 So last time we talked about this assignment, if you start with a scheme S locally, almost a finite type, you attach to it this category of incoherent sheaves. So in fact this, so the reason this was invented, first of all, this was invented simultaneously or quasi-simultaneously, but many different groups for different reasons. And so, and the way I came across it was not because of Langlands. So I didn't know that this was fix the problem in Langlands correspondence. My reasons were different and I'm going to explain what they were. So the reasons are the following. So yesterday, I mean, no, yes, last week we said that if you have amorphism X to Y, there is a nicely behaved functor of direct image in this context. Direct image, yes. And so you don't need properness? No, it's direct image and it's defined. We gave a kind of a definition, namely you take an object of co, you regarded as an object of quasi-co, you move it, apply the usual direct image and then you notice, aha, it actually is in D plus and therefore it can be lifted to inco, that was the formula. So it's designed so that this diagram commutes where I remind that psi was this natural functor going from inco to quasi-co. You embed co into quasi-coherent sheaves and then you extend. So this functor is nice if you behave nicely with it. If you start doing something not good, it will betray you. Namely, so remember that in the eventually co-connective case, there was also a left adjoint. So if you embed quasi-coherent sheaves into coherent sheaves and this upward looking diagram will not commute. So nice functor, but be careful with it. Okay, but in addition to pullbacks, to push forward, one actually wants to have pullbacks. So if f is proper, you want to create this adjoint pair, so proper. And of course, you can do it within quasi-coherent sheaves, but something bad will happen. So f of Pashrik, so let me write it. Let me write just usual quasi-coherent sheaves. So this is the usual direct image and this is its right adjoint. It exists for some general reasons, however, so however is what's called not continuous. And I remind you what continuous means. So continuous means that does not commute with infinite direct sums. So there is no problem on D plus, I suppose. Pardon me? On D plus it is on D bigger than or equal to something. Yeah, but D plus is not closed under direct sums. Yeah, okay, you take D bigger than or equal to something. Yeah, but I don't want to do that, I want to do the whole thing. And then, so let me tell you why it doesn't commute with direct sums, infinite direct sums. So namely, we have the following lemma, I actually stated one half of it already. Suppose we have a pair of adjoint functors, f and g, as usual, one that's above is the left adjoint, and I'm always in the world of categories with arbitrary direct sums. So if g is continuous, let me say it like this, we said it already, then f sends compacts to compacts. And b, if c is compactly generated, this is if and only if. And so in this situation, it's easy to see that if you have a proper map, there is no reason that it would send compact guys named the perfect guys in x to perfect guys in y. Take the inclusion of a singular point in y, and then you'll get the image of a skyscraper will not be perfect. But the question of, you can limit the category in some way, and no, but what is the point with respect to the original problem of the geometric long-lance? So let's see if you take a curve of jnus at least two. So this is impossible to have a less general, maybe less complete statement by bounding categories on both sides. It's impossible because this functor is of even in arbitrary genus, the Langlans equivalence is a unbounded, yeah, it's of unbounded amplitude, yeah, well, at least one in the bad side. So if you take the constant sheaf on bungee, it will go on the spectral side to something which is goes infinitely to the left to D minus. So therefore you don't want to cut D minus because you will lose one of the main players. You lose the constant sheaf. So you don't want to do that. So for that reason, we want to work with categories, well, these big categories. Okay, so therefore this will not be continuous, but in the analogous situation here, so again the right adjoint will exist for tautological reasons, and it will be continuous because of point B of the lemma. So now let me say a few words. Why do I insist so much on having continuous functors? Why do I care? So there are two reasons. One is very practical and the other is even more practical. So why continuous functors? So answer one is that if the functor discontinues, it's very difficult to compute anything. So namely, so say it's your functor and suppose C is generated by a bunch of objects. That means that every other object can be obtained as an iteration of procedures of taking direct sums and cones. If your functor is continuous, so you have the algorithm of how to extend, if you know what your functor does on the given collection of objects, you know how to extend it everywhere. Well, just because I have control, yes, it's manageable. If the functor discontinues, if I know what my functor does in the generating set, I really cannot say much. And also, I suppose there are some, I'm not sure, but in usual algebraic geometry, the functor, of course, is like in modern times, where it has many problems, in particular, obvious ones, like you can restrict two opens and you get the same upper shake and I suppose when you do this infinite thing, you can lose some of the trivial effects, like if you can restrict an open on the source or on the thing. That's still fine. These things are still fine. We'll get there, actually, in a second. So let me just say that have more control. But here is kind of, well, maybe a better reason. So you can consider this world of DG categories and continuous functors. So the main observation is that these guys can be tensored up. So this is actually a symmetric monoidal category. So if you have C and D, you can create their tensor product, which is another category of the same kind. So let me give an example. So if C is A modules, where A is associated with algebra and D is B modules, then C tensor D will be A tensor B modules. Again, I'm talking about all modules. And so it turns out that this operation of tensor product gives you a lot of power. So you can do a lot of things and later on in today's talk we'll see some of these things. So for example, typical example, if you have an algebra object inside this, i.e., it's associated algebra within this monoidal category, if you have a left module and a right module, you can tensor them up over A and get yet another category. So this gives you a power to produce a lot of new categories. Sometimes construction that, well, in this case you knew what you wanted to get will be A tensor B modules, but sometimes you don't have an explicit formula. And some categories that you wanted to exist will exist because of this thing. This is an operation which is well-defined, though. Yes, it's well-defined. So I'm saying that this category has a symmetric monoidal structure. And it's really functorial only with respect to continuous maps. If you have C1 mapping to C2, then C1 tensor D will map to C2 tensor D if the functor from C1 to C2 was continuous. It will not map if the functor was discontinuous. And what is the tensor over A? Same thing. So this is a monoidal category. If you have an algebra object, if you have a left module and a right module, you just may attempt to take the tensor product. It's a well-defined operation in a monoidal