 Thank you very much. It's really an honor to give this course, obviously. So thank you very much for the invitation. Well, I'm sure I'll disappoint you. I'll do my best to minimize the damage. Okay, so one could say that this whole idea of singular support is technical. So it emerges from a rather fine point on homological algebra. So, and let me tell you the source, the origin. So there are these things called spectral sequences, and we're used to them converging, but sometimes they diverge. So the spectral sequence doesn't compute what you want to compute. And so this whole idea of singular support is a measure to which a spectral sequence does not converge. And you may think, well, the spectral sequence doesn't converge. Who cares? It's boring. But in fact, some mathematics does emerge from there. And I should say that it's not the first time. So the exact framework that I'll be discussing has appeared before. So one is, I'll mention it not in a temporal succession. One is what's called matrix factorizations. These things were discovered by physicists. So these people are smart guys. And another context was much, much earlier. So this is tate cohomology. So they're also smart guys. But it's the same idea that we'll be dealing with today. So as I said, we'll be measuring the divergence of spectral sequence in its kind of functional analysis. So we'll be doing functional analysis within category theory. So now as to the framework of these lectures today, well, let me say a sort of introduction. So Gelfand used to say that if you write a paper, the introduction must contain it all. So like you must say everything you want to say in the introduction, and then there are few technical details left for the remaining 100 pages of your paper. So I'll basically try to say it all in one today. And I don't know what will happen. Okay, so let me begin. So, well, see, I already screwed up. No, I don't know how to lift it. This needs to go. Yes. Yes. Yes. Yes. So, yes, all of this is joint work with Dima Rinkin. So for us, the motivation came from geometric planks. So let me begin from there. So X is a curve, smooth, complete over a ground field K. So K will be algebraically closed, characteristic zero. This is because mainly, just because I'm a coward, I'm afraid of characteristic P. And G is a reductive group. So geometric leg lengths begin as follows. You consider bound G. This is the moduli space of principle G bundles on a curve. And you consider the category of D modules on that. Derived category. Well, when you talk about geometric length, 90% of the time you talk about this. But in this course, we won't. So it appears a motivation. So I won't spend much time talking about D modules and what this derived category means. So on the other side, you have the stack of local systems on your curve with respect to G check. And this will be of interest for us. And I'll actually spend some time on it. Define it more precisely in my next lecture. And so we consider the derived category of quasi-coherent sheaves on that. And the naive form of geometric leg lengths says that there must be an equivalence. And so this naive form takes place as is when G is commutative, i.e. when it's a torus. So it's given by the Fourier-Moucaille-Lomond transform. And it fails when G is non-commutative. And this is the failure is exactly, well, our interest in this course. So when you state geometric length, of course, you don't just throw it like this. You want to specify what properties this equivalence has. As usual, well, we're doing length lengths, the most interesting part is the cuspital part here and the irreducible part here. But again, for us, this doesn't, it happens not to be the of central interest for these, for these lectures, because, well, as you know from the theory of potomorphy functions, cuspital functions present no, well, they're mysterious, but they present no analytical difficulties. They have rapid decay and analytically they're fine. And as I said, we'll be dealing with functional analysis. So it's the Eisenstein series that cause you analytical troubles. And that's our interest. So again, we'll be doing functional analysis in algebraic geometry. So I will now state the property that this equivalence is supposed to have with respect to Eisenstein series. And from there, we'll see where the trouble is. So here is this property, it's compatibility of languines with Eisenstein. So let P be a parabolic. So it sits inside G and it projects onto its levy. And we have the corresponding parabolic inside the languines dual. And so we have the corresponding diagram of moduli spaces. So in here, so let me call this map P and Q. And this I call P spectral because we call this spectral side and Q spectral. Well, corresponding to these diagrams, we define functors, let's say going in one direction from the levy to G. So here it's the Eisenstein functor. So we pull back the means of this map Q and push forward the means of P. And here we, I call it spectral Eisenstein, we pull back, well, do the same thing. So if you all were gubbers, you would ask why this is defined. Because we're dealing with non-holomic demodules, so this function is not defined in general. It's a left adjoint. It might or might not be defined. It happens to be defined in the central image here of this. So again, I'm just saying this for honesty. Not that it's... Exactly. I'm applying it to non-holomic demodules, but once, so not all guys here, but ones that come back by means of this, come as pullbacks by means of this map. And you can show that to these guys the left adjoint is defined on them. This is the left adjoint of... Pia Pashrik. Pia Pashrik is always defined on all demodules, but P. Lowashrik may not be defined. And no, your notation is a Pia Pall star. We had the elections of Kashiwara. My notations are standard. It's like I'm using the most standard notation. Pia Pashrik is the function. It's the pullback for demodules that's always defined. And Q upper star is the... I don't... Okay, Q upper star. Okay, you're right. So Q upper star, you may ask why is it defined. It's this... In the demodule world, it's not always defined, but this morphism Q is smooth. So it is defined. Another question because if you take a levy, if you lift the levy inside G. There is another functionality coming from the fact that M check is contained in G check. Correct. And so you have a long-range functionality which is well defined. So there is a... Yes, there is a map. There's another puncture here and you can also describe what happens here. But long-range functionality for Pitch check, what does it mean exactly? Because classically I don't know. No, no. But I'm not doing for Pitch check. I'm going from here to here. I'm not even trying to go from Pitch