Spectral Sequences Live! 17: The Grothendieck spectral sequence

[00:00:00]Hey, look.

[00:00:02]The people want spectral sequences.

[00:00:05][clears throat] You got to give them spectral sequences.

[00:00:08]So, good morning mathematicians and spectral sequence addicts.

[00:00:17]Um, we're all here together in our addiction on this fine day in November.

[00:00:47]Um, sorry, I just paused because I was thinking about stuff.

[00:00:54]I was thinking I don't know what I was even trying to sound like there. [sighs] Um, oh, I forgot to like my own video before we started. tragic.

[00:01:08]Okay, where are we going? What are we doing?

[00:01:15]So, if you remember a while back, now that we have some experience, if you remember a while back, I was interested in some statements in this paper about representations of of piotic groups. [sighs and gasps] And there's a spectral sequence argument somewhere here.

[00:01:41]Um, [snorts] where is it?

[00:02:02]here.

[00:02:04]Proof of crawlery 2.

[00:02:08]Uh oh. So this is this had to come sooner.

[00:02:16]I guess I could just control F spectral sequence.

[00:02:19]Could I not?

[00:02:22]I did. I folded and went spectral. Well, part of it is cuz one of my one of my friends uh IRL who said he needed to uh um started watching my spectral sequences stuff cuz he needs to know some spectral sequence stuff and is also interested in this paper in particular.

[00:02:44]So I was like okay that on top of the other people. I mean I still think my normal office hours are more popular. My normal office hours get more views.

[00:02:54]Hello, Castaka Corey.

[00:02:56]My my normal office hours get more views. Uh but yeah, lot lots of people uh are asking for this. Good morning, Mom. Good morning. [sighs] You've got uh you've got yourself a crown today, says number three.

[00:03:17]But you're number one in my books.

[00:03:20]Okay. [snorts] Um, for every pair of smooth G modules, uh, well, for most spectral sequences would be, uh, for most spectral sequences would be inscrutable.

[00:03:38][snorts] We get for every pair of G modules.

[00:03:43]Um, hello hello Vic Vicron.

[00:03:51]uh right we get a spectral sequence like this. So we need to understand what kind of spectral sequence. So this proof is the same as in the classical case. So maybe we could look up castleman. This is a castleman reference. Uh it starts with the observation that the co-invariant functor is left adjint to the exact functor viewing a smooth GN module is a smooth G module.

[00:04:18]Therefore VN is a projective G mod N module uh if V is projective and then by vignos I believe >> [snorts] >> uh we know that the restriction functor preserves projectives using arguments applied to the growth and spectral sequence. we obtain the first part of the claim.

[00:04:39]Okay.

[00:04:41]So maybe in order to better understand this, we should first look up applications of the growth and spectral sequence which hey by the way isn't that uh isn't that the spectral sequence that one uses to prove um the composibility of the derived functors? Yes, it is. Those first two things can be exercises.

[00:05:16]Uh let's see here [snorts] that the co-invariant functor is left a joint uh to this. Yes, that's probably just that's just an exercise.

[00:05:30]Um and then yeah, projective if this is projective. Yeah.

[00:05:37]Um, I don't know if this thing I don't remember how easy that is to prove. My guess is that it's not hard.

[00:05:46]Um, oh wait, never mind. This is the one that uses growth and spectral sequences.

[00:05:54]So, okay, maybe we should go uh let's go look at Vineuras.

[00:05:59]Uh, let's go look at Venuras. Actually, can we click on it? No.

[00:06:11]V1. Oh, it's in French.

[00:06:16]Uh, okay.

[00:06:24]Let's go ask Let's go ask my good friend Claude to translate it for me.

[00:06:43]>> [snorts] >> Oh, those weird characters got copied there. I wonder if it would uh Oh, okay.

[00:06:50]I got to get it directly.

[00:06:55]Um, you're an undergraduate studying math. Are there any books or texts I would recommend to learn Lee algebbras?

[00:07:04]Um h sort of depends on your background and what you want to do with them.

[00:07:13]Um yeah, I mean Sarah's book Sarah's got this little book. Where is it? Do I have it easily accessible on my shelf? It's somewhere here. Sar's got this little book. It's called like complex semi-impoly algebbras. Um, it's short. It's sweet.

[00:07:36]Yeah, look at this thin little thing.

[00:07:39]You could take it with you anywhere. You know, you could read it when you're out for a walk. Um, that's probably a good place to start.

[00:07:49]Um, Fulton and Harris, of course, is a classic. There's a lot of other stuff in that book that's good. So, uh, you could go read Fulton Harris. Um, yeah.