category. If you're- So A is a kind of an algorithm. Yeah, A is, so it's an algebra object inside the DG cat continuous. Well, Aka, monoidal category, monoidal DG category. So, and if I were asked to define what I mean by a monoidal DG category, I will refer to Luri's definition of this tensor product. I will say it's an algebra object in this monoidal category. So monoidal DG categories are algebra objects in here. Algebra object in the monoidal category of DG categories. But when you say monoidal category, it's not in the very old sense of usual category. No, no, I'm saying that this is a monoidal category because of this operation. No, but you see, there is an old sense of monoidal category. It's one utility monoidal category. Yes, it's, yeah, it's, we live in an infinity world. Yes. Can I see it? Yeah. In some model, for a little question. Well, in the infinity world, and then again there, yes. Sorry, what's the link with content function? So the operation of tensor product is only functorial with respect to continuous functions. If you have C1 to C2, you will not be able to tensor of this morphism unless your function was continuous. Okay, so going back to this, so we still want to define, if you have a map, we want to define f up a shriek from int co x, int co y to int co x. Well, in the following way, so you want that if f is an open embedding, you want f up a shriek to be the left adjoint of the functorial direct image. And if f is proper, is a proper map, you want f up a shriek to be the right adjoint. And in general, I mean, when you want to attempt the construction, of course, what you will do, by Nagata, you will factor your morphism like so. And you will say that f up a shriek is first f bar up a shriek times j up a shriek. And you know what they are in each case. And of course, you can't do this in infinite setting. I mean, you just can't define functors but say, okay, just choose a factorization. There is infinite number of homotopic consistencies that you have to check. Yet, you can set up a machinery using these factorizations. You can organize a big, big thing and make it work. So, this has actually been done in the world of constructible sheaves, so there's two papers by Yifeng Liu and, I'm forgetting his first name, Wei Zhenzhen. So, they did it. They constructed this f up a shriek. So, it is a functor. You're taking schemes, almost a finite type. It's a functor to dig a cat. Yes, so, yeah, according to my conventions, I don't even say that they're derived. So, I'm currently writing a book with Nick Rosenblum, where we'll give a larger framework for this construction. So, I'll be very happy if you ask me a question about this, but maybe in question time. So, in this way, this looks like a construction, but in fact, you can state and prove of the existence of uniqueness theorem. So, it won't be some arbitrary thing. So, you can make a statement that such a functor exists and is unique given some more structure. And again, I'll be very happy to talk about it, but not in the kind of main time of the lecture. All right, so now I want to go a few more steps in this direction. So, we know what up a shriek is on schemes. So, then I claim that we can actually extend it automatically to pre-stacks. So, let me be slightly technical. So, I'll talk about pre-stacks that are called locally almost a finite type, left. So, and these are, by definition, these are contra variant functors. I'm taking a fine derived schemes that are what we call eventually co-connective. They have finitely many co-homologies. And these are arbitrary functors from this to group oids. So, these are before we define just pre-stacks to be arbitrary functors from all affine schemes to group oids. And here I'm considering these guys, I just didn't want to lie. So, I'm giving this formal definition. So, these left pre-stacks are actually full subcategory among all pre-stacks. So, there is a fully faithful embedding. However, it's not either a right adjoint or left adjoint anything. So, it takes little thoughts to say what it is. And again, I would be happy to say it, but again in question time. Just, you don't need it for this lecture, but I want to say that this notion, this notion are connected. So, I'll now extend int co to a functor from all, from schemes to pre-stacks to djcat. So, and it'll be the same procedure as how we extended quasi-co from schemes to stacks. Namely, so int co of a pre-stack y will be the limit over, well, it's the same procedure, int co of s. So, now we can talk about int co on pre-stacks. So, let me give an example. It'll be a test of alertness to see if you guys are asleep. So, for example, I'll say the d-modules on the scheme x are int co. So, last time we introduced to any scheme we introduced x deram. It was a pre-stack, such that, so let me remind that home from s to x deram was by definition home from s reduced to x. So, this is a pre-stack, and I'll say that d-modules on x are by definition int co of that. Arbitrary. Are people happy or not? What? Yeah, we're supposed to be, is this supposed to get tested when we're awake or is that coming? No, no, no, it's supposed, it's already tested if you're awake or not. Yeah, so this is already supposed to provoke some reaction. So, let me give you, so let me say what should have aroused your suspicions. So, last week we gave a different definition of d-module. So, this is today, yes, last week we gave a different definition. We said that it's quasi co. Okay, so you can ask what's going on? Why am I changing from here to here? You can ask, are they the same actually? Or of what? Okay, so let me explain what's going on here. So, there are two more pieces of data. So, note that on an affine scheme, if you consider perfect complexes, this is a monoidal category, you can tensor them up, and you can tensor coherent sheaves by perfect sheaves. Yeah, just tensor up. And you can extend it and you'll have an action of this monoidal category on this category. And the way the definitions work for quasi co for int co, the same will be true that on any pre-stack quasi co will act on int co. This is on the one hand. On the other hand, so if you have any scheme or stack, there is a, oops sorry, there is a canonical map to the point, and you define the dualizing to be the pullback, of course. And so using these two pieces of structure, you define a functor, I'll call it Upsilon, for any y, it's a functor that goes from quasi co of y to int co of y, namely action on the dualizing, f maps to f tensor omega y. So for any pre-stack, you have a map from quasi co to int co, and the theorem is that if y is Durham of anything for any pre-stack, well, within our finite type world, then this Upsilon is an equivalence. So what is omega? Up a shriek from the point. We now have the formalism, so we can up a shriek. And so y is any... So y is obtained from a completely arbitrary pre-stack by taking Durham. So the Durham is smooth. Durham is smooth, let me put it like this. And if y is a scheme, then Upsilon y is an equivalence if and only if the scheme is smooth. So for schemes, this Upsilon is an equivalence only for smooth guys, but the Durham thing is smooth. Smooth means classical and smooth. So y is a derived scheme, and this functor will be an equivalence only if and only if y is classical and smooth in the usual sense. And so just a moment about the upper shriek. So you said that you can define it for... Any morphism. Any pre-stacks. So I've extended... So I had my functor defined on schemes, and then what I do is technically... So I gave a formula what intco is on a pre-stack, but