check. Yes, but on the other side you have the lock system. On the right hand side. Yeah, so I go from here, pull back, push forward. The point is when you do the automorphic induction, you have a representation which is usually not irreducible. Correct. And you have a quotient and a sub-object. And so you have to choose which one you... Your presentation? I mean, between the Steinberg and the Trivole for GL2, you have to decide which one you want. Correct. In terms of functionality. So here you are thinking in terms of taking the whole induced representation. Well, so what you're referring to is the local theory and here's the... Locale or globale. You have also globale. When you look at the discrete spectrum for example, for GLN, you have the trivial representation. Yes, but you see, correct, but I'm not trying to decompose automorphic representations into sub-representations. So it's not even representation. Think about functions. I start with a kind of function on this induced space and I consider that function. So I'm not even interested here at the questions of breaking it up into... So considering the G-check, I know that the dual for root data, but I don't know how to do it for groups in terms of... If I choose a splitting, I mean, what is the... What's P-check? No, G-check. Yes. It's G-check. So I know if I choose a maximum... So I know there is a morphism, but it's not the P-check. Here I'm in trouble. Choose. Choose the pinning. Okay, so I choose the pinning of everything and then everything is defined with split. Yes, let's do that for safety. Otherwise... Yeah. Okay, so we're more than one governing the audience, which is great. So we'll talk about it a lot. So if the morphism... So there is this theory of Hapraschik for quasi-kirin sheaf, this kind of extraordinary direct image. It's well-behaved for morphisms of finite or dimensions, such as this one. Yeah, it's Apashrik that... Apashrik from the point produces usually dualizing. It's this kind of Apashrik. And again, we'll return to this point, but I... So far, I'm really glad people are paying attention. They're not just taking symbols symbolically. And P-check is... Are you working on kirin or quasi-kirin? That's exactly my point. So far, the conjecture is falsely stated. It's the quasi-kirin category. Okay, P-spec is not proper. It is proper. P-spec is proper. This guy is not proper, but this guy is proper. So this actually preserves coherence? It preserves coherence, but you're jumping ahead. We'll get there in five minutes. Okay. It preserves coherence, but it doesn't preserve something else, which is crucial for us. And bounded coherence? Yeah, it preserves bounded coherence, but that's not enough. For what I'll say soon. Give me five minutes. Okay. So the promised compatibility is the following. So let's write the same diagram for the levy. So here we have this Eisenstein functor. And here we have this spectral Eisenstein functor. And we want this diagram to commute. And this is the compatibility of Langlands between the group and its levy subgroups. So what I'll do now, I'll explain why this thing cannot hold. This is impossible. Claim I. And again. And this has to do with, as I said, divergence of spectral sequences. So let me throw you into the cold water, okay? So I'll start talking about the essence of the subject. So it'll, well, it's becoming technical. So first of all, what kind of categories are we working with? So as Offer asked, these are, well, when I write quasi-co, I mean the unbounded derived category. No finiteness conditions whatsoever. So definition, a triangulated category is said to be co-complete if it contains all direct sums. Okay, so you don't have any size bound. Exactly. I was expecting this question. So let me please ignore set theory. But it's enough to do countably generated things. Yeah, yeah, just a little. Let me not just, not even go there. Yes, kind of for all practical purposes, countable is the word. Okay, so now I give the following definition. So let's C be co-complete category and object C is compact if home from C commutes with the direct sums. So let me give you a familiar example. Let's first do it, not in the derived category. Let's do it in the abelian category. So let's take, instead of C for a moment, we'll take a ring A and we consider the category of A modules. So this will be my notation for the unbounded derived category. I don't like to write D, but so when I mean the abelian category, I'll put a little heart. It's Luri's notation I didn't invent it. So when you see a heart, it means the heart of the T structure. So I'm deviating from this context. So let's ask ourselves the following question. When does an object M, I mean a usual A module, well, is an object such that home from M to something, commute with direct sums? I'm now confused. A module half is the category of A modules or the derived category? So heart means the heart of the T structure. It's abelian category for a moment. So I'm deviating from this context. I want to ask the same question in the abelian category. So let me actually turn to Gabber. So what do you say? No, but you see there is a difference because for the architect with a module of M, find it presented. I know, but no, I'm asking a very concrete question. When does this commute with direct sums? Find it generated. Find it generated. So question, answer. If and only if M is finally generated. Good. Okay, but let me modify this question. Now I'll take indeed the unbounded derived category, but M will still be in the heart. But I now ask, so when is M compact? Is compact and indeed, if and only if it's a perfect complex. If it admits a finite resolution, finite projective resolution, whose terms are finally generated projective modules. So compact question. So, and then it becomes silly. Let me not even write in the heart. So when an object complex is compact, if and only if it's perfect. So IE by definition admits, well, equivalent to a finite complex of finally generated projective modules. Okay, so now let's specialize to the case when A is a finely generated K algebra. So we denote the S spec A. Pardon? Oh, commutative, yes. So when I write quasi-co of S, it's by definition means what I denoted a mod. So unbounded derived category. So from here we see the following. So let me call it lemma. An object is compact, if and only if it satisfies the following two conditions. So two conditions. One is that F belongs to what I denote by co. So co means finite complex with coherent cohomologies. Again, it's something that more classical would be denoted by db co. Again, I don't like to write d. So finite complex with coherent cohomologies and there is an extra condition. So that is that it has a finite torque dimension. So for every point of S. So let's