[00:08:03]Then I'd maybe move on to Humphre. Oh, you know what else is good? Um, actually, uh, like Fulton Harris. Okay. Lee algebra are in the middle of the book, but um, uh, Lee algebra are in the middle of the book, but you can just skip to that section. Another good one is Hall actually lee groups lee algebbras and representations by hall. Again there's a lot of stuff before there and this is mostly just focused on the complex case but this is a good book for an undergrad um for a first pass at these things. So those are some things but maybe start maybe start with sir.

[00:08:52]Uh okay AI overview gave it to me in French. Uh vineas venuras where is your paper?

[00:09:13]Is this it?

[00:09:16]I think so.

[00:09:19][cough and clears throat] I need like an easily accessible version.

[00:09:29]Uh, I might have to go dig it up through my library. We'll we'll maybe look at this later. Oh, wait. Is did I see an English translation here?

[00:09:41]No. reducible modular representations of pi group and simple. Okay, whatever.

[00:09:46]Let's just go back to the paper.

[00:09:49]Uh, never followed up about the Lee theory stream. Then maybe soon you'll email. Sounds good. Hello, ciao.

[00:10:02]Uh, okay. Let's just go let's just go back to this paper. Let's just look at anyways. We need to understand the growth and de spectral sequence. I think in general that's the thing that is the thing. So let's just go back to that.

[00:10:27]What is this paper? Uh this paper by ORL you mean?

[00:10:34]uh on extensions of generalized Steinberg representations.

[00:10:52]Uh okay, maybe we start at the wiki page.

[00:10:57]Oh, wow. There's YouTube someone else's uh YouTube videos.

[00:11:04]about the growth and spectral sequence.

[00:11:06]Maybe I'll just have to go watch those.

[00:11:08]Uh yeah, it's probably what I should be doing.

[00:11:17]Okay, let's look at this one. And there we go. We got a nice paper there. Okay, great. Now we've got some material set up.

[00:11:27]Uh oh, this was in the Tahaku paper.

[00:11:30]Okay.

[00:11:32]Um, copy strike them. No competition.

[00:11:40][snorts] I think they could copy strike me.

[00:11:44]They're uh in in before K theory got to it.

[00:11:56]Five term exact sequence.

[00:12:01]uh little array spectral sequence local global x sequence. Okay, this looks most similar to what we're to our applications.

[00:12:19][sighs] Um okay, there's a derivation here. Okay, maybe we can read through some of this derivation then.

[00:12:33]Um, it doesn't exactly give a general statement, does it? Oh, I guess I guess this is the general statement. I guess the general statement is that if you define your E2 page to be the composition of these right derived functors, then it converges to the right drive functor. Okay.

[00:13:01]Okay. So let's see if K is an injective complex in an ailion category such that the kernels of the differentials um [sighs and gasps] are injective objects.

[00:13:16]Uh so the complex is injective but also the kernels are injective objects.

[00:13:22]Then for each n the coology is an injective object and for any um are we wait are we disproving the SLMA? No, we're using the lema.

[00:13:38]Uh it's so difficult.

[00:13:42]What is so difficult? spectral sequences kind of um then the co-homology will be an injective object and for any left exact additive functor we should maybe go back to viable actually maybe this maybe maybe we've learned enough to go back to viable uh is an injective object and for any left exact additive functor G on C Uh we have this right. Sure.

[00:14:22]Sure.

[00:14:25]Yeah. This looks like the proof of the lema though actually.

[00:14:35]Uh we now construct a spectral sequence.

[00:14:38]uh let a be an injective resolution of a writing phi for the induced maps here.

[00:14:49]We then have these short exact sequences, [sighs] right?

[00:14:58]Then we can take injective resolutions of the first and third nonzero terms.

[00:15:09]Um right wait we can take the we can take injective resolutions of the first term.

[00:15:21]Oh we can take injective resolutions of anything I guess. uh horseshoe lema homological algebra. The horseshoe lema also called the simultaneous resolution theorem is a starting is a statement relating resolutions of two objects.

[00:15:45]uh uh says that if an object a is an extension of a prime uh by a double-p prime then a resolution of a can be built up inductively. I see. I see. Right. Right.

[00:16:06]Right. Right.

[00:16:18]Okie dokie.

[00:16:20]Uh so then we can take this direct sum.

[00:16:23]We get a resolution of the thing in the middle. Hence we found uh an injective resolution of the complex.

