Well hello and welcome to Jonathan from the Heart. I'm Jonathan Asley of JonathanAsley.com and I'm so excited to be doing this short video of you today. Our topic we're going to talk about male midlife crisis and what to know about it. Now really quickly if you're brand new to my YouTube channel please hit the subscribe button hit the bell so you can be notified of new videos and if any time during this video the content resonates with you please hit that like button so I can be seen in the YouTube algorithms. Also these are my weekend videos I shoot out on my balcony very similar to the videos I shoot my private group called Midlife Love Mastery. This is a group where you can have direct access to me on a regular basis and if you post questions in the Facebook group I shoot personalized videos just for you. So if you need my support and you can't afford coaching you might want to check out my groups so check out the link below. Alright we're going to talk about midlife crisis but before we do I just have a real quick rant because when I wake up in the morning and I check my analytics on YouTube a lot of videos will pop up and what's interesting is I've seen this huge proliferation of young women who are giving dating advice you know women in their 20s maybe early 30s giving dating advice all centered around how men are supposed to be chivalrous and women sit in their feminine energy and men will just claim you because men traditionally when they like someone they're going to go after them and that's certainly true to some degree we men biologically speaking when we are physically attracted to a person and we want sexual connection with that person we may go after them like you know like a you know be going to honey kind of thing that may be true but does that really signify a man being in a relationship? Can you apply a lot of that and what's interesting is a lot of the rhetoric I hear especially from my contemporaries you know that men when they are in love they're just going to do whatever it takes to make you happy and that men are these certain creatures that are just so perfect and you know what and a lot of this advice will tell women to do their own personal development work but takes no consideration about what might be going inside a man so I'm here to step into that conversation because my audience is midlife and I basically say midlife is after baby making years and before retirement so if you're 42 to 69 that's the demographic I speak to primarily because what I'm about to share today and it centers around the multiple things happening at midlife that cause human beings to practically unravel. I call it where our blueprint collides with our reality, our blueprint collides with our reality and for example my blueprint growing up was really simple it was my folks would tell me you know after high school you're going to go to college you're going to go to college you're going to go to college and after college you're going to get a job, you're going to get a job and then after getting a job you're going to meet a gal and then you're going to get married by a house and start a family that was the blueprint the revolutionary road if you didn't see the movie with with Kate Widslett and oh my Leonardo DiCaprio you know it's back in the 50s that centered around this whole belief of this fantasy of how relationships would be and then he has a midlife they both have a midlife crisis in fact in this conversation this isn't singular to men this is both for men and women alike. The reason why I'm leaning into this conversation because folks I went through this on a horrific scale on so many different levels and maybe you might glean something from this part of the conversation so I was told to get a job meet a girl get married by a house start a family I did all that and roughly right at around age I think I want to say age 39 or 40 I mean just you know I was living what I thought was an idyllic life but quite frankly I really wasn't I was starting to experience a lot of things happening inside of me centered around you know who am I why am I here all that kind of stuff started to happen and then the worst thing that happened I lost my quarter million dollar a year job and it happened at the same month that my wife asked for a divorce so imagine these two colliding things happening going through and quite frankly I was I mean in all retrospect I'm grateful she asked for the divorce she had very valid reasons for asking the divorce and I don't want to get into the particulars of that but imagine going through a divorce and losing your job at the same time and I had a quarter million dollar a year job in fact right around 2008 there was all of these you know we had this financial crisis that happened and lots and pizza people lost their job and then we just had a pandemic now and lots of people have lost their jobs so I'm going through this divorce and job loss and I found comfort in doing drugs alcohol and cereal dating that's where I found my comfort was in this area now I didn't realize it was preparing me for what I'm doing today but I'm just telling you I used to go to bed wishing I didn't wake up for the longest time and I had two young children and I was struggling on the inside and this is true for so many men they go through and mine wasn't just what was happening in the moment it was also an unraveling of what was happening in what happened in my childhood because why I'm sharing this part of it today is because if you're not familiar with the work of Brene Brown I highly recommend checking out her books called the gift of imperfection the gift of imperfection but she has a article she says it's called midlife unraveling I highly recommend googling midlife unraveling this is where all the armor that we built up through our childhood wounds and traumas and then of course our adult traumas as well all this armor that we've built up starts to unravel from an emotional perspective in fact the real reason why this oftentimes happens is because we have repressed emotions and then we can only take so much as a human being men and women like we can only take so much when we have these repressed emotions that it's going to unravel it's going to implode it's going to explode and this happens you