denote by K sub S the size skyscraper. We form this derived fiber. It has finitely many cohomologies. Homopolis S is a part of dimension. It's usually, of course, here it follows. It's independent of the point. Yeah, but this is enough for these purposes. So we denote this compact object by perf. So perf of S sits inside co of S. And the inclusion is an equality if and only if S is smooth. We are over an algebraically closed field. Okay. All right. So now let me explain why this is trouble for this. So we have the following very basic lemma. Suppose you have a functor between co-complete categories. In fact, we have a pair of adjoint functors. So we'll have lots of functors in the upcoming lectures. I'll be using the topologist's notation. The one on top is the left adjoint, and one on the bottom is right adjoint. That's the notation they use, and I became convinced that it's actually a convenient notation. So let me also introduce a piece of terminology. A functor is called continuous if it commutes with arbitrary direct sums. If it takes direct sums to direct sums. Yes. I mean, when I talk about functors between triangle categories, I mean what are usually called exact functors, exact triangles to exact triangles. I'm not referring to any kind of t-structure. It takes direct sums to direct sums. So I want to just add a word of warning. It's that it's not guaranteed. Not every functor takes direct sums to direct sums. Namely, if you take some kind of infinite product, that would fail to take direct sums to direct sums. Okay, here comes the little lemma. Let me... So this is stuck. Yeah, so let me see. I'll use it once and then I'll... So lemma is the following. Is it acceptable to give exercises to the audience in these lectures? Let me do it. So a left adjoint is always continuous. So if your functor appears as left adjoint, it automatically takes direct sums to direct sums. So it's an exercise. And b, if g is continuous, then f takes compacts to compacts. Okay, so now I'll explain why this is trouble. So the claim is that this Eisenstein functor on the automorphic side is actually a left adjoint. So is the left adjoint to another functor that I call constant term and, well, you just go the other way in this diagram. So you pull back with the specter up a shriek and that's a functor that's always defined on z-modules and you push forward by means of the star with respect to q. So hence, i's takes compacts to compacts. But now let's look at what's happening on the dual side. Why is this place continuous? Yes, so these standard functors on the category of demodules there by construction continues. Yeah, they're essentially tensor products of something. And what is CT standing for? Constant term. I'm just mimicking the definition of the level of automorphic functions. So, well, I discussed compactness for affine schemes. We will return in much more detail in the case of algebraic stacks such as loxies. So believe me for now that things work in the same way so that compacts are perfects. So the claim is that the functor i's spec. So first of all, as Gader remarked, it does send coherence to coherence. So there is no trouble with that. The reason it preserves coherence is because the morphism p-spec is proper. So this is okay. But it does not send perf to perf. Those are stacks and we'll return. Pardon? There are Arten's stacks. So you have to use this load. Arten's stacks there. No, no, no. I'll get to this point in a moment. You use which kind of topology? You use the smooth... To define this? Yes. I'll return to that. So these categories can be defined using any kind of topology. You can use smooth coverings or flat coverings. You'll always get the same. We'll get there in the next lecture. So to answer Maxim's question, loxis is actually not in. It says quasi-compact. Because if you become very unstable, you won't admit any connection. And the reason for this is that this morphism fails to be a finite or dimension. So it's easy to write an example for p-1 that this morphism immediately produces for you objects. You start with structure sheaf. You get something which is not perfect. It finishes the story. So this cannot hold as stated. And so my goal over this lecture and the next, I'll correct the right-hand side to make the conjecture, well, at least not self-contradictory. The most point of view on coherent sheafs is perfect on something else. Like lonesome... Can you repeat that again? No, the most point of view on coherent sheafs is perfect on something else. That's where we're going. But you can also correct the author's size for a coherent one. One can? Yes. It's more difficult to work with because, see, we'll correct it by some kind of phantoms. And these phantoms on the coherent side are easier to work with than these phantoms on the demodule side. But you can. OK. So now comes the next piece of bad news is that you can't stay with an algebraic geometry. If you want to do any of this, you have to go to derived algebraic geometry. So, and there are two ways to learn derived algebraic geometry. So either you take Luri's books or dot-con-abriela and read them all, spend three years, pass an exam, and in the end you'll still not know it. What are the authors? Tonya Vitosi. Yeah, so the only way to learn it, I think, is just to believe in its existence and start using it. Are you doing it just for simplification and defining the structures, see for general stuff? See, if it was the day of judgment and you were God, you would ask me, what do you mean by derived algebraic geometry? I'll say, I don't know and I don't care. I just use it. Seriously. I'll tell you what I mean. No, no, I got the impression that it is a part of the algorithm I learned which goes much to a much older thing like in the U.S. series there are simplificial rings so one can do some things with structural replacements with simplifications. So I'm not sure exactly what pit because I didn't read those things that you mentioned exactly but I'm just asking whether it is in the context of simplificial... So because we are algebraic, well, because when we create 6.0 it's enough to do commutative CG, CDGAs. So because of this, it's enough to stay there. And other things just like simplification just on one side or on two sides? One side. So I will explain that in a moment. So let me say like this, well, I still make a lot of mistakes in derived algebraic geometry and in higher category theory, lots. 99% of these mistakes happen to be at the level of usual categories. I state a false lemma for higher categories and it happens to be false for ordinary categories. So somehow it's... somehow if you know the usual category theory you end up making correct statements about higher categories. That's my experience. Okay, so what do