[00:16:33]>> [sighs] >> such that each row satisfies the hypothesis of the lema. Did we uh injective complex such that the kernels of the differentials are injective objects?

[00:16:57]Uh uh what the kernels here?

[00:17:10]Why why are the kernels injective objects of which complex? Now I'm confused.

[00:17:27]Can I help you with probability and stat statistics? Unlikely.

[00:17:32]You can ask your question anyways, but it is unlikely I'll be able to help you.

[00:17:48]Um, take injective resolutions.

[00:17:55]Is F left exact here? Good question.

[00:18:00]Um, yeah, I guess so because we're taking a right derived funer. So, um, I'm going to I guess we have left exactness.

[00:18:17]Is that enough to preserve the injectivity?

[00:18:21]I Um, since we found injective resolution such that each row satisfies the lema, why didn't I enter end yesterday's stream with a QED? I did.

[00:18:46]I end every stream with a QED. Did it cut off before?

[00:18:54]Absolutely I did. I never I never don't I never don't.

[00:19:00][snorts] Also in one one month you got to start studying for your written math exam in probability and statistics. Any good advice for studying for a written exam when you've never done it before in uni?

[00:19:16]Uh I don't I mean presumably you've studied for a written exam in high school and I think the basic idea is the same. Uh yeah I I don't know if I have any special advice for your first written uni exam. just uh study lots and have lots of ex you know all your definitions exactly as they're written in the textbook have them memorized uh have many examples with proof on hand.

[00:19:59]Yeah, I think I think Boris they just meant the the exams say they have done this is their first university exam not first written exam.

[00:20:12]So why do we construct this injective resolution in this way?

[00:20:18]Well because we want uh because we want to we want to use this lema here, right?

[00:20:23]We're going to use this lema here.

[00:20:26]So let's just move on for a second.

[00:20:30]Uh so each row allegedly allegedly the kernels of each of these things should be injective. Oh but I guess no I guess maybe wait maybe that is true by the construction.

[00:20:48]Maybe that is true by the construction.

[00:20:52]Uh so we take E 0 to be this double complex and this gives rise to two spectral sequences.

[00:21:01]Oh we have a horizontal and a vertical sequence which we're now going to examine. Uh on the one hand by definition [clears throat] EP prime one is this guy which is always zero unless Q is zero.

[00:21:25]Oh, since FAP is ga cyclic by hypothesis.

[00:21:33]Um, what FAP is GAYlic?

[00:21:42]When did we say that?

[00:21:45]By hypothesis. Oh, wait. Is that Oh, between two a billion categories with enough injectives. Oh, and we're assuming that F takes injective objects to gylic objects. I see. So, that is quite literally uh that is quite literally part of the hypothesis.

[00:22:10]You mean Kurfi and Mfi are both kernels?

[00:22:16]Uh, you're right. I should draw these resolutions, but I'm not going to right now. I'm not going to right now.

[00:22:27][sighs and gasps] Um, hence uh which is always zero unless Q is zero.

[00:22:37]Uh, right. Okay.

[00:22:42]So hence the E2 page is this.

[00:22:48]Uh maybe I should draw something.

[00:22:57][clears throat] Uh maybe I should draw something. So uh wait a second. So our E1 page was uh so we had E doubleprime 1 PQ is H Q G IP dot uh R Q GF AP And right. So the E2 page.

[00:23:45]Uh, wait. Why is the E2 page that?

[00:23:58]Why is the E2 page that? Wait, E2N. What do What do you mean by N? Is N the total degree? E2N.

[00:24:10]Uh yeah. What is E2N? Is that the sum of E2 PQ with P plus Q uh being N?

[00:24:21][snorts] Anyone here interested in theoretical computer science?

[00:24:28]Theoretically, not not particularly.

[00:24:41]Well, I [snorts] mean, I don't know.

[00:24:44]Questions about Oh, yes. You know, we were we were yapping yesterday about uh Busy Beaver and stuff. I'm not not interested in these things, but those are the things I'm more interested in, I would I would say.

[00:24:59]Uh, but you know, I've dabbled in a little type theory and maybe we'll get back into proof assistance. [sighs] Um, right, E2N is supposed to be this, but Why? Because like E2 PQ should be the kernel of E1 PQ E1 what? P + one Q modulo the image of E1 PQ uh sorry P minus one Q right uh yeah I just Don't see why this should have this should we should get the right drive functor of the composition here.

[00:26:37]I mean if most of these are zero, if these are mostly zero then what?

[00:26:50]Uh these are zero unless Q is zero.