know for some people in their late 30s for some people are even 40s for some people in their 50s oftentimes there's a traumatic event that's happened like a divorce or a job loss or a physical issue and so I'm sharing this all because we have to look at human beings from the bigger picture not from this fantasy of masculine does and feminine receives and all you have to do is sit in your feminine it's all just going to magically work out because I'm here to say humans are riddled with issues with flaws and it requires an understanding and not this listen I'm not here to suggest that you want to invest in a boat broken piece of property I'm not here to suggest that you want to you you want to purchase a fixer upper but we all have some fixing up to do this is why I'm such a big proponent of humans doing personal development work this is why I wrote my book what the heck is self-love anyway what the heck is self-love anyway by the way there's a link below to get my book and why I'm sharing this with you why this conversation is so important to understanding men in particular is that they are not these perfect creatures just like you're not a perfect creature we are all struggling on the inside in some way shape or form so the question now begs what are you going to do about this okay first I want us all as human beings to come from this place of you know compassion instead of this expectation way of dating the reason why I judge and criticize my contemporaries especially these younger ones it's all based on expectation that a man is supposed to do this to make a woman feel safe well here's the thing ladies we men when we start going to this unraveling we don't feel safe and to then and then oftentimes a lot of coaches judge it well that's just a man who's just an insecure weak man and I can't be with an insecure weak man I have to be with a alpha male so I can feel good and I need that alpha male to pursue me so I can feel good about myself we are here in the United States in particular suckling on the nipple of I need you to love me so I can feel good about myself and imagine somebody who's suffering on the inside how are they going to be able to support you how are they going to be there for you if they're struggling on the inside so this is why I'm such a big proponent of individuals doing their own work while there may be exploring a relationship together if you're not familiar with the book the Hoffman process the Hoffman process I highly recommend this this is a if you bought this book and did the work in here this is a 50-hour deep dive into healing your own stuff and what's going to happen through this process is you're going to come at it quite more compassionate towards yourself and compassionate towards men okay why is this so important I want to get to that piece okay I'm so anxious I'm by the way I'm really quickly my shirt is Batman today and my coffee mug is the Hoffman process what I just shared why this is such why I wore the shirt today because Batman's a perfect example of this he lost his children or he lost his parents at I think age 11 or 12 imagine the imagine the traumatic effect that has on a child and how he channeled that was to become I mean he became a recluse and then he became a champion for others because he was really I mean I don't want to get into the picture but particulars of that allegory if you will I'm just here to share he's a perfect example where he became very withdrawn and this happens to so many human beings this is why if you're not this is why the fantasy of love that many of these you know coaches are trying to sell you on doesn't contemplate these two other important factors you must consider and that's love attachment style and if you're not familiar with the book attached by Amir Levine and Rachel Heller I highly check recommend you checking that out and also the Amago why we tend to choose people like our parents you have to check out the book getting the love you want by Harvelle Hendricks and Helen Hunt why is this so important because a lot of what we believe is love is either we're falsely attached to another human being or we're chasing the love of our parents through another human being so this is why you many of you ladies get hung up on men when you're not really experiencing true love you're just experiencing some level of attachment or desire to heal okay so here's what I recommend for everybody going forward is if okay so I want you to decide do you want a casual relationship or do you want a serious relationship I repeat that do you want a casual relationship or do you want a serious relationship the reason being is a casual relationship basically is companionship connection and sex without all the expectation of where is this relationship going to end up and without any expectation of future partnership and for some people they may only be capable men and women alike may be only capable of a casual relationship so I want you to get really crystal clear on what it is you want okay now a serious relationship means that there is some sort of destination you want to either move in together get married or have some level of partnership with one another that's a more serious relationship and I think it's important to identify that very early on in fact a lot of people in midlife women in particular might only be capable of a casual relationship just like most men are only capable of a casual relationship so identify that first now this is the tricky part oftentimes you might say I'm only want a casual relationship when you actually want something more and this is where you get into a lot of conflict because here's the problem you both agreed right from the get go you want something casual not all of a sudden you want something more serious and that space in between that's called drama that's called drama because that's what happens when it there's a unmet expectation from another human being okay so again coming back to being on the same page for those of you who want a serious relationship and you want to vet where is this man on his journey of his midlife crisis where is he in the tunnel is he entering the tunnel like the first you know like the first five years of my after my divorce was strung out on drugs and alcohol and that sort