I mean by derived algebraic geometry? So in ordinary algebraic geometry you know if you have a commutative ring A you can attach to it spec A, some kind of topological space with a shift of functions. So what you have to know about derived algebraic geometry is the same thing. But A is now a... what do you call it? Cdga. Commutative dg algebra which lives in non-positive comological degrees. The latter is important. It starts from zero and goes to the left. So you can ask what is the topological space? So of spec A is exactly the same as the topological space where if it's zero I eat topcohomology. And it's exactly the... like scheme versus varieties. The nilpotents don't matter for your topological space and you really should think of this A as some enhanced nilpotents. Maybe you can... it's like in the army or this cdga. You cannot write it. Yes, I... Yes, it's... Absolutely, so... How do you know? Have you been... See, I have... No, I mean... No, no, no. How do you know it's in the army? But how do you know that? I met the main military service. You see, the reason I invaded mine is because I actually... I just saw these abbreviations. I said to myself I can't handle it. Commutative... abbreviations. Differential graded algebra. Maybe you say non-positively graded. You said a lot of things. Non-positively graded. So it sits in degrees zero and below. Ah, zero and below. Yes. So... So the topological space is the same as of its topcohomology and you really should think of this A as some version of nilpotence. So you would... So we are used to having algebras with nilpotence. These elements of this algebra do not... Well, they give rise to functions, but these functions may be non-zero element of the algebra, but the function may be zero if it's nilpotent element. So the same thing you should think of this A. It's kind of a richer fluff of nilpotence. Okay, so here are... So in order to do algebraic geometry, you really need one thing. You need to be able to localize. So in ordinary algebraic geometry if you have an element f in A reduced, so killed in nilpotence, in this case you will be able to localize f... localize A with respect to f. So you localize not really with respect to elements in A itself, but actually with respect to elements in the corresponding reduced ring. And the same happens here. If f is an element in H0 and even take reduced of that, you will be able to produce a new algebra, a new CDGA. And so we have the risky localization and out of that you build your algebraic geometry. But instead of morphisms of ringed spaces, you will have to introduce some quasi-asomorphisms. Yes, so I'll say I'll say something about this. So say you have two derived schemes. So derived schemes form a category. They do. So you have S1 and S2, you can form the set. And it's fine, however, this may lead you to a dangerous place. So let me say what else you have and why you want to consider that something else. What you really have is the homotopy type of maps S1 to S2 such that this home set is Pi0 of this homotopy type. So these derived schemes, they form what's called the higher category. In which sense of higher category? Exactly. When you want to actually do it, do you have a regular way to Yes, I mean so what I do, I have say I have Luri's book on my desk, I put my left hand on the book and I type with my right hand. And that's the best I can do. Or just believe in the existence of higher categories. There is a way of using certain kind of simplicials. Yeah, that's what Luri does. Other people use other approaches on the internet, you can find all kind of references. Yeah, exactly. Exactly, so I kind of in honest answer I don't know, I don't care. So I believe that there exists one I believe that there exists one just one theory of infinity categories. But there are people who have pattern comparing certain things. Yes, let them do it. So if you want, okay, so let's say I use Luri's model for me, I use that. But honestly, I don't even use it. All right, so let me say a word why you want to consider this space instead of this set. So here is a typical way and so maybe if you've never seen higher categories, maybe this is the point of entry. So what I explain now gives you motivation why you want to think about higher categories. So let's do fiber products. So say I have let's say even a fine derived schemes and I want to form this. So let's say this is spec A1, this is spec A2, this is spec A3. So on the one hand I know what I want to get. This is going to be spec of A1 derived sensor product A2, A3. But I mean well, Gorothendrick taught us it's not good to just give definitions by explicit formulas. When you define an object you want to say what funcrate represents. So in ordinary algebraic geometry we know that HOM from T to map to the fiber product is to map like this. So this is what happens in ordinary algebraic geometry now, this is false in derived algebraic geometry. What is true and this is kind of when I said like it's easy to get yourself used to thinking in higher categories never write this, write this. And this is true it's fiber product of homotopy types well now you can take pi not of both sides by pi not of homotopy types does not commute with this. Of course you need that not just homotopy type but actually well, you need something Yeah, so I mean it's fiber product homotopy type, it's a fiber product in the category of spaces homotopy fiber product so And derived tensor product is by using a flat replacement of one factor of something like this Yeah, and again so I know how to compute it, I don't have the list of the tensor product if you consider it as a CDGA it's unique up to some equivalence or something. Exactly, so I mean contractable space of choices, I mean this higher category theory produces such things so and if you're a practitioner you won't ever bother about I mean when you compute derived tensor product you will not really find the flat resolution you'll all you care about how this thing looks like and exactly know how to compute it. So this is the setup for derived algebraic geometry so let me let me give you one example of what it buys you What you imagine the whole kind of compatibility is in doing this where you have to check that some guy from you to the other Yes, and even in usual in derived geometry we have spectral sequence assassins problem and when you're doing this without knowing the exact definitions how can and that's what I said it turns out that one is very unlikely to make mistakes in that if you just buy yourself into the ideology of infinity categories kind of knowing how this theory is set up is really unhelpful Can you do all the science science? Because there's a category of super rubbish places All right, so let me give an example so for