[00:26:59]So what Uh, I mean, I guess we try plugging it in when Q is zero.

[00:27:22]And I'm mostly Z. How dare you?

[00:27:27][sighs] How dare you come on my live stream and and say these things.

[00:27:37]Um, okay. So, here's where it's not zero.

[00:27:42]Uh, and so what?

[00:27:53]Oh, I see. Because then uh right we have this here right this is GF this is just GF that feeling when have a GF am I right um uh okay but now actually here's a question why why were these two why were these two things equal I just accepted that uh without questioning it. Uh on the one hand on the one hand uh the cuth cohomology of this complex should be the right derived functor of this composition. And now why was that?

[00:28:52]Why was that exactly?

[00:28:58]Uh, also what's up with these primes and such?

[00:29:02]Yeah, I don't know.

[00:29:06]Could have used V and H subV preub presubh.

[00:29:14]Uh, now the double complex. Um, I see.

[00:29:21]I guess we're using that lema.

[00:29:26]We're using that lema, right?

[00:29:29]Uh cuz this will be G of H, N of K.

[00:29:46]Right.

[00:29:49]Yes. So I think yeah this is by the lema then okay wait no that's up here this is this says just by definition okay whatever I believe this part of it on the other hand by the definition of the lema okay this is the part that comes from the lema.

[00:30:22]Uh but now we're going up vertically the other way. Okay. And since this is an injective resolution of this uh E2 here, E2 prime uh is this.

[00:30:46][sighs] I guess that's it.

[00:30:55]You think we need a third dimension of resolution so that we have cubes to crank?

[00:31:02][snorts] [clears throat] Why not infinite dimensional? Why not a bon space of spectral sequences?

[00:31:15]Let's, you know, let's get crazy with it.

[00:31:19]Let's get real crazy with it. Okay, I I think I get this. I might come back and uh go over this proof again in detail.

[00:31:28]Uh follow follow this wiki proof and write things out.

[00:31:35][snorts] Uh but let's look at some other examples in nature. Let's see if this document has anything for me.

[00:31:44]Uh, let's look at this. This is This is more, you know, 10 pages. Maybe more of a leisurely stroll. Maybe we'll get more out of this here.

[00:31:59]Um, [snorts] uh, didn't find the English translation.

[00:32:08]I think I have the English translation.

[00:32:16]Tahoku English. Here we go.

[00:32:23]Yeah.

[00:32:24]Oh, 120 pages.

[00:32:27]Yikes.

[00:32:30]Um, [snorts] let's see the exact sequence. array spectral sequence page 54.

[00:32:45]I guess that's what we want, right?

[00:32:51][snorts] I remember many years ago I decided I was going to learn French math uh and I was going to try and translate this paper and I was going back and forth between the English and French.

[00:33:10]didn't get very far. I was a young man then and I'm an old man now.

[00:33:16]Uh flurry spectral sequence of a continuous function.

[00:33:23]I see.

[00:33:29]Yeah, maybe there was other things.

[00:33:35]Yeah, maybe maybe this was covered before. It follows from the crawler proposition 3.3.1 the E of conditions of lema da da da and we can apply theorem 2.1.4 maybe that's what we want 2.1.4.

[00:34:01]Here we go. Nice. Okay.

[00:34:11]Uhhuh.

[00:34:13]Let's see. Should I read this or should I read the other paper?

[00:34:17]I mean, this paper is probably getting even more Let's Let's take a look at Growth and De's original words first.

[00:34:29]Um and by original of course I mean translated we consider the coariant functors do same hypothesis as before and we have this the second assumption about the pair means that the functors are ephacable. Is that how you pronounce that?

[00:35:00]A fasciable.

[00:35:01]I don't know.

[00:35:06]This is how we'll usually verify this hypothesis.

[00:35:10]We immediately verify that in order to calculate the second spectral sequence of a composite functor, the one in question in 2.4.1 4.1 is sufficient to take a resolution da da.

[00:35:26]Um, okay. Yeah, this isn't much of a proof.

[00:35:31][sighs] Uh, you once read a mathematical economics paper that said infinite dimensional ban space. You're pretty sure just to scare economists.

[00:35:42]Um, good. We must strike fear into their hearts. It had no reason to mention this thing. [laughter] The space it was working with was a finite dimensional manifold. Interesting.

[00:35:54]Uh, however, the model it was using made no sense and simple changes to the model to make it more sensible made [clears throat] the paradox it was describing completely disappear.

[00:36:08]Sometimes you just got to scare the economists.