of thing or is he in the five years you know coming out of the tunnel and it's okay to meet someone who's on the way out they don't have to be completely out but so long as they're on the way out and let me tell you how men show up when they're on the way out and they want a serious relationship they actually are seeking some level of partnership with one another a collaboration with another human being and they're not driven by the testosterone leads and the feminine just sits back they're not driven that way they're seeking someone who wants to lean into a co-creative kind of relationship the relationship I constantly talk about where it's a two lane street where you're traveling together okay it's not him pulling you along or chasing you so you feel validated okay and a lot of young girls want to teach you and I said girls I feel bad but a lot of young women say they they're teaching advice is all centered around all about princess energy that a man is supposed to prove his worth to you that doesn't happen in midlife I mean just be grateful that you have someone who's interested and wants to explore something with you and what I'd like to invite you to do is if you two decide to date to spend some time getting to know one another and then you decide you want to explore a relationship with one another I highly recommend ordering two copies of this book called eight dates by doctors John and Julie Gottman if you've agreed you want to explore a relationship together and before you have sexual intercourse with one another I highly recommend getting this book and reading it together so you can get a sense of a framework is is this something we want to explore together and a man who's in the middle of his midlife crisis I mean he's just beginning or he's in the middle of it he's not going to want to do this he's not going to want to do this someone who is in the process of healing and they might be doing some therapy they might be doing some inner work they might be exploring themselves from an introspective perspective it might be that they're reading books they would welcome this kind of additional support because that's what this provides this book provides that additional support by the way there's a link to Jonathan recommends books below so why does midlife crisis happen because when we have repressed emotions when we have a repressed feelings there is going to be an implosion at some point and a lot of people go down the drugs the alcohol like I did along with you know a lot of different things there's sex addicts there's porn addicts there's gambling addicts there's all these things ways to cope and at some point some men never get out of that and then some men push through and my hope is that you do a better job of being intentional in the early stage of dating rather than the fantasy way of dating because that's going to set you up for failure I want you to approach this from this understanding that human beings are riddled with flaws and it's okay you are they are let's stop expecting this what does the book say you know let's stop expecting perfectionism and understand that there's an imperfect world out there and by the way I am not here remotely suggesting you compromise your boundaries or accept bad behavior I'm just here to say let's have compassion for another person's journey and be intentional right from the get go because that is a better way to start off to find out if you're on the same page or if you're in two different boats going in two different directions okay I think you get the gist of where I'm going alright I'd like to hear your thoughts on this please post a comment below I do my best to read them all and even respond if you like my shirt please tell me about it if you're if you read any of the books let me know about it if you feel like let me just share this if this works for you let me know as well I want to hear about it so post a comment below alright I'm going to wrap up this video as I always do first off give myself a big gigantic Jonathan bear a hug of self love I'm going to reach into the camera and give you a hug of love if that's okay I'm going to ask you to turn to someone a pet a teddy bear a pillow and give it or them a hug of love because hugs are a great source of love and let's face it we could all use more love in our lives thanks a bunch bye bye now


So in today's session, we will have two talks by Irina on uniform art in this lemma continuing from the last time. So, you know, you may begin now. Thank you. All right, so our goal is to prove the weak version of the uniform art in this lemma. So I'm restating a few things that we did last time. We don't know all the conditions under which the weaker the strong version hold. In my notes I wrote it wrote up a few strong version uniform strong versions that necessarily true for all ideals but when you restrict ideals to maximal ideals or the principle ideals then that has been proved. But what we did for a week version. For the week version we already made some reductions. We don't have to worry about all ideals. We it suffices to prove it for the primary for ideals that are primary to a maximal ideal. That would not be good enough for a strong version that the inclusions just don't go the same way. So strong version is very different kind of animal. So anyway, our goal is to prove. I'm trying to show one screen. No. Okay, so these are right here. If why isn't this working. There. So if we want to prove, we cannot possibly prove all three parts in all the detail because that would be much too much but I will make some indications. Another thing that we reduced to last time is we had this wish definition. We related this uniform Martin we slam our we made a digression that we could just prove that we have this analytic on ramifiers that are not zero. And there exists elements that multiply the end power of indoor closure, when I deal into just the ordinary and minus case bar if we can, if there exists one K that works for all ideals are all and that would be key of our. And so the first goal today is to prove so this is. The first one is T of R, not zero. So no we haven't yet proved the week uniform Artemis lemma, but we're making a distraction which is actually an important ingredient for the proof as