the future let me say that so s spec a is well here's a kind of derived algebraic geometry word almost of finite type if really it's your it's your definition right? almost so H0 over K is a finite type in the usual sense and all the other HIs are finely generated as modules over H0 so this is kind of the derived algebraic geometry analog of being finite type that's where it thinks finite yeah so without almost it means that actually it's actually finite but okay so in this case you can define the tangent complex so if it's not finite type you have to talk about cotangent complex by the way quasi-co has the same meaning it's a modules so well illusie of course has a definition but let me give this definition like this in the world of derived algebraic geometry so rather I'll define the tangent fibers so let me take a point of s oops what do you mean by a point the point of the actual scheme yeah yeah it's a it's a map from spec K to this thing i.e. it's a map from A to K when you map A to K you automatically kill all the lower columnologies a point is a point on the underlying classical let me say so for s you can attach classical namely if s is spec A this is spec H0 of A classical is just completely completely analogous to passing to the reduced but you're not killing all the new potents I'm killing all the lower columnologies as a point of a scheme is the same as the point of the underlying variety the same thing the point of a derived scheme is the point of underlying classical scheme but you say that the topological space is you define it as a topological space first and you did not say that there is another space there cannot be something else no but it's even written on the topological yes it was unambiguous from the beginning the rest is not even a set right but I'm saying that if you think of points as maps from spec K to A you recover the same thing namely just you recover the close points K points so I want to define the tangent space well the tangents derive tangent space so it will be a complex so by the way my notation for the category of chain complexes of vector spaces is Vect if I write a heart I mean I mean just vector spaces as usual when I write less or equal to 0 or greater than or equal to 0 I mean complexes situated lower degrees degrees below 0 and above 0 so the tangent is a complex that is above 0 so H0 of that is the tangent space of the underlying classical as we know it who knows how to define the tangent space of a scheme at a point dual of that but I won't really yes so it's home from spec epsilon K to us such that well the close point is sent to your point does anybody have an idea how to if you send a classical to a derived scheme special or not so in that direction home from a classical to derived is the same thing as a form home from classical to the underlying classical the same thing as map from a variety to a scheme is the map to the underlying variety same no no no so that's home from classical to derived is discrete in that direction yeah so yeah home from classical derived this map space is actually discrete so who has an idea of what that is supposed to be go for it exactly but now change the meaning of epsilon so it will be now a CDGA where the degree of epsilon is negative I and that recovers so like you lose these tangent vectors have a geometric meaning it's also maps from a version of dual numbers but alter the degree and now it is by zero home I wrote home which is which is by zero of maps see and of course you can recover the entire thing so let me give another exercise to the audience recover not the individual co-homologists by the entire thing like give me a formula that gives just the cotangent fiber as is so if you really want to do the exercise I'll be very happy to talk to you all right so in almost finite type these two pieces of information for information are equivalent so I'm not losing any information I would not do this thing in the infinite type and it's in general it's better to do it it's better talk about the cotangent the reason I talk about the tangent is because it has this immediate geometric interpretation so I am proceeding in a pace kind of well significantly slower than I thought I would and I'm really glad that I am because I really like that people are active it's been going on for an hour shall we make a break do you think it's a good idea yeah so let's make a break and we'll 10 minutes something as I said we're interested in measuring the difference between perf and co will return to these in the derived setting in a moment I will not be able to really measure the difference in the general case but there will be a specific class of derived schemes for which I'll be able to do it these schemes are called quasi smooth so let me give the definition so I first give the following lemma so all my schemes from now on will be assumed locally almost of finite type so s is a smooth classical scheme classical I mean that only h0 exists, hi below 0 then a 0 if and only if it's tangent fibers only exist in degree 0 so this is something you know for classical schemes if you take a classical scheme you know it's smooth if and only if the tangent fibers are take illusives, cotangent complex take fibers, take dualized and you will only see 0th homology but it's also true in the derived setting with the same proof so now comes the definition s is quasi smooth if these guys vanish for i greater than 2 so smooth difference from quasi smooth but in that you allow one extra homology so this is a very convenient definition as a definition let me reformulate it in maybe more comprehensible terms so first of all let me just say that this is equivalent to the following so cheekily equivalent to the following if you take the cotangent complex of s so then locally it can be written but is a look so free it can be represented by length to complex consisting of free guys locally now let me you are speaking now about your over a differential greater algebra so have you you didn't really discuss the kind of architecture but you know what it is there are 3 modules different senses just direct sum of several copies of a itself no shifts take several copies of your ring a put in degree negative 1 take several copies of degree a put in degree 0 have a map take the cone and that's what I want this to be what do you say free locally okay but now let me give you geometric reformulation s now is not classical no yes this is this is derived so here is lemma which is parallel to this lemma so s is quasi smooth if locally it can be written as a fiber product like this when I write a n the affine space I really mean a n it's a spectrum of the polynomial algebra classical one so a map between a ns as we said a map from classical to derived is the same as to