this formula has a name. It's called the right con extension. I extend from schemes to pre-stacks in some automatic way and I obtain a functor from pre-stacks to DG categories. Yes, yes? Relative to... Shriek. Yeah, everything with shriek pull back. No, because if you have a morphism, because I understand that... I'll explain. The shriek don't pull back, right? Okay, I think I understand your question, I'll explain. So I think here's what Oprah is protesting against. This has nothing to do with pre-stacks. You can ask this question about schemes. So say we have a morphism of schemes, and then you have the following guys. There is quasi-co of x. There is quasi-co of y. There is intco of x. And there is intco of y. And then there are the following operations. Here, there is a usual star pullback. Here, there is this function, epsilon, tensoring up by the dualizing. Here, there is an epsilon for y, and here is the shriek pullback. The claim is that this diagram commutes. Intco y. So the shriek is the final intco of... Schemes, but then by extension, any pre-stacks. But when you take the derived kind of inverse limit of intco s, this means that if you have s-prime going to s, you are using the final intco s to intco s-prime, which is the usual pullback. Which is the after shriek. Yeah, it's the up-a-shriek that I said that I erased this blackboard. I said that up-a-shriek is, by the work of Liu and Zheng, up-a-shriek is defined as a functor from the category opposite of a fine scheme's almost a finite type to digit categories. No, but this was for the lattice sheeps. No, no. So they did it for lattice sheeps, but the same construction works for intco. Okay, so the lim is not relative to... Up-a-shriek. But it seems that when you define in previous lectures what was intco, it was lim. I don't remember now. Correct, and I'll comment in a second. So it's another notion of intco. No, they are equivalent. I'll explain it in a moment. So that's where we're going in a second. If y is a regular smooth scheme, so what are the two categories? Not derived, something less simple. So what is a statement? What does it mean? I'll explain it in a moment. Let me just do it right now. Is it omega? Yeah, omega is what you think it is. So let's recall what happened. So that's another moment to test your awakeness. So let's live in the world of schemes. I just said that if you have a scheme x, you have the sphunkter omega x from quasi-co to intco. So this is the sphunkter and it's tensoring by the dualizing. It should arouse in you in the sense that we are in a zoo. So do you know what I mean? Well, I claim that we already had a bunch of functions between these two categories before. So what we said that always you have this sphunkter in this direction and if x was eventually co-connected, you also had a sphunkter that I call xi. And here I come and I say that in addition to all of this, you also have this sphunkter. And what is going on? Well, indeed, you have all of these guys and so, well, you have all of these guys and they're different sphunkters. Okay, but what are they? Okay, so the simple example is the zephyne line. Example, if x is smooth, so then let me say that these... Well, the way we set up intco, it will be just the same as quasi-co. There is no difference. So xi x equals omega is the identity sphunkter. Nothing is going on. In this case, this sphunkter is tensoring up by the dualizing line, y, by the canonical bundle. Shifted to the line. Canonical bundle... Shifted, so it's the dualizing. Shifted to the minus the dimension of x. So, Ofer actually asked another question. He detected a discrepancy in how I set up intco in the case of algebraic stacks. Let me suppress that for a moment because it'll take me too far afield. Again, it'll be a question that I'll be very happy to answer kind of in question time. So, we have only zoo. And let me just say that it's only this sphunkter that actually makes sense for arbitrary pre-stacks. So, these guys are actually a feature of schemes and it's this guy that extends to arbitrary pre-stacks. So, this is the more fundamental of the more fundamental sphunkter. And here I stated in the theorem that this sphunkter is an equivalence on anything which is deram. But if x is a locally complete intersection scheme, I mean defined by Gwendolyn and Schwan, then it's already interesting. Yes. So, in this case, and yes, and actually I should have said this. Okay. So, we have these two categories and now there are two functions. There's this xi and there is the epsilon. And you can ask how are they different? And they differ by tensoring by a line bundle. So, if it's locally complete intersection, this dualizing still makes sense. It's a line and these are two functions from one category to another and they are not the same, but they differ by tensoring by a line. So, that's how it looks like. We just asked this. So, the deram is a pre-stack. Yes. But is it ever anything else? Are there any conditions under which it's... No. It's never represented. No. It's cotangent complex is zero. So, on any pre-stack you can talk about the cotangent complex and for anything which is deram this cotangent complex is zero. Because the cotangent complex is tested by mapping infinitesimal thickenings to your stack and deram is set up in such a way that doesn't feel any nilpotent. So, to define the cotangent complex on the stack this is a very general kind of stack. Yeah. So, cotangent complex is defined in the way we talked about in the first lecture. You're taking this square zero extensions and you're mapping. And you... Yes, thank you. Because this condition is not sufficient. The project is zero. It is. Yeah. All right. Isn't deram of these things what we get? Deram of deram. Deram of deram is deram. Yeah, because zero is a deram. Okay. So, let me just summarize what just happened. What just happened was a crash course on Upper Shriek. So, now I'm coming back to the notion of singular support and I want to comment on how it behaves with respect to Upper Shriek. Again, the reason I want to do it, again, because we're looking at this diagram, loxis parabolic, loxis for the levy, loxis for G. This is... We call this P spectral. This is Q spectral. And our Eisenstein functor, I spectral, was the composition of Q-spec Upper Shriek. And last time I said that we'll have to talk about it and now we have it with P-spec int-co lower star. And what I wanted to explain is that this functor maps the category, so it maps int-co nilp on loxis M to int-co nilp loxis G. What we did explain last time was the following. We did a half of it. We introduced, so int-co nilp on loxis P, and explained that the direct image maps it to int-co nilp loxis G. Today, I want to explain the second part. I want to explain that Q-spec Upper Shriek maps int-co nilp loxis M to int-co nilp loxis P. Just the other half of what I did last time. Okay, let me first remind you where this came from. It's a general assertion. It's not specific to int-cos. It's something general about singular support. So, reminder. So let X to Y be a map between quasi-smooth schemes. So then we have this functor of int-co direct image on the one hand and also we have this correspondence. I will go from sin Y over Y. It maps to sin X and projects down to sin Y. And I call this map sin of F. So the theorem from last time was the following. So fix n sub X inside sin of X and n of Y inside sin of Y. that If you take the singular Co-differential the sync of f take the pre-image of Nx it would be will be contained in NY so in this case the claim was that This direct image maps