well as, I think it's of interest of its own kind. So first of all, for T of R to be non zero. R needs to be reduced. And the reason is that. It's not reduced. All internal closures always contain the, the nil radical of the ring, but ordinary powers do not do not contain the nil radical in fact high powers do not contain it at all. So we really need the ring to be reduced. So the theorem that proves so let R be reduced be reduced. No theory and ring. And of characteristic P. Then T of R is not a zero. If any of these conditions happening. So, the first one is our has a test element, and the cruel dimension of our is finite. So, I do not know if there exists. There might be a theory and rings of characteristic key that have a test element but are infinite dimensional. There might be a theorem of that form but I'm not sure. So I am imposing this finite dimensionality right now. In the next cases are is finitely generated over an excellent. We know that such a ring is finite dimensional. Or the third condition, if R is a finite. And it is. If it is a finite then there's a theorem of points that it has finite dimension. And so, both of these two imply the dimension of our is finite. And then the other thing that's also true conditions. Part two and three imply that our has a test element. The proof of of for part three was already given by Kevin Tucker this was theorem seven. I also wrote it up in the notes and the proof of that, in case to our is finitely generated over an excellent local ring but then have a test element. That is, a lot of pages, many of the 14 page section six in humickel hockster paper the first one, the first big one. So I cannot possibly justify this so take two and three was now somebody would have to check the proof but I'm not doing it today. So, so what essentially I have to do is. This is all I have to prove. I've reduced to the proof of only part one, and the rest is by previous work by various people. So we have a test element and we have a finite dimensional ring. So, So, the first thing is and one thing I in my notes, I just 15 minutes before my lecture realized I need a little bit more work. The first thing to do is step one. We can assume, suppose that there exists a K impact actually. We will not prove that K is to be where D is dimension. So D. That is the dimension of the ring. Suppose there exists case such that for all zero dimensional ideal zero dimensional primary ideals. We have this inclusion, I, and for all. And David milk of the pain, we take the inter closure by to the end that's containing it to the end minus K. Then it's, we already proved last time the uniform and the weak uniform martini slama. It suffices to prove it for and primary ideals or and there is over maximum ideals. It's the same kind of proof. So what we actually have that the same holds for all ideals so that means that T sub K of R is actually the whole ring. And one multiplies the inter closure and arbitrary of the inter closure of the end power of an arbitrary ideal into n minus K power of that ideal. Okay, so that is, I'm leaving that. Step two step. Step two is, we need to find some kind of reduction of eye. So that we will have a uniform D. So the first thing is, is by prime avoidance. So, so now we only have to prove it for that I be M primary, arbitrary and primary, where M is a maximum ideal. And now we want to find some kind of reduction of our eye. So first of all by prime avoidance. For prime avoidance, I don't need M primary at all, and this would work for any ideal. There exists a D plus one generated ideal J containing I such that the radical of I equals the radical of J. So this is just prime avoidance and I will assume that you have seen something like this. Or, I will leave it as an exercise. By the way, the exercises there perhaps too many exercises I just added a few which haven't been uploaded. But I'm also saying don't do all of them. And I write what I think of exercises. Good for the soul. So then. So for this we don't need the M primary ideal. The what we need for M primary ideal is that now we can localize after localization at M, then I is M primary after localization. So if we know that the ideal is, if we know that the ideal is, sorry, if we know that the residue field is infinite, then we can find the degenerated reduction, but I didn't assume that the, the residue field is infinite and there's a little bit more work. And so here I will also leave it as an exercise that we can we have that the integral closure. Sorry, I don't understand why this happens sometimes. I must be touching something here. This is in this side. What did I do with it. I started you know, okay. All right. By an exercise. What we have is that the, if we take the integral closure right to the end so this is locally. That's contained in the tight closure. I guess I should be locally. Tight closure after localization, and mine is deep. And even though the dimension of our localize it and might be smaller, but it's at most D so this D works. Okay, so this is the usual grievances code of theorem. So we know this. We know, Ian talked about the grievances code of theorem, and it works whether they have an infinite residue field or not because it can first pass to the infinite residue field so it's again one of those tricks. Okay, so now what when we left to our Then, oh, I didn't need this at all. When you lived to our, then the inter closure of Oh, and this is to fall in the inter closure right to the end. Well, we know that the closure does not commute with localization, but it does commute with localization. For all and primary ideals, there's only primary ideals primary to maximum ideas. Okay, so actually, either made a mistake right now I discovered this only after a gap. So maybe I don't need this at all. Anyway, so we have the inter closure right to the end is contained in the title right to the end minus D. And then what we did assume is that our has a test element. So we have a test element. This is true for all and. And so let's. Okay, we're now. Now this is final step, let, let C be a test element. And then C times. I to the the inter closure C times inter closure right to the end is containing C times the inter closure and minus D to the title closure of this power, but C is a