[00:36:14]I see. So there isn't really a proof here. He just says do it and he points to somewhere else which fair. Which like to be honest like that's totally fair.

[00:36:30]Um okay. What do we got here? This is just basic basic baby stuff over here that I'm obviously confused about.

[00:36:42]Let's look at how they deal with it here.

[00:36:45]Now we are ready to construct the growth and de spectral sequence. So we have two uh left exact functors. If gm maps injective objects to fyclic objects then for any object a and c there is a spectral sequence starting on the e2 page given by this.

[00:37:09]This has stood as economics paradox for 40 years now and you've only found one reference talking about a resolution but there is nothing to resolve the model makes no sense.

[00:37:22]Well, is it possible that uh you're misunderstanding some part of it, Leo?

[00:37:29]It's like what? Every economist is an idiot and just didn't realize the model makes no sense for 40 years. What is this paradox that you're talking about?

[00:37:41][cough and clears throat] Now I am intrigued.

[00:37:46]You have piqu my interest. H shall we say h we could say many things but perhaps we could perhaps we should just say no there's no possibility of that. Okay.

[00:38:01]Well worth asking nonetheless.

[00:38:06]worth worth asking nonetheless.

[00:38:09]Uh a bit cursed here using G. Oh, what?

[00:38:14]Uh G for the first functor. Yes, that's true. That is true, Nicholas. Little bit cursed.

[00:38:23]Uh you're so cryptic, Leo. It is It is not a famous one, but there is a popular YouTube video about it. Okay.

[00:38:34]Uh, okay.

[00:38:41]I guess I'll just work from that stringent piece of information alone.

[00:38:47][snorts] Yeah. Okay. So, let's take our cursed setup here.

[00:39:01]You forget the name. Oh, okay. What's the name of the YouTube video?

[00:39:07]What's the name of the YouTube video about it? Surely you could find the YouTube video and tell us the name [snorts] of Uh, okay. For a general complex with an increasing filtration.

[00:39:48]Uh-huh.

[00:39:50]Okay.

[00:39:53]Uh we'll take the graded piece of our complex whatever it's increasing.

[00:40:10]Okay.

[00:40:12]Then we have a spectral sequence of a filtration here.

[00:40:19]Uh, will just tell us the physics with the birds on voting. Okay.

[00:40:28]Physics with the birds on voting.

[00:40:32]I'll watch it afterwards.

[00:40:45]Oh, I think I've seen this thumbnail before.

[00:40:49]Okie dokie. We'll see. We shall see.

[00:40:53]Leo, we shall see. Are you smarter than every economist ever born? Ever.

[00:41:00]That's what it comes down to. The final fight. Actually, I want to see you fight every economist at once.

[00:41:08]Uh, physically.

[00:41:11]That's the only way I'll be convinced.

[00:41:14][sighs] Uh, then we have a spectral sequence. E0 PQ.

[00:41:21]Uh well, we define our E0 page to be the P grading of P plus Q and this allegedly converges to this.

[00:41:34]Actually, I think that's true in general, right? I believe that it converges to that. Sure, that I this much I can believe. [sighs] That's general spectral sequence nonsense. Uh they mix a model for utility of goods, mix it with preference for voting, artificially add a whole plus negative preferences.

[00:42:01]I don't I mean I don't understand any of what you're saying.

[00:42:06]And you need all this to get the result.

[00:42:10]Okay.

[00:42:13]I don't know what adding a hole has to do with voting, but whatever. Okay, you can fill in the zero and it goes away.

[00:42:24]You can remove the marginal utilities and the paradox goes away.

[00:42:33]Okay. I mean, I'll have to watch the video. I don't understand anything you're talking about right now.

[00:42:46]Maybe you can publish your paper in an economics journal and get get your get your Nobel Prize in economics for resolving the get your Nobel Prize in economics for resolving this problem like uh John Nash and they'll they'll make a movie about you and an okay looking mind [laughter] burn.

[00:43:17]I'm just kidding. I'm just kidding.

[00:43:20][snorts] Uh the growth and dee special the growth and de spectral sequence is just a special case of this arising as a spectral sequence of a double complex.

[00:43:34]Uh the cartan islandberg resolution. The double complex we'll use in question is called uh ah so that's what a carten island break resolution is.

[00:43:45]Um a lot of people argued about the math in the comments but all the math is right but the fundamental but fundamentally the model models nothing.

[00:43:58]Okay. Well I mean yeah I just I I can't actually uh comment one way or the other until I watch this video.

[00:44:08]Uh, let A be a chain complex in an Aelian category.