underline classical this is really given by how many m polynomials and n variables and this is a point so like this side of this diagram is completely classical but you form it in the derived setting so you form derived map of rings so quasi smooth means that it's a derived so derived locally complete intersections and do you have to pass to the rescue operand with affine spaces so the rocker models are full of affine spaces I don't remember but in any case it doesn't matter so I think let me not even think of it so let me actually prove this thing in one direction it's because we'll use it a bunch of times suppose you have in general suppose you have a fiber product like this call this map f for y1 and y2 anything so let me write the tangent of s tangent shift is the dual of the content let me just say what it is so it's going to be the following well it is what it is what it's supposed to be so you're taking tangent shift of y1 restricted to s and the differential of f this is df maps to the tangent shift of y2 restricted to s and you take co-cone or topologist like to call it fiber so in this setting you'll see that the tangent shift of s is the tangent shift of am is just the free module of rank m so you'll see it'll be os power n mapping to os power n fiber of that so it's explicitly a complex of length 2 put in degrees 0 and 1 so this was co-tangent this is degrees negative 1 and 0 this is dualized so if you wish this is the proof in one direction and the proof in the opposite direction it's pretty much the same as this so who remembers the definition of locally-compete intersection in classical algebraic geometry no a scheme I give you a classical scheme when is it a locally-compete intersection so no find a type over k ah ok co-tangent complex that's true but differently okay look at it you know really it's rich close some scheme of something smooth defined by a regular sequence in one of the equivalence senses of a regular sequence equations is equal to the co-dimensional yes or you can say it like this let's note that classical classical lci is equivalent to the following it's a quasi-smooth plus classical so if you take a classical scheme regarded as a derived scheme so it just doesn't happen to have lower homologies it's lci if and only if it's quasi-smooth so that partly explains why lci's are so nice so basically kind of classical schemes that arise are either smooth or they obtained from smooth schemes by you taking the fiber of such some map so when you take the fiber of such some map so what's a fiber? it's a fiber product and when you're doing it in the classical setting as follows you're taking the derived setting and then you're throwing away the lower homologies so you're doing something horrible so lci's are those classical schemes that you didn't do anything horrible to them there's also the question it's bounded cotangent complex equivalent to lci is it true or not I'm forgetting it was Krillin's conjecture it's a kind of conjecture I'm forgetting the status of this is there something like that in oh no no no in derived algebraic geometry you can well you can have a cotangent complex of any length that phenomenon is exactly kind of passing through the classical you're doing something very non-trivial from the homological point of view at truncating so now we'll play another game which is we'll need it for our definitions but I think it's fun in any case I find it fun ah I'll play this game here so let's do this take the tangent fiber and shift it homologically to the right by one do you know well who of you knows that this has a canonical structure of the algebra yes why do you know that who chose you invented you told that already some wire I Krillin it's basically it's Krillin's reality that's Krillin's reality in fact derived geometry was already in Korea but we didn't we didn't say the word Lee yet yes so ok so let me correct so Krillin tells us this but let me give a different interpretation not a difference same but it's kind of it's it's very intuitive in the but what is the point with ts it can be anything take any scheme s take any point no no but it can be a vector space so far it's just a complex of vector spaces this kind of interpretation oh if I can realize it like in this way is that the question no no but what is specific to the fact that it is t of the time ground space yeah yeah class maybe so I'm saying that if you take this particular complex of vector spaces shifted by one unit it will have a canonical structure of the algebra but this is only for this specific thing well I mean if you ask me can I realize any complex of vector spaces in this form no no no what do you mean canonical structure of the algebra of the complex djla another army word differential graded algebra canonical object in the infinity category of differential graded algebras so I'll construct it you don't have to specify a point oh I specify yes I mean I'm doing it point wise so here is the construction you have your point consider the fiber product point times point over my scheme s in the derived sense but now well because it's a point it's actually a group object derived schemes and it has a name it's called inertia so now in the right algebraic geometry you have a group you can take it's le algebra and if you look at this calculation like of this what happens under fiber squares you'll see that the le algebra of inertia it's underlying well le algebra well it's a tangent space it's a complex so le algebra of this will be will be this so if you want to think of the le algebra structure it's really le algebra of le group namely this one the inertia again and I'm not saying anything different from quillen I'm just re-interpreting and what is quillen's originality quillen's original definition what was the result of whatever I know quillen it was almost an equivalence of categories between let's say commutative rings commutative algebras and le algebras contra variant something but let's ignore that for a second so this is my definition okay so here is another fun fact about this so consider remember we denote by k sub s the skyscraper let's consider r home so I usually don't like I don't write r for derived I'm just if I were to write home it will look a little weird what I mean is derived and the morphisms of my skyscraper so what structure does that have it's an associative differential graded algebra I'm sure you've seen this one in commutative algebra question why do I mention this after this but the arm is over what oh depending on what you do math from skyscraper skyscraper it's an associative dg algebra I started talking about inertia and then I skipped to this why like no it's not how I'm at a point okay so and now we are ready to define actually singular support no but