test element. So this is just in I to the end minus D power. All right, so now, if we go back here. What I said, in this side I have to correct. So we didn't prove if we, we didn't quite prove this in step one. We just prove that there's C times, I to the, the inter closure is in I to the end minus K. And so then we can conclude that for some non zero C. And then we know that this year is also not zero. So a little bit of back and forth. If we know that this T sub K of ours, not zero when we only restrict to and primary deals then it's not, it's not zero for otherwise. So, so C is in this T sub D sub D sub D of R, and if I go back here under here, I, I was also wrong on this side here. Where did I say it was just D. Okay. I should be minus one. So that proves this. And then I want to state another theorem. And this one, again, we cannot prove this is due to Lippman and satay. And it's quite restrictive I write the statement in 4.2 in the notes. And so the previous theorem the previous pages we started with the reduced string of characteristic P, Lippman and satay as handle. As handles characteristic zero. So let R be a regular domain with infinite residue fields, and whose field of fractions field of fractions as characteristic zero. It's not saying that our contains a field of characteristic zero. Just, it has field of fractions as characteristic zero. T of s is not zero for any finitely generated our algebra as that is that contains. And as finite crew dimension. So a proof of this. Lippman satay theorem, the T of s is not zero. This would take us a while as well. So, sorry, Lippman satay proved some other things but then he unique approved. The T of s is not zero. In this case. So this is this ingredient that we came stumbled upon last time. And now we need another ingredient that will help us prove the uniform art and re-slamma. Okay, so these are now do new section co and the qualifiers. It's not an official name but obviously it was meant. So, this was. First, we need the definition. So we start with the complex that has F zero F one. And it stops. So we have F sub T, F sub T, and then zero. So this is a complex of finitely generated free modules. So each of the capital F, I should call these G sub I so so the F's are finitely generated free modules and the G's are maps in the complex. So we know what the rank of a free module is. I guess ours over the ring are. So we know what a rank of a free module is. How about the rank, rank of a map. So you have a free module and you have a map here. This is the maximum of R such that the, we can think of G as a matrix. So if we take the ideal of R by R minors of this matrix that is not zero. So this is rank of a map. So let's say that two things. The complex F satisfies a standard conditions on rank. So first of all, over here at the steep spot, the rank of F sub T is the rank of G sub T. And everywhere else, the rank of a rank of G sub I plus the rank of G sub I plus one is exactly the rank of F sub I. So this, if we had vector spaces here, this is exactly the exactness of a complex. But we don't have vector space of fields. There's general not a field. And then we say that F satisfies standard condition on height. If the height of the ideal. So the ideal of maximum possible minors of G sub I that don't give us the zero ideal so this by definition is a non zero ideal. If height of this is at least I, you might have seen F satisfies standard condition on depth in the depth of these ideals is bigger than equal to I. R is going to call it then height is the same as rank. So we have this would be an exactness, a criterion, but then the second part of the different or the next definition is under these conditions. So let's say that the set C M of R the column home coli piers of R. This is the set of all elements X in our such that. So it's the set of all elements X in our such that for all complexes at satisfying standard. And height conditions. C kills this X kills all the homologies. This is zero for all I bigger than equal to one of course. So, in the column of coloring, one would be an element of this column of color. This ideal versus. This is actually an ideal. It's not hard to see. So in a column of coloring, this ideal would contain one, but in general, it may not be so big. Okay, and the, I will start this. The next theorem now, but I won't I will be able to finish the proof only after the break. So here's our lemma is the level.  Let P be a prime ideal in our. And let C be an element of the ring not in P such that its image in our mod P. Lies is in the coma qualifier of our mod P. So let J be any ideal generated by a one so the elements. And what we want is the height of this ideal. When we go mod P is at least D. And suppose that there exists an integer K such that for all N.  So in other words what we were saying that at least for this ideal J would have uniform art and reads for the module M is the whole ring and M is the ideal P plus CR. And the conclusion is that then we have at least for this J. And we have the uniform art and reads also for P with the same. With the same K. Oh. Oh, so my iPad is showing is it 931 or is it 941. How much 31. Okay. Oh, so my iPad is showing the wrong time. Okay. All right, so I have plenty of time to read this. All right. All right. So, proof. So, what do we have to show? Well, this is certainly contained in P plus CR. Actually, let's, let's make it equal. So this if I also intersect with P. And then by assumption this is containing J to the N minus K P plus CR intersect with P. Now, by the same trick we've done several times this is J to the N minus K P. So this is J to the N minus K C intersected with P. But, but this year, this ideal here, I don't hurt anything if I write intersected with C. So this year is J to the N minus K P plus. Oh, C times J to the N minus K. Oh, this here is just C times P right because P is a prime ideal and C is not contained in it. So we can factor C out so we're left with this. So we have to show to show C. And this, I have to erase this bar so you don't see into it's not into a closure. I had it underlined. So we have to show is that C multiplies J to the N minus K intersect with P into the J and minus K P. So this is where the column of qualifiers coming so C is a column