[00:44:15]Cartan Islandberg resolution.

[00:44:18]Okay, so we've got a chain complex.

[00:44:24]Uh, Carten Islandberg is going to be a b-graded object uh consisting of uh injective objects [sighs] with a horizontal and vertical.

[00:44:42]And we're going to suppose that right horizontal vertical have by degrees and they commute. [cough and clears throat] Okay. Also the columns. So this is so we're going to say the IP dot delv uh is an injective resolution of AP.

[00:45:25]Um, and the induced complex is wait B is the B is the images. Yeah, what's B here? I got to go check their notation. You've actually thought about doing a project paper on how to artificially make social science paradoxes using topological ex uh obstructions.

[00:45:57]What are social science paradoxes?

[00:46:01]Although the n equals 2 case is covered in the video and that problem was done in your first topology class which is also your first ever uni math course.

[00:46:12]Okay.

[00:46:16]What is B?

[00:46:18]What is B?

[00:46:25]What? I don't see any B's here. What is this notation?

[00:46:32]Like certainly this is supposed to be like boundaries or something, right? be for boundaries or co-boundaries or something, but I don't see it.

[00:46:42]I don't see it anywhere.

[00:46:45]I don't see it anywhere.

[00:46:49]Yeah. What the hell is B? Yeah. Okay.

[00:46:52]So, sure. Sure. Sure. Sure. Okay. Yeah.

[00:46:54]B. Okay. [sighs] Certainly B is Yeah. Okay. B is the the images.

[00:47:02]Okay.

[00:47:05]Certainly B is the images.

[00:47:07]And what else are we claiming? The induced complexes.

[00:47:12][cough and clears throat] The induced complexes.

[00:47:17][snorts] H I um from the horizontal are injective resolutions of BAP and HAP.

[00:47:38]Okay, [sighs] let's see.

[00:47:45]Okay, we're just stating this. We're not going to prove it.

[00:47:53]Uh definition. Oh, right. We're sorry.

[00:47:57]This is the definition of a carton island break, right?

[00:48:01]Okay. We're just assuming that all of these things uh this is a definition.

[00:48:07]Okay. If C has enough injectives, a Cartan Islandberg resolution always exists.

[00:48:13]Um, right. We have two exact sequences, right? We're just using the horseshoe lema here. Again, two exact sequences for every P. Using the horseshoe lema, we can pick injective resolutions for each of these objects and add them together together to get an injective resolution for ZP.

[00:48:40]Uh okay.

[00:48:50]Uh then we can do the same thing for the second exact sequence uh to get an injective resolution for AP and the map between injective resolutions is induced by this composition.

[00:49:08]Why would we prove anything? H this is a good question actually.

[00:49:16]Uh I have no other answer [sighs] than uh a neurotic compulsion, a nervous tick, if you will.

[00:49:26]Proofs are just nervous ticks.

[00:49:31]Uh Leo, it's just it's just you and me in this chat right now. How intimate.

[00:49:39]See, this is the thing. No one every everyone asks for for more spectral sequence videos, but but but who's who's here uh yelling random things in chat? It's just us.

[00:49:58][snorts] Um, it makes sense that it's just you and me or it makes sense that uh that proving is a nervous tick.

[00:50:18][clears throat] [snorts] Okay. The map between injective resolutions of these two uh Not really. You're here, too. Traveling number is here, too. You've just been silent. Okay. Well, nice. Nice to uh to see that you're here.

[00:50:42]Feels less lonely now. Uh to be fair, you don't find spectral sequences that interesting. Fair enough. Fair enough.

[00:50:52]Yeah. You think people don't have as much to comment on? Yeah. Maybe not.

[00:50:57]Maybe not. That's true.

[00:51:01]H [sighs] but I need to know this and my friend needs to know this. So So here we are.

[00:51:10]Okay. Do these details even matter actually? Like maybe we should just go jump over to uh calcin to put this in play to see to see how it goes. We basically have the idea of the proof, right?

[00:51:26]uh to construct the growth and de spectral sequence take functors f and g satisfying the hypothesis. Yeah, I guess it's just the same thing all over again.

[00:51:37]Uh perhaps it's not entertaining if I do this. Uh you know, this is how I feel about the whole the whole series. I'm like, uh, I just need to carefully do these computations like again alone in my room, but whatever.

[00:51:54]Whereas YouTube awards you king for being the most obnoxious commenter.

[00:51:58]Yeah, that's true.

[00:52:03]Yeah. Okay, let's let's Yeah, I don't know. I feel I I feel bad for not writing out more of this in detail. I should write out more of this in detail.