behind there are some concrete statement on classical statement on the on the top right is there behind that oh there may be there is a classical statement no in a sense no it is a classical statement so you can take you can realize this by the differential graded algebra take its universal development in a very classical sense take its co-homologies it will be just a graded associative algebra and it will be the algebra of x so it's a true statement yes but it's also true in this fancy language of in some derived sense some equivalence you mean what do I mean by equal sign canonically equi- canonically isomorphic in my world of again I have the infinity category of associative dg algebras and it's a canonically isomorphism in that in that setting so it means that you have models of both sides some kind of zigzag diagram between them or something contractable choices for example the algebra means a bracket we should see a bracket somewhere you mean the leib bracket here yes I mean so but I mean in a more naive way I mean I should be at least detect the homology group I don't know of course you can construct yeah so yeah so we'll use the statement for one of the definitions single of support it's good for theoretical definition so like it in my experience this is not a kind of object that you'll actually want to realize by explicit co-chains some of them some of the objects we'll encounter you want to write down explicitly like you want to write down really model for differential graded algebra for some of them you don't experimentally empirically this is not the kind of thing that you'll want to write down explicitly at the end you will so if you go with later on your singular support you will see a concrete statement on scheme and we will not be able to understand no no no you will I mean kind of I'll make sure the things are computable singular support yeah so this is good for a theoretical definition and for actual computations you do something else my next lecture I'll give a bunch of different definitions some of them are good for proving things and some of them are good for computing things so this one is good for defining things alright so but an example would be good I mean else is the affine so I really want to get into an example and I hope it's still realistic so I might not if I don't have time to do it now I'll start from an example next time alright I now want to define singular support so S is almost a finite type and so let me say what I mean by co-S so these are objects in quasi-co such that the individual cohomologies are finitely generated over H0 of S and bounded finitely many non-zero cohomologies for all but finitely many so let's start starting from this S I'll define for you a subset somewhere and that's what I'm going to do we take an object and we take a point and we consider that's acted on just this thing acts by endomorphism of the source therefore it acts in particular so the DGLA differential graded Lie algebra in particular let's just now let's pass to individual cohomologies so if it changes the graded graded Lie algebra acts on so now assume that S is quasi-smooth so then this differential graded algebra will have only two pieces it will have a piece in degree 1 that corresponds to H0 of the tangent space piece in degree 2 well, we have this shift that corresponds to H1 of this in particular in degree 2 it will have this abelian Lie algebra which is just H1 so if you consider H1 yet note this shift it placed in degree 2 it's abelian Lie algebra acts on this module which corresponds the symmetric algebra acts on this module symmetric algebra but the generator is placed in degree 2 and now you can take the support in the most naive sense just support in the sense of commutative algebra have a graded module over a graded polynomial ring so we defined singular support at S of F to be the support is still awake the support will be a subset of what? which? great conical because things are graded the support of this direction as a module over sim what kind of module is it? it happens to be that is the extra 0 for large no no no it's just a finely generated module over this algebra I'll state theorems to this effect next time so when F is clear in your sense of course it's not difficult theorem it requires proof but when you say place in degree 2 it it acts from X i to X i to X i plus 2 yeah yeah so it's a risky closed conical subset now we want to take all of those S's together where do they live? these guys use again this blackboard so when you first come to the first course of differential geometry they define the tangent space to you tangent bundle how do they define it? for manifold no they say the tangent bundle is the union over all points of a manifold of tangent vectors you don't like it, huh? okay sing of S is the union over points in S it's been brutal from this definition it's not very clear why this is a point of algebraic variety that's okay we'll write it's isomorphic to a relative spec it's a spec over S who is still awake? it is inside something it's an honest no but when you say union it's inside something no as a tangent bundle like you know the tangent bundle it's a union abstract union of points and tangent vectors that's what I mean it's like no it's not inside anything it's like yeah so this is a classical scheme no derived games here I didn't take H0 but for the moment it's just a second yeah so you take the tangent sheaf shifted conlogically by 1 it lived in degrees 0 and 1 you shift it take H0 of that and it's no longer vector bundle it's a coherent sheaf take its sim over OS and take the relative spec so now you are describing S classical S classical well it would be the same because I took H0 so you mean it has even non-radio structure for a scheme structure for our purposes we don't care because we will be only interested in support in that no but if you write that it may have it may have the importance as I wrote it's a scheme I could have replaced it by the reduced part for my purposes all I care about is that its points are defined in this way it's the same okay and okay I'll have to sacrifice some of this so finally so definition for F in coherent of S the singular support of F well it's the union for S in S of this point-wise singular supports so each of these fellows sits inside each of these fellows so therefore it makes sense to get upset in single S so proposition that I'll explain well fiber-wise it's the risky closed as I wrote it's not obvious it's the risky closed but you have to, well it's not difficult to prove is no that's not so we'll get there to the fourth lecture we cannot get, you absolutely cannot get anything on this scheme it will only have support along these guys we'll talk about this in detail so that is impossible