compatible wild ramification where the recent  work can be applied to.  That will be all of my talk.  Thank you.  So time for questions.  Please, can you give a hint in the question of the application of the application theorem?  Well, so at least one ingredient is to use an equivalent form of the Gavir theorem.  So we have to work with Dion's alterations in the equivalent form.  I have the paper stashed in my backpack.  I believe that if you have some alteration, something that makes ramification tame for  one KL, Ki, then probably you can prove that the other ones are also tame and then you can  prove that the other ones are also tame.  So we are speaking about this different notion of compatible and compatible in the boundary.  Compatible means that all localy close to each other.  So you have compatibility.  Let's say for least sheaves.  So you have a compatibility with some normal crossing device or maybe I don't know, the  generic one was over all of us.  So suppose you know that one of them is tame, then it follows that the other is also tame.  Is it the case?  The others are tame.  But in any case, we only have a finite number of Ls.  So we can make everything tame.  And then you can study the action of the group and look at the, but I believe that probably  the decision to find that number is not needed.  If you can indeed get the, I mean, if you prove.  Yeah.  The one tameness implies the other.  Yes.  And then you can probably put all of, then you can find a comb on the alteration, the  alteration, the alteration that works for everything, guys.  Okay, so another technical question about this kind of problem with ser and so on.  So we work with the Bragg variety, let's say proper smooth or the field of fraction,  but we can instead work with digital analytics spaces or formal scheme.  So of course you can put the same conjecture for proper smooth digital analytics spaces.  So of course some of the results on the work monodromes thing is used as a bright nature  because you are, so you don't have this, but the other part of the kind of independence  of early results.  So is it, how much does it extend to this case?  Did you, so?  There were some results by Mieda, I think.  But we're seeing.  But otherwise I don't know about these particular questions there.  Are there places where the trace, the greater trace is rational but not integer?  Integrality is another problem because you can naturally have, for example, when you  undertake dual it will not preserve the integers, when you take duals or it will just take twists.  So they naturally occur.  No, you just take twist.  You can have q to some negative power as eigenvalues.  But there are estimates on how the integrality is preserved both under the usual operation  for both the.  Yeah, for both the integrality and for inverse integrality.  So for single file, it's a problem.  So if we localize that, so if you invert p then both integrality and inverse integrality  is preserved by all the operations, sorry, except that some of them exchanges the two.  And for the p-divisibility you have some bounds on all the operations.  So if not, it's time to thank the organizers of this beautiful conference.  So first I met who I think is the main actor of organizing.
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