intco Nx on x to intco NY on y So it maps this subcategory to this subcategory. That was the theorem from last time So now I want to state the corresponding result For a pashrik, so Let me ask some student Who will volunteer the theorem for me so so same circumstances Nx and xy so when Does f up a shriek send intco and y y to Intco no apple up a shriek Okay, so Pierre wants to give an answer, please Give me so it has to do with the singular No, give me the answer it happens. In fact, it will be for no leave, but give me sufficient condition Transfer cells it will be some other place you'll see we'll gather in a second just in one minute. It's not transversality. It's containment It's an estimate from above. So anybody yeah some x should contain some Yeah, we want this to be small as compared to this Like the bigger n-axis the easier is for this to hold. So what's your estimate that's that's to Smooth schemes Yes, there will be something there will there is a parallel assertion about demodules, right you can estimate the characteristic variety of Upstairs in terms of the downstairs. So give me the analogous thing here. So what if you have a demodule downstairs if you want its Singular support of its pullback to be contained somewhere. What's your condition? No, no, no transversal containment Transversal will be in the moment Yeah, it's the same thing in the opposite direction so theorem So this happens when provided that containment Just in the opposite direction that and why Why is small enough as compared to an X ie? Something is contained inverse image something else if sing F of This is contained in an X same condition just inverted containment and It's an exercise Apply it to the morphism Q and obtain this containment deduce From theorem But here we wanted more so we wanted this functor eyes not only to respect single supports. We actually wanted To send coherent things to coherent things. We wanted to preserve compactness For this piece back lower star it happens because the morphism was proper Now we are doing this kilo upper shriek enough upper shriek will not in general pervert preserve Coherence you're doing up a shriek Like if you have a singular variety and a point mapping to it. I if you do up a shriek Well, you'll get all the X's there'll be there'll be infinitely many of them So it's not doesn't preserve coherence so So you can you may want in these circumstances ask so when is F upper shriek of F coherent if F was coherent and this is Well Pierre was saying transversality this will be transversality So we can actually give a criterion when coherence would preserve in terms of singular support Map map between quasi smooth schemes pardon No This is what you're saying is too weak. We can give a we can give a finer condition. Let me Do something more general well after all up a shriek is a Version of tensor product. Let me just say something more general for tensor products. So here is a theorem so X is quasi smooth and F One enough to our coherent sheaves. So then the following are equivalent So one is that if you take the usual tensor product, of course derived one is coherent. I Mean coherent means it has finally many cohomologies B You can take their shriek tensor product By this I mean so what's tensor product? It's upper star under the diagonal map of the external tensor product This will be upper shriek Is coherent then it will be Internal home In either direction It's coherent D Internal home in the opposite direction is coherent Shriek you take F1 times F2 on X times X and you shriek pull back with respect to the diagonal map And of course, this is all boring, but all of it is equivalent to the following Namely you take the singular support of F1 It's a subset inside sing you take the singular support of F2 it's a sub thing inside sing you intersect and you want it to be zero section Now for for a map between Usual schemes, you know if it is local complete intersections and or even finite all dimension Anyways, then the F upper shriek sense could bound current about yes Now in your context, let us say if you have usual Complete intersections, of course the map between there is not necessarily complete. Exactly. So your condition In this case Is not it's it's non-trivial So you didn't state exactly the condition. So I said I stated something more general instead of taking up a pullback, I talked about tensor product of Sheaves so you can this pullback can be expressed in terms of the tensor product Pullback Shriek pullback can be expressed in terms of this kind of tensor product and store pullback can be expressed in terms of this Kind of tensor product on something else on something else. I'm currently I'm blanking on what I think on x times on x times y or something like this On x times y because you Yeah, so you will be You'll be doing F tensor The structure sheaf You'll be tensoring with the with the structure sheaf of the diagonal something like that I was speaking to the diagonal something like that So I'm saying that if you know this you can give a criterion for when coherence is preserved There is there is there is sergility. Yes Yes Yes, it's isomorphic to Df1 tensor of those Yes, yes, that's what I'm saying that if you want if you talk about that for Pashik you you'll be using point B of the theorem But not the structure sheaf, but the dualizing. Yes For for quasi smooth. It doesn't matter because the dualizing is a line bundle And then the singular support that you are you will be using is on x cross y Yes But because the other thing is is just the dualizing its singular support will be zero time to single support of the original thing But now when you have usual schemes You could use your singular support was using the fact that it is quite a smooth That it uses that for usual schemes. There is no there is no What is the singular support for usual Computer section, it's the same as you regard the usual computer section as derived one and You create its its sing and it is upset there Usual case is a particular case of this So maybe let me maybe it'll easy if I just give it if they write the formula so so You know what I'll write this formula. I don't want to do it while I'm standing I'll do it during the break So I said that you can give a criterion Deduced this criterion from here. I'll do it, but let me not do it. Just standing up. All right Okay, so This is well These two theorems are kind of basic basic functoriality properties of in of singular support So that's how they behave with respect to push forwards and pull backs All right, so But your Eisenstein thing so if I take g equal g and 2 The the ball is a borel. So I'm is a torus. So I take a local Unlock the sink of the torus. So it's not so complicated. I take the punctual sheath at some point. Yes, what is the image? You asked this question before Yeah, but I at least on on an open subset an open sub something of Loxy And the curve is of genius bigger than two. So there is a irreducible local system Let me understand the question again. So you take the nicest open subset that you like Do you prefer of Loxys lock system of G? Okay, so we want to do the computation. So you're starting with Sigma It's a point in Loxys T Loxys T torus. Yeah a particular point skyscraper Okay, let's be careful So in our category nilp is zero for the torus, right? So this is actually in quasi co Loxys T ie in co nilp But that was zero Okay, we start with this and then we do Eisenstein and You're restricted to a nice. Yeah, so and this is this is an object of int co Loxys G It's a particular object there. So So let's ask a question about it. So