of qualifier mod P. So we can just show that now we have some math or some complex of finitely generated free arm modules. Free arm of P modules. So we need arm of P modules that satisfy the standard rank and height conditions. And that maybe. So here is J and minus K intersect with P J and minus AP is this equal to H sub I of some F of satisfying standard rank and height conditions. So this is the question. And that's the goal for when we do that, then we'll be done. So, let's see. So proof of this proof of this that it actually true. So, so now let's start with. We started with complex and I'll call the maps, maybe I should call them G. F's get map G's and G's get map ages. This be a minimal homogenous free resolution of the ideal generated by D the nth power of the ideal generated by D variables in the polynomial ring. So T, T is generated over the integers by excellent through XD. So, there are many ways of producing such a resolution and take for example, the Egon Northcott resolution. I'm, that's another thing that you might have seen it if not. If anybody knows more recent recent reference like Bruce Feather have a pretty nice proof but there might be another one. I'm not in print some other nicer proof. So if you know a nicer proof and want to type it up and the great, we can include it, or we put it in your next expository book. And so what we know is, at least for this Egon Northcott resolution. We know that if we take the ideal. So, for each, for each of the maps. So if we take the ideal of maximal minors so I omitted writing that this is the ideal of maximal minors just implicit that it is the ideal of maximal minors. So we know that this ideal, at least up to radical, it's equal to X1 through XD. So that's what makes this Egon Northcott resolution very powerful. And. And so now let F be the advantage of this ring T is that it maps to every ring in particular maps to our R. So not arm of P but R well it could also map to arm of P but we're tensing with our, and the map is that X sub i of course goes to our a sub i sub i sub i is J is this ideal generated by base. So this is F. And so what do we do is now we compare. We compare. So on top we have G the bottom we have F. So all I have to really write is there's are the ego Northcott resolution. resolution, I should have said of r mod. So here is r to the n plus, wait, we're doing f here. This is f. So I already answered and d minus one choose d minus one. And here whatever it is, let's just call it f two. We will need more details than this. And then, so this is f. We don't know if f is an exact complex. We know that because the a's are unlikely a regular sequence. So this is not necessarily an exact complex. But down on the bottom, we will have a similar thing. But h here in this row is a free resolution of r mod j to the n. So and here we have h sub two. Again, it doesn't matter what the rest are. Now these two are equal and whatever the map here is, we have the same map here. We call it g one. It's the same map. And this bottom thing h is an exact complex. And f is a free complex. And whenever you have a free complex going, well, at least the start, we have here a start of the map from one complex to the other. We know that there exists a map going down here that makes everything convenient. So at least we have a map from f two to h two. So now we really need something mod p because c is a column a column by a mod p. So tensor with r mod p. And what do we get? So f tensor r mod p. And at least the start of the map r mod p. What we have is we have a surjective map. This map is a surjection at least in the second spot. In the second spot. Not second. So this is here. This is in the first spot. First spot. So we have a surjection that should be a different color. No, this is color. We have a surjection after tensing with r mod p. I should just do it here. So we have its identity here, its surjection on the homologous. So we have h one of f tensor r mod p surjects onto h one, g, sorry, h tensor r mod p. So we have a surjection. But what is happening with this thing here? This thing here is still is a complex of finitely generated free modules, free r mod p modules and the height condition holds. So the radicals of all the tensored maps. Well, before tensing the radical contained all of the contained the all the x is so now after passing to r the radical contains all the a's. And so now it still contains all the a's. All the a's. But these are the radicals in r mod p. So we have the standard height conditions are satisfied. Right. Condition is satisfied. But also the rank condition and rank conditions are also satisfied because they were satisfied by the original Eagle North by complex. And the maps didn't go to zero the and the ranks stayed the same. Okay, so what that says. So what this says, thus, c kills this homology of the tensored f. So this is zero. But we have this surjective map here. And by the surjection see also kills this. This is the red complex. That's also zero. So what we have to figure out what is this homology here, but this is exactly j to the n intersected with p over j to the n. So that requires a little bit of work. All right, so let's go back to where did we start. So what we have on this side, we have to show this. Well, we showed it for all powers of n. So we conclude that j to the n intersected with p, we follow through over here. This is contained in j to the n minus k p plus c times j to the n minus k intersected with p. We just now prove that this is in j to the n minus k p and that does it. So let's go back to what we proved. So remember, on Monday, we had some cases where if we had some kind of reductions mod p and I call them j there too, we had some kind of reductions mod p and if we can get a good handle on them, then then we can we get some leverage on ordinary other ideals, or at least in primary ideals. So here this lemma really gives us that leverage. And by the previous section on trying to find t of r is nonzero. So if if we know the t of r mod p is not zero and the these column of qualifiers of r mod p are nonzero and we can do this for all primes p, then we then we are in pretty good shape for being able to prove the uniform art in the Islam. Okay, so when is the next section would be when is this column of qualifiers not zero? When is c of m of r not zero? So that's the other big question. And for the t of r to not be zero, remember we needed a reduced strain for the column of qualifiers we don't need reduced strings. And the next section section six in the notes has quite a few results where it is not zero. I give fairly complete proofs where the coma qualifiers are not zero. But I will probably skip that part. It's kind of technical to get to the juicy part in the next section is when is the when can you prove the uniform art movies lemma? Okay, are there any questions? Okay, well, then we can take a break now. Yes, yes. Okay, we'll meet after 10 minutes. Yeah. Okay. All right. See you.