[00:52:15]Um, okay. They do some they do things maybe a little bit different than Wikipedia.

[00:52:24]So, so maybe it is worth reading. But I do want to look where's this Castleman reference here. Um, right. Castleman 2 8.9. I think I have this paper downloaded.

[00:52:40]Uh, nonunitary arguments. Let's see. Non unitary.

[00:52:47]Come on.

[00:52:50]Damn.

[00:52:51]Let's see.

[00:52:54]Castleman.

[00:52:58]Come on. I hate Windows search. I should be doing some like command line search or something. I know I have this paper somewhere.

[00:53:09][sighs] Actually, I think I have a folder called something like X project or something.

[00:53:18]Yeah, surely.

[00:53:25]Is this the right Castleman paper?

[00:53:31]Nice. I knew I had it.

[00:53:35][gasps] Windows can't fool me.

[00:53:43]Oh, yes. I'm much too smart for that.

[00:53:47]Um, you would have guessed stream ain't really for views, though. No, it's not.

[00:53:53]I don't know why I do stream. Why Why do I spend Why do I spend so much time every morning streaming? I don't know what it does. What does it do? What does it do for me? Why do I do this? Why are you here? Why am I here? Why are any of us here?

[00:54:10][sighs] I have no idea.

[00:54:15]Have I heard of Google?

[00:54:18]Googla Googla? No.

[00:54:23]And I don't want to hear about it. I guess it's too late cuz you've you've written those words. But I haven't heard it actually. I've only I've only read about it now because you typed that.

[00:54:35]Um, yes, that's true, Leo. This is uh for you. The words of Lana Del Rey. For you, for you. It's all for you.

[00:54:48]Every spectral sequence I do. I converge it all the time.

[00:54:55]Heaven is a place where um I was trying to think of some other good pun that rhymes with you or not pun but anyways.

[00:55:09]Okay, what do we have here? Uh okay, so this is just the application. I see this is just this is just Castleman.

[00:55:20]This is just Castleman stating this theorem from Warlick.

[00:55:24]Uh where is it now?

[00:55:44]What we really need to understand though I mean this is a great tool. So, this is a great tool, but uh I would like to find a way to general because the resolutions they come up with here only works for these so-called generalized Steinberg representations. I'd love to find a way to do this for other representations.

[00:56:05]Greetings from Brazil. Hello.

[00:56:08]Hello, Leonardo.

[00:56:11]Are you late or did I start early? No, I started late today actually.

[00:56:18]Uh, suppose H to be closed and normal in G. Wow, that's too much for me actually.

[00:56:27]I mean, this is an important thing because we just mod out by the this is how we get results. Uh, killing off the center.

[00:56:35]That's what I think that's what I think this theorem is really serving in ORL.

[00:56:41]Uh smooth G modules with H acting trivially then there is a spectral sequence uh right converging to this.

[00:56:58]Uhhuh.

[00:57:00]Consider [snorts] the functors here and here.

[00:57:08]The composite functor is just h in this case.

[00:57:13]Uh since h acts trivially on v.

[00:57:19]One is co one is contra. The first takes projectives to projectives.

[00:57:25]Uh and its derived functors are homology.

[00:57:28]A standard argument gives a spectral sequence. Okay.

[00:57:32]You think YouTube blocked your meme? You had a meme. Uh, you must head off. Uh, I think I must as well right away. Well, thanks. Thanks for joining and yapping.

[00:57:52]Um, I think I kind of understand this slightly a little bit better. Yeah, I guess this is basically the growth and deep spectral sequence here.

[00:58:07]Let's maybe just go back to orlic one more time to review the last little bit of this. [sighs] Um, bonjour.

[00:58:23]Uh, >> [sighs] >> Right. Similar facts about contradiance.

[00:58:34][sighs] Uh some duality theorem. Parabolic induction makes our lives easy. Yes, this is I think this is I think something that we're using in particular.

[00:58:49]Uh it's going to sound random, but honestly sub the stream in your recommendations not subbed yet.

[00:58:55]Oo, pausing. Hesitating.

[00:58:58]Hesitating.

[00:59:00]How can I tip the scales?

[00:59:03]Uh, you unfortunately have a shabby math background. Uh, and you're looking for advice to improve at maths. Uh, definitely sub. That's the most important thing you could do to improve at math. [laughter] [gasps] Uh, to sum up your situation.

[00:59:22]uh to sum your situation up. You're at uh algebra level at the very basics with linear single variable equations. You want to improve through a decently fast route.