what kind of question would you like to ask? So there is a very interesting question that one can ask But okay, so I'll let you ask a question. So we get we get a well-defined object So Yes Yeah, it's a sub stack. Yes indeed. So and it's easy to say what it is It's either the local system is reducible or when it is reducible This two one dimensional guys are non isomorphic. This is open subset on that open sub stack This int co nilp is just quasi co and then what what you get is what you think you'll get So it'll be just direct image Well, I mean there can't be anything because you're just dealing with the usual quasi co but on the interesting subset where nilp is not zero you're getting some phenomenon and You asked this question before so No, no, you asked a bunch of years ago with this there was this mystery of Eisenstein and it's that's where it lies and it's exactly It's mystery of Eisenstein has to do with remember I wrote this treacherous diagram that one diagram did not commute it It has to do with that All right, so you take the constant She from all on Luxus why nux of T Again so There will be phenomenon so it's The open part where nilp is zero you got you will not get anything interesting, but then something will happen Okay, so I Think I should make a break. So let me just say what I had in mind for the last hour There are two themes that I wanted to cover and I think realistically only I will only have time for one of them So, let me just Say what these are So theme number one is the following So we said that if F is a Into coherent sheaf you can assign to it as its singular support of F which is a subset It's really it's really a subset. There's no scheme theoretic structure, but you can ask is there a scheme theoretic structure Can you do something slightly better? So and let let me tell you what kind what kind of better thing one can do so Let me consider an object not an intco, but I'll make quotient intco by quasi-co. It's called the category of singularities So in general if you have a I'll denote this with a circle. So it's intco Modulo quasi-co So well by the same and now it's well the same principle to it You can attach an object in the projectivization of the sing Both remember so we had this adjunction so So you can think of this quotient as either the kernel of this functor or The quotient by the image of this functor. It's the same and the image This is fully faithful You can take this and modulo the image of this functor or it's the same as the kernel of that functor Perfect complex and some And So quasi-co gets supported on the zero section therefore if you take an object of the quotient It will have a well-defined support on the projectivization So let me say to this you can assign let me say singular support with the circle of F Which is a subset in this projectivization but so What you do have you have more so here you have not exactly scheme theoretic structure, but you have the following piece of structure So it will be the following theorem that you consider the category of D modules on This projectivization and this category. It's a monoidal category And now we remember what I talked about in the beginning. So it's a monoidal category in the world of DJ categories As such it will act on this category of singularities So not only do you have it's a dumb dead-way singular support something act on something and so and This is actually of central technical importance for full england's and I can explain that so It kind of it's a central result that allows you to do things So this is theme number one and theme number two is the following It has to do with local language So local language is still very highly conjectural. We don't unlike Global language. We don't have a statement, but we kind of know what it's supposed to be So it's supposed to compare the following so global language was an equivalence of a Pair of categories this category was supposed to be equivalent to this one Local angles is supposed to be an equivalence between a pair of two categories This two categories supposed to be equivalent to these two category So what are they one is? categories equipped With an action of the loop group Yes, yes, I mean it's a DG categories equipped with pieces pieces of structure When I say category I mean DG category So these guys together form a two category an infinity two category The totality of these forms an infinity two category No, no, no, no, no, it's different the word to means that I allow non-invertible two morphisms There is there is book in the in the process What Yeah, you know Lurie did infinity one categories and these guys are infinity two categories Yeah, so This Maybe not so bad Okay, so this is the geometric side and here it will be categories Well, let me say it again Don't understand it technically yet sheaves of categories over Local systems with respect to language dual group on the formal puncture disc and Again, it's a mess. So I've been working for many years trying to make sense of what it is but even if I Am able to make sense of it what it is this cannot hold For the same reason as the global guy was wrong Namely here. There is more stuff than here. So there are there's a phenomenon Whitaker degeneracy index G-check Langlund's dual Yeah, I look so let me just If I'll go in either this direction of this direction the next hour and then I will explain whatever I can explain So but what I want to mention is that even man managed to do this It's still wrong because we have to account for the same phenomenon In the global case. We had to replace quasi co Of locks sis. We have to make this replacement of int co nilp So we enlarge the spectral side and then it had a chance to be Equivalent to the geometric side. So here so sheaves of categories over a pre-stack You can think of it as some kind of categorical to categorical quasi co So then we should be able to also somehow enlarge it To some version in categorical and the only problem is that no one knows how to do it So in the case of quasi co you said, okay, just take coherent sheaves in complete and that's your guy Here it's just completely not clear in complete. It's just completely not clear So see this is a two category and then you're enlarging this two category and the worst is that Let's just relax for a moment So I'm sure you guys have seen two categories in your lives But all the two categories that I've seen are of the following sort. It's a cat Each two category has the following shape. It's a it consists of categories equipped with an extra with extra structure its categories with With extra structure So any two category that I've seen in my life comes equipped with a forgetful functor to the two category of categories Well, if you can think of any other, please let me know but I haven't seen one So see we've seen categories that are not equipped with natural functor to sets So that's why I was so so Yeah, we know categories that are not sets with extra structure like sheaves on the manifold. It's not set with extra structure. It's more and plenty of other examples No, I don't even if I did it will be the same problem. Yeah, okay, but I can do it But I mean it's so these are my dad Yeah, these are technical problems of this but even if I resolved all of these it will still be not enough for the following reason that I'm creating a two category without and this cat this big two category is really doesn't have a forgetful functor to To the two category of categories If you take a simple level Like you already level or something you can expect that you know the answer I know the point of view of