All right, very good morning. It is Friday the 4th of September. I hope you're doing well. I thought I would start with Splash of red New color here going on given the NASDAQ after rallying for 11 of the past 13 sessions Yesterday things coming undone somewhat the NASDAQ 100 finishing down over 5% The other indices not quite so bad, but again some of the biggest down days We've had in a number of weeks, of course some of the mega cap tech names Which obviously have been leading this market higher pretty consistently now for for the last couple of weeks came off The most sharply so Apple was down about 8% you can see here Microsoft down 6.2 Amazon down about 4.6 Google about 5 Facebook about 4 Some of those names which have really outperformed recently like zoom for example came off sharply Tesla Someone and so forth. So yeah a little bit of reckoning say for the relative calm, I think that's been Really observed in markets of late and I think that in itself is quite telling It's interesting always is after a day when we have price action like yesterday and people start to You know, it's quite so quick people's kind of Interpretation that this is the beginning of the end and that's it now the bubble's gonna pop. We can't sustain this I would say look you've got to there's got to be a little bit more to it before I think you can start calling that type of activity, but they're definitely a few things I think we need to be aware of going forward for today's session. So with that being said, let's have a look at a couple of equity charts Because there are some first of all some key technical levels of which the market has been responding to of late So I'm looking here at the NASDAQ 100 and I'm going to switch us to a daily Continuation chart and in particular both the NASDAQ and the S&P 500 are quite a key Technical level, which is the 21 DMA now the 21 DMA you can see here as far as the NASDAQ 100 is concerned has been a really quite telling signal for people to Buy into that any dip that's occurred The most recent one was when we had the bottom end of this trend line coming in with the 21 DMA when we had a Brief pullback of the initial break to all-time highs and what was then also superior of profit-taking And that then created then the next Entry point for then the push in the eventual break Which was quite a key level of course around 11 283 and then the kind of resumption of this really strong upward trend that we've had Now the posh push down yesterday and then in the overnight session Asia was down But certainly not as bad of what we had on the close of Wall Street The losses were slightly more contained in Tokyo Shanghai in Hong Kong overnight But the 21 DMA pretty much being tested to the tick Is quite a key area now to keep an eye on here in the NASDAQ You can see just putting a horizontal line here as well. It was the error support on the 25th of August So it will be an important level Equally, so I'll come back to the Dow in a moment because I just want to bring up the S&P first because the S&P 21 DMA also warrants strong vigilance if I was going to adopt a kind of ECV phrase For the equity market because the 21 TMA as well Is in very close proximity to where price is trading at the moment So that would be my first kind of area to keep an eye on If we were to get some further pressure to the downside and then the next real key area for the NASDAQ or the S&P Excuse me would be 33 97 basically the 3400 level just below their 97 and a half encapsulates that previous high that we printed on the 19th of August that then was Repeated test on the 21st before the break and support level for the commencement of the push higher And that of course was also the prior all-time high previous to the pandemic So for me the 21 DMA in the spools, but then the 3400 and 33 97 I know for one The likes of Sam on the desk He's been calling that figure for a while that if he gets a pullback then and that's an area You know all things to be considered Obviously re-evaluating at the time when we get there But an area where technically he would like to then just pick up some more for re-entry of the long again so Once more the the kind of thinking being that you know corrections in markets after we've had a really sharp run-up like what we've had Some would say is a necessary Component of a sustainable market rally You know if a market just goes up consistently higher and higher and higher It's almost then the inevitability around a fairly significant and sharp sell-off at some point So what is healthy then are minor corrections along the way and and I guess the question mark is out now And how today will perform will be quite telling is today the commencement something a little bit more deeper in that correction Or is it just that and you start to see these dip buyers come in and certainly the 21 DMA will be quite key in those indices As far as the Dow is concerned the one i'm looking at here is This rectangle box here so it in the futures on the daily continuation chart around 28,000 28,052 actually is the figure but what i'm looking at here is the initial low that we printed In yesterday's session and that coincides with the low of support in 25th And then that was the high on the 11th of august on the recovery Which puts us back down and also a key level We can see here at the beginning of the year in late jan before the breakdown on the pandemic situation So for me, this is a really key area of support as well So you can see there's some really important areas technically that are very close by here in the