[00:59:36]You're already a legal adult but didn't make it to college or anything because of your health situation. Already a legal adult but didn't make it to college or anything because of health.

[00:59:46]Yeah. Um well, yeah. I mean, there's faster ways to learn math, arguably, maybe, and slower ways, but uh, you know, at the end of the day, I know you want to try and go through a decently fast route, but there's just no there's just no royal road to learning math. Uh at the end of the day, you just have to sit there and do lots of exercises, uh solve lots of problems and read lots of books, and it just takes time.

[01:00:26]Yeah. Um so, uh linear single variable equations. I mean once you feel like you have a handle on that once you've practiced doing that a bunch then you know the next thing to go to is linear equations with multiple variables. Uh so like any standard linear algebra textbook uh I think would be would be the next would be the logical next step um in improving your algebra.

[01:01:04]And that's really all there is to it at the end of the day, you know. Um uh uh yeah, that's that's really all there is to it at the end of the day. You know, there's always finding yourself a tutor.

[01:01:26]I do tutoring, just to throw it out there. But um you know even with that uh you know you can find people to to help guide you. Uh I'm definitely not the cheapest tutor in the business. That's for sure. Um but you can always find uh Yeah. Yeah. At the end of the day, you just got to sit down and do lots of problems. That's that's kind of it. And if it takes you longer than some other people, I mean that's the way the cookie crumbles sometimes.

[01:01:59]>> [clears throat] >> Okay.

[01:02:03]Yeah, this is an interesting theorem. Uh this the exterior algebra of what like the character lattice or something of G.

[01:02:14]Uh this is from Burell and Wallik.

[01:02:18]Uh first case G is semi- simple and simply connected.

[01:02:26]Yeah.

[01:02:27]And then here's where things get really interesting. I don't understand it all.

[01:02:30]The result is a constant coefficient system on a base chamber inside the brewha tits building which is contractable.

[01:02:39]Yeah. Like none of these words make sense to me and I really want them to.

[01:02:49]Yeah.

[01:02:54]The question is I guess how we generalize it because we use HG1 here.

[01:03:01]Like I maybe want to put a different representation there than the character and I don't know like how do we get how do we get um and actually why is this a bulg and the other one's not? I don't know. We'll need to go back and look at notation.

[01:03:22]But [sighs] uh quotient is compact.

[01:03:31]Let's see.

[01:03:34]Yeah. What's the essential step here?

[01:03:36]Now it is known that the coalology of a finite rank free commutative discrete group coincides with the coology of the corresponding Taurus.

[01:03:51]Oh, what do you mean?

[01:03:57][sighs] God.

[01:03:59]Okay.

[01:04:04]Yeah. What else is the essential thing here? [clears throat and cough] They use this. This is a very specific, right? See, and here. Yeah. This is this is a key point. This is a key point that they're using here.

[01:04:17]So they're getting they're getting what they're getting HG they're taking the cohomology of G acting on I mean this is some induced representation this is like parabolic induction uh and then this should be the same as some thing on the levy which is how they're relating this to the character lattice.

[01:04:48]um of the levy using this previous theorem.

[01:04:53]But now I wonder you know the interesting question to uh to ask here is what happens if we put like a different character uh what if we put a non-trivial character of the levy here? Is there some way to is there some way to change like some obvious way that this changes?

[01:05:14]Um, I wonder and I guess the way to answer that question is to really understand this proof, which means really understand the brewh hot tits building. So, I don't know. But I guess we're getting away from spectral sequences [gasps] here. So, um, I don't know. Maybe tomorrow we'll we'll do another episode. Maybe we'll go look at Botton 2, try and understand those spectral sequences. I don't know. Let me know what you think. You subbed. Well, great. I'm I'm glad I uh I'm glad I convinced you to to cross cross the threshold.

[01:05:53]You hope one day you'll be at the level that we're all at, chat included. Uh well, just keep keep grinding. Keep at it. Um it's just a matter of time and patience and dedication. Appreciate the advice, but you feel you'd be intriguing the stream by asking more questions.

[01:06:10]Well, no. You can always just ask. We can always just ask. Just always just ask away. Ask away. I'm about to hang up stream now, but you know, in future streams, uh, you know, just ask random questions.

[01:06:23]That's that's why I'm live streaming this stuff, basically.

[01:06:28]So, feel free. Uh, but that's going to do it for today.

[01:06:33]So, let me be really clear. I'm about to say QED in case it cuts off early. Uh, you know, I just don't want Boris calling me out here.