local long-run. Yes All I want to say is that there are two options for the next hour and I would like to know which ones of which one would you like to meet to go to? No Which one is supposed to be in the first week Okay, so We can take a vote and then I can do in question time. I can do the other There is this kind of at T Okay, let's let's take a vote who wants one The first Okay, who wants to seems like there is there's parity So, okay, let's take five minutes. So one is It's content of another paper with a rink in it's called the category of singularities and global Springer fibers Okay, so I Said that I'll talk about the following the demodules on the singular the projectivization of the sing act on The category of singularities, but I will start not from here. I will I'll motivate it by language. So I'll say what problem we were solving and then how that kid that thing came up So let me remind you Something that I stated at the end of last time So we said that you take all this spectral Eisenstein series functors and You apply them to quasi-co Don't even do the modification. So this quasi-co sits inside in co you get some collection of objects inside in co nil and see what these guys generate so the theorem Was that what they generate is exactly in co nil of loxious G Okay, now, let's take a look what it means what it means to generate Suppose as usual you have a pair of our giant funkers F and G So the lemma is that the image of F Generates G Including M equal to G the bulk So the image of that generates D if and on if the functor G is conservative conservative in the world of DG categories means just it doesn't send anything to zero Pardon except zero But so you can imagine that to prove something this conservativity would not be enough So kind of what we want to do we won't turn this kind of statement That some functor G is actually fully faithful. So this is So D will be This in co so we want to map this category fully faithfully into something has to do with quasi-co's button parabolic's and So that is this is what this is where we are going So, okay now a scary creature will appear So don't be afraid Okay, say have a morphism F from X to Y. I want to talk about So who remembers what how do we define D modules on X? But what on X the rum in co quasi-co Yeah, so either and it doesn't it doesn't matter, but say I want to define relative D modules I want kind of quasi-coherent sheaves or in coherent sheaves I don't know on X with a connection along the fibers. How do you define that? Okay, give me the definition So vertical D modules on X with respect to the map are We'll then figure out if you want quasi-co interco. Give me the geometry. It'll be sheaves on what But what could I do to fix the fiber but the fiber is nasty you can think like you can take a point mapping to Y Yes So you're considering this creature X the rum over Y the rum with Y So this kind of thing Okay, now, what do you want to consider quasi-co in co? well The claim is the actually want to consider both Yeah, so you see if X will equal to Y. This is just X so and We know that there is canonical map this Upsilon and Well, you can show that in this case, it's a fully faithful embedding But what happens is that the category that we actually interested in is in between it is genuinely in between and neither nor so note that there is a map from here and From why to this thing For the moment not but soon it will be sorry, there's a map from X to here so therefore there is Now we'll so that's why I talked about the streak pullbacks And that's why I preferred theme one because we're gonna use what we talked what we developed by the streak pullbacks so Shriek pullback defines a map from here to into co X Shriek pullback with respect to this map kind of think of it forgetting the connection Forget relatively module suggest all modules. Now, let's assume that X and Y are quasi smooth again. I assume X and Y are quasi smooth in this case there is well Also known as quasi coherent sheaves on X Things with zero single of support and let's define This as the pre-image So these are those objects here such that they forget the connection They have zero single support and it's fairly easy to see that the image of this functor actually Lies in here. So there is quasi code. There is intco and there is something in between and It turns out I'll say in the explain the moment that this is the beast we're interested in And let me give it a name because it's too much notation. Let me call it. I X over Y Okay, and so what we're gonna do We're gonna consider this morphism locks his P to locks his G and We are interested I claim in this fellow so you can ask Why are we interested in this fellow and not in some other fellow? The reason is that so we are still trying to prove Langlans and Well, what you want to do you want to translate things from the spectral side to the geometric side And it turns out that this guy if you assume Langlans well If you assume induction this guy can be translated to the Well can be fully faithfully embedded into something explicit on the geometric side so let me just say that By induction on semi-simple rank this category Fully faithfully embeds Into what's known The p degenerate Whitaker category You don't have to understand what this p degenerate Whitaker category is. I'm just saying this because to explain why I'm interested in this category Okay, so these are my building blocks and So, let me state the theorem and then I'll explain Its meaning so the theorem will look as follows It will say that a certain functor from ind co nil on locks his G To well, I want to take them all of them together and I'll write the following symbol. I'll rate right glue This Categories where P runs through the whole set of parabolic of G So the theorem is that is fully faithful. So what I have to explain is that What do I mean by this glue and what is the functor? So well So it's kind of easy to see that For the left a joint what The essential what's generated by the central image of this is the same as what's generated by these guys So the generation is the same, but I'm getting something fully faithful and so this theorem is central to Langlund's because Yes, this is what I'll explain On the geometric side for GLN we have an analogous embedding of D of bungee on to glue Well, we have a map in general glue and will be this With degenerate Whitaker guys So we have a functor in general. It's known to be fully faithful embedding for GLM and If you believe what I said you'll get a fully faithful functor in one direction So that's kind of that's the the mechanics of the proof of the proof of Langlund's So that's why we're interested in this assertion So you Yeah, I will I will not be talking about the geometric side. I'm saying I'm just giving motivation Why I'm interested in this kind of assertion. I'm interested in assertion because I have a parallel assertion On the geometric side, I have a functor Between these categories which I can prove that is fully faithful. That's what I mentioned by Induction on a semi-simple rank and then I can show that if I on the generators if I go this way The generators are in the image of this functor and this gives me a functor in this direction Which is automatically fully faithful. So this this is this is how the proof of Langlund's for GLM goes No, no, no, not at all. No, this is our Langlund's right. We're trying to compare this and this If I put Langlund's dual So we're trying to prove that this category is equivalent to this What I'm trying to do I'm trying to construct the functor in one direction and moreover