so you really cannot produce anything that lives, see scheme theoretical here like with respect to the scheme you only have something satiretic but it has multiplicity so this is something I don't know so we only know single support we don't know anything about cycles this question hasn't been investigated by us, it might have been investigated by other people I just have zero knowledge and it might be interesting so I just don't know okay ignorance but I'm interested so I just don't know so to prove it's a risky closed you need to do something not just point-wise yeah I have to generalize this and we will do I just wanted to give a definition of single support which I gave we'll the next lecture will be devoted to five different definitions this will be one of them but you can imagine for the support the characteristic variety you have two definitions definition with the module and you have also definition grade up things and you have also definition with vanishing cycles the point is in the support in the characteristic variety is there something like that yes well not exactly vanishing cycles yeah so yeah we'll choose a function and then something will happen along this function yeah there is definition like this all right but so let me state so single support does indeed measure something so in one direction it's very easy from here to here but if it just happens that single support is zero your thing was perfect so single support does measure the degree of reflection of your object just in the same way if you look at this thing this variety measures to what extent your s was not smooth so f as I stated this lemma that that lemma you see if these guys are all zero then s was smooth so this measures how much s is far from being smooth and f measures how much f is far from being perfect okay so so this has a meaning even if s is classical and it's hard to understand so it has no it has a meaning if f it only makes sense for smooth sorry for quasi-smooth schemes so if s is if it's LCI that it makes perfect sense but otherwise for just classical the most basic example of LCI but so the most basic example is coming up and it's not yes but it's not classical that's the thing the most basic example I will do it I think well we are going to write equal to zero you mean empty or equal to the zero section no the zero section zero section then because the zero model I think have empty yeah it contains zero section alright so what do you say if it is f is zero the singular support what did you say if f is zero single support it's empty sorry yeah that's the way I should say it so let's check if f is zero if it is empty or not empty okay so okay now we are ready for an example I still want to keep that okay and the singular support is empty if f is zero yes empty is zero so here is the most basic example of a quasi-smooth scheme classical and it's the kind of scheme that's used all the time so namely take point times point over a vector space i.e. it's a spectrum spec of sim of v star okay so let's this is my s okay so before we talk about the thing what are the points of this thing how many points does it have use classical points there's only one notion of point k points how many no so what's the classical scheme underlying this one sims yeah but so I'm first asking what are the points of this guy one point okay now what's the sing of this what do you say v of v star okay great so now we'll analyze this example in detail so I claim that cove s has a very explicit description these are so this is causul duality and it's so causul duality usually breaks down because some finiteness conditions don't hold but they exactly break down because of the difference between co and perf so as written is exactly true and this function in one direction is given k sorry f goes to well r home from our unique point f I should have said that this what do you mean well this is a cdga I'm considering modules for this such that their cohomologies are finally generated over in the classical sense but now v minus 2 is placed in degree and this is not like the convention that you had originally for cdga correct so this is not a legit object of derived algebraic geometry and I'm not regarding it as such I'm not trying to take the spectrum of this okay so so now you are taking a finally generated module in the right to consider guys which are I'm taking the right category of these I'm taking modules if I take their cohomologies they become it's a graded module over the graded ring and I want that to be finally generated okay so this is the analog of d finally yes yeah so or it's the same as perfect module so by this so by the way inside here we have perf and somebody guess what that corresponds to yeah that's that supported zero so this is and let me write kind of zero support cohomologies are supported as at zero so this means that the cohomology actually is five dimensional okay yeah that's that are fairly many cohomologies yes and so well basically I'm just applying the definition but this gives you a way to think of the singular support so you start with an object here you transfer it here look at the cohomologies it's a graded module over sim v and you take its support so like this causul duality well in this particular case just exhibits you by definition well the graded is what is graded oh it's a it's a module over this differential graded algebra I take its individual cohomologies when you say the sim the module is graded module yes yes yes I mean I always that's my notation I call it a it's a differential graded algebra when I write a module I mean cohomologically graded so this is if you wish well in this case this causul duality gives you singular support explicitly that's what it does and it turns out that all the theorems you prove at the end of the day you reduce to this case so this is the most basic quasi smooth scheme this is the case when you take the point at the point of all smooth scheme a classical smooth scheme yes well that example discover it's the same because it's all local it's the same as a vector space so kind of if you want to have an example you want to have this example all right maybe I'll end now there's a concrete description of the laws of perfect perfect is just whatever can be constructed as finitely many operations on the free guy so that's a definition well I mean on the other side so these are the guys well I can tell you what they are they're not functional so the support at zero means it's artinian so that's what happens to perfect so under causul duality goes to artinian but then you can say and you take the support that's the support it measures how much non-artinian you are okay so I think I did maybe let me see I did two thirds of what I wanted to do but it's fine yeah not so bad