major three indices And one thing I would say is that don't forget it is the labor day holiday in the us on monday now just given The fairly stretched nature of this run up in equity markets Is this an opportunity for some book squaring to kind of rotate out some of these large cap tech names And also not wanting to carry any risk over the weekend Do we see some more kind of closing out of these these positions? I kind of profit-taking and consequently if that does break down again Then i'd be looking down around 27 6 24 that prior high We had at the beginning of june and an area the bottom side of the kind of consolidation phase that we saw In the early part of august would be a potential next area of support So you know could we have another heavy down day? Sure But there's got to be some key breaks and what i'd be looking to see that more heavy move down and certainly the 3400 in case the s&p Is a coordinated kind of synchronized break would need to happen Um, obviously we've got payrolls later. I don't really think payrolls is too much Of a big deal, but could it act as a catalyst perhaps? under the right scenario But again It's this idea that I don't think that we're We can call at the moment that this is the beginning of the end of the stock market Kind of rallying. I think far from it. We've got to see some more evidence Of consistency behind the selling And I think today will be quite key and I I would say on the balance though What makes today a little bit more risky Is the fact that it is a long weekend in the states and typically In my mind that tends to add a little bit to the nervousness of wanting to hold onto any position So you could see some more of that book squaring taking place A few other things then to to look at from a Equity market points of view A couple of charts here for one that it was the biggest route that we've had Since june as far as the major three indices in a coordinated just kind of push lower Overall selling taking place and again, hence the reason why it's caused quite a lot of media attention Obviously stocks like apple given just how sizable they are nowadays an 8% move is equating to around $150 billion worth of market cap and You know these news organizations love spinning these numbers and you know, how much is bezos loss in one single day these types of things I really think they're quite insignificant and it just kind of sensationalizes the move I don't really think that that those types of numbers are particularly meaningful Also as well The other thing is you've got to you got to look at the world in a bit of context I mean this is looking at the the rally we've had obviously it's been an almighty one since the trough of the Bottom of the pandemic seen in late march and since that point, you know the performance of apple shares So, you know if we take back what we've just seen it only puts us back to where we were kind of mid august anyway, so As I said, I think we've got to see a little bit more here Before we can draw any real conclusions behind just one day's price action Feel like i'm sounding a little bit like a central banker when I say that One thing that I thought was quite interesting and I just wanted to to point out was this This was looking at Something in the ft And it was talking about insiders offloading stocks now if you're not familiar with this you can actually Track insider activity, so this would be looking at data Based on filings with the us securities regulator Excluding sales made for tax purposes through executive pay plans So basically what this is telling you then is major top level us executives who hold Perhaps some significant shares within a company. What is their activity and a lot of stock traders? What they look at is insider dealing and what we've seen here is that insiders So this would be us executives The figure here capturing stock sales of at least A minimum of 10 000 us dollars in their stock holdings in companies with market capitalization of at least one billion dollars And what we've seen is us executives have sold close to seven billion dollars of stock in their own companies over the last month That comes in at over 1 000 executives. So as we've started to rally and we've had this Incredible move in markets over the last couple of months It says quite a lot when those in charge of these companies at the helm Are actually offloading quite a lot of their stock holding And you wouldn't do that unless you thought that the future was particularly uncertain and that you'd want to cash out your chips at the most optimum level So I just thought it was quite an interesting thing here because Insiders are offloading stocks. We haven't really seen this type of volume Since the 2015 So perhaps then a little bit of an insight as to what you know these these execs on the ground for see about the Ability for this market to continue to punch high from where it is at the moment Right, okay, some other things then obviously we've got non-farm payrolls coming out later um My overall take on non-farm payrolls is Not expecting too much from this particular one as i'm about to discuss And similar in the u.s. Into the uk as well I think the coming months are particularly important And a lot of that is tied to the expiration of various different stimulus packages that have been implemented And help support those recovery you've seen in markets since the onset in in march april As some of that comes to expiration. That's when I think that we're really going to see the telling Situation of the underlying labor market and how should and how fast or not is the economic recovery going to be in in real terms So just having a look then at payrolls for today Looking at the actual details the expectation here on the bloomberg median consensus is for 1.35 million jobs The unemployment rate is expected to drop below 10 percent for the first time since in march when we saw the initial pop Expectations for unemployment at 9.8 percent and average hourly earnings year and year Expected at 4.5 percent is kind of the headline readings um a few different things then uh If I look at this here, we've had Obviously the initial reopening of the u.s. Which was occurring. Obviously they they did that Very early comparative to other areas globally And