one that is fully faithful and The way I'm doing it. I'm embedding both into some larger categories and I'm constructing a fully faithful functor between this larger categories but then I Still have to construct these functors Finally, it'll be an equivalent finally it'll be an equivalence, but I mean you there are various stages of the proof and this stage will show that The functor is actually fully faithful Jill to there'll be two pieces. I have to say what glue is in a moment You've been insisting that Well, it's well, it's not of course, it's not boring When you say you have a map on this on the right side That means that for P equal G you are able to do something for P equals G What will happen is that I'm dealing with quasi co and so Quasical G. Well, okay, let me just say it again all these are scary words. I'm not What happens it goes to embeds fully faithfully into the run version of Representation to G check and this is the Whitaker category. So it's a run. It's a run version. So you're putting again, so I don't want to it's a It's a category in Run is the person Zivran The student Okay, yeah, so so I'm asking I'm answering Lamont's question why this is what is useful because I mean this this embeds into something local Yeah, so the way you do it you you can exhibit generators you mean this oh so this is completely geometric So you then you Produce this collection Representation yes, this these are the Whitaker coefficients So there is a functor of Whitaker model that takes goes from here. It's a paper dedicated to you So it's Yeah, yeah, I know So It was just a motivation why I wanted this theorem So let me just maybe let me just go ahead with this. Of course this language story is very interesting, but Okay, I will just to remove the temptation. Let me erase this Okay, so what is glue? So here's the setting for glue Say you have some index category a and you have a functor from a to DJ cat So to objects of a you have some DJ category c and you have when you have an arrow from a to b You have functors. Let me call them f a to b So in this case, there are a couple of things you can do one is something will been doing all along You can take the limit So this is a category whose objects you should think of as follows. These are objects little c a for every a and For every arrow You you're given isomorphisms. So the limit is just really kind of a compatible system of objects in each of your categories But there's also another thing. It's called lax limit and I'll explain give an example in a moment This is also collections of objects like this But now we no longer require isomorphisms. We only require maps in one direction in this case this direction and There are two there's lax limit and the co-lax limit Yes, this is what you meant to say yes, it will be and this is what I call glue and Alex well, yes, that's people also So and let me let me give an example exactly Maxim's example what I mean by glue So consider the following example say a is this category. It has two objects zero and one and one arrow and consider the following example of this C a's let y be a topological space and Y zero being open and Y one will be a complimentary closed So C zero will be Sheaves on Y zero again, I mean derived category C one will be sheaves on Y one and the functor zero to one will be I up a shriek J low a shriek. So then I claim that this glue Sheaves Why I will be sheaves on Y itself So all I'm saying the sheaves on Y are glued in precisely in this sense In term from sheaves on the strata. It's not a limit. It's the lax limit. So So this is what I mean by the right-hand side Yes, and then it will be arrows in the opposite direction It'll be co-lax. I wrote this example because I wanted the arrow in this direction Yes No, no, no, there's no shift, but it's as as written but I'm when a I'm doing derived categories Functors is the functor is as written You take all the parabolic I'm take all the power. Oh, no Postset of standard parabolic's conjugates standards conjugates a class with parabolic's Yeah, it's a finite post set. All right, so This is what I take so This is what I mean by glue. Let me say what this functor is So if you look at the definition of what it means to glue what I have to do have to produce a family of functors loxious G to these categories that are not Compatible under pullbacks, but they are lax compatible. So I don't need if I go from one parabolic to another I won't don't want the functors to strictly compose. I want I need them to compose up to up to natural transformation So, let me say what these functors are So the functor is the following. I go from int co nil loxious G I Even forget that it was nil I embedded into the whole thing then I do up a shriek to int co loxious loxious P the rom over loxious G the rom loxious G oops Problem I've gotten here. I wanted to land in a subcategory. What do I do? No, I landed. No, I'm not here. I'm here. I really land here What do I do a Joint so so and then a joint What's called I? loxious P over loxious G So if I didn't have the a joints, I will land in the strict limit, but because I have these adjoints Strictnesses broken and I only have natural transformations. So these are my functors and the theorem says that this this The functor into the group category is a fully faithful embedding Okay, so in the remaining 15 minutes, I'll try to explain the Why what I said about D modules helps to prove this theorem Oh, it it exists, it's the right adjoint it automatically exists by general Louis stuff And moreover is it continues because this fungus sense compacts to compacts This embedding Yeah, and it's all of these categories a compact generated Yeah, so there is a map from here the stack maps to why Yeah, yeah the projection just pull back right adjoint right adjoint left adjoint does not exist You said that sometimes you have a joint and both so this is right adjoint It is continuous because the initial functor sense compacts to compacts okay, so I'd like to explain not the proof, but the idea of the proof of this theorem and That idea is the following so I Mentioned the following theorem that there is a canonical action of D mod Projectivization, I mean let's say sing Fly on intco with a circle Why which is by definition the cat what's called the category of singularities and again here So now I'm gonna use what I said before that this procedure of tensoring up So here are the features In terms of this action I can say I can express the singular support So for that M be a closed subset Inside this projectivization Well, I can take Intco with the circle with supports in M On why so before to a subset in sing itself We attach the subcategory, but now I'm taking a subset in the projectivization And this subcategory will live in the category of singularities So this category turns out to be we can express it in terms of this action. Namely you take The entire guy And you tensor over D mod on the projectivization with D mod on M So there is a restriction map from m sits inside this projectivization There is a restriction map tensor up and you recover what the category with given supports is This is point a of the theorem and Point B of the theorem pertains to that situation when we have not just y but x over y You leave that for a second So let me also introduce the notation I not X over y It will be I X over y where I kill the boring part. I kill the quasi-code part So point B says the following So that this category I not can also be expressed So again, it now will be a question to the audience So it will be intco Of y Over D mod of the same thing Sing why with what can you guess? So it's the difference between This and this it's a claim. It's also gotten by this procedure of tensoring This procedure tensor product with something So let me say that it's D mod projectivization of Something so I want you to guess what this what that something is We put it on so okay. I'll just call it. It's not contingent. Let me call it relative sing So let me say what it is. Okay. Pierre is well equipped to answer this question. So Let's look at our picture again. So we have y to x It maps to sing y it maps to sing x and Here, let's take the kernel. Let me call that Y over x it's the guys here that die So this is what I want to put here So this is a map to fiber-wise linear map and I'm taking its kernel I eat so I'm taking the pre-image of the zero section. Oh, sorry. It's x over y so This is what this category looks like so I want to say How this kind of thing helps to prove the theorem written on the blackboard? Well, you prove it in several stages first It's quite easy to see that It's enough to prove to mod out everything by quasi-co. So it's really Enough to prove this Why It's the co-differential like if you like look usual differential geometry in usual differential geometry have co-tangent of y times co-tangent of y over y with x Have this diagram Which should be familiar this is the just derived version Okay, so I want to explain how this kind of thing helps to prove this kind of thing So so now if you look at the left-hand sides, what you'll see is that so now Both sides are obtained from intco Luxus G by tensoring over so Luxus over D mod sing Luxus G with something I'm in all cases. I'm killing the quasi-co and that's what the circle signifies So a claim in this Both sides are obtained With something so let me explain what these some things are so on the one hand we have this projectivization of nilp, so we have D mod on the projectivization of nilp and here it'll be the category glued from the following things Will be it'll be projectivizations of the relative things relative things on Luxus P Luxus G So D modules on that So we've replaced the original statement that some functor that has to do with intco About a functor that only has to do with D modules The advantage of the letter is the latter statement is topological so It's the statement is that certain homotopy types are have trivial homology Let me say very explicitly so so The latter statement is that Yeah It is supposed to be no to The fully fatefulness is that certain Homotopy types have trivial homology and in the remaining five minutes, I'll say what these homotopy types are Groups yes, now we'll see Usual sense No, nothing it just homotopy types as we know them So and else let me say very precisely what they are So fix a point a comma sigma in nilp So Sigma is a local system for G so Sigma G is a point in Luxus G and a it's a section Durham section of the adjoint So nilpotent and the morphism So now let P a be a parabolic to it you attach the following scheme Which are called global springer fiber because you have the following things it'll be Reductions Sigma P of Sigma G to P such that this a Belongs not just to be the parabolic it belongs to the unipotent radical and So they then you can form a homotopy type by gluing them and there is a so these reductions Let me call it and the claim is that this homotopy type So so We don't know if it's the homotopy type is contractible we know the homology is trivial Yes, so there will be a diagram of spaces In this case the diagram will be in this case the diagram will be strict Yes, you take the column call limit homotopy call limit to call limit It's some it's some sorts of push-outs True yes, yes, yes Homology of a call limit is the call limit of homologies. So this It's not a great idea to do it because the way Kind of an object was a point in each plus morphism between images a kind of a pass So this looks like doing homotopy limit not calling But now you all do the same with now that there's a homotopy call limit instead. Oh, I don't understand that analogy because You see so this what I call lacks limit you can equally call it lacks call limit so this So a limit really has a universal property for mapping in and call limit has a universal property for mapping out So lacks limits and call limits. They have lacks universal properties and they have both So you have a lacks limit You can say what it takes to map out of it is the same way as what it takes to map into it It's just the same natural transformations So when you talk about lacks things in this context, I could have called these guys just the same way lacks call limits categorical definition It's not you love the call you usually call it of a category Junction of objects and then filled with morphisms. Yeah, it's a relation and and and so and limit is obtained by families for every and so What you Can maybe after yes. Yes, I can well I'll so I don't want to kind of keep people but I mean I'll tell you exactly which homotopy type it is And the end of the day this homotype homotopy type is very Well, it's no no, it's not something that we had seen before but we But at the end of the day after gluing so we certainly conjecture this homotopy type at the end is trivial We proved its homological triviality so you glue a bunch of Springer fibers for different parabolic see glue in them some way and the claim is that at the end of the day No, so what what is involved is the following so you have a springer fiber Let me say it like this so you have a datum of a local system datum of a local system you can think of a curve It's just a bunch of a homomorphism from from the fundamental group to To G so then How should I say it so if you have a so the springer fibers I'm talking about are the intersection of the springer the springer fibers, you know it Plus the fixed point locus for these endomorphism acting on the flag variety So it's these kind of things so the curve only appears via its fundamental group namely the fixed point loci of of the generators on the Oh Yeah, this part is just you take a strict diagram of topological spaces The maps that are involved are not in bedding So you have to when you're taking this co-limit you have taken the homotopy co-limit because this But the claim is if you take the homotopy co-limit at the end you'll get we conjecture. It's a point We haven't proved it but For each you'll to this kind of thing for you to is a triviality but well Well after we massaged it up to here it became a triviality When we got so there is stuff, but it has to do with all these theorems about Action of demodules kind of this is a non-triviality for g2 But this kind of statement already when you're down to parabolic because you only have two parabolic it becomes very very easy For g3, okay, I'm tempted to say what the diagram looks like for g3. It looks as follows so you consider the flag variety the complete flag variety for g3 and then you'll see something like 2p1's meeting at a point and this diagram will look as follows. It'll be The post that will be will look like this zero going to one going to two And it will be a one map that collapses one component to a point It'll be another map that collapsed the other component to the point when they glue them together You left only with the point. It's something like something like that so it's Again, I didn't want to be very precise here. I just want to say how The statement allows you to go from intercoherent sheaves to actually topology and then you attack the problem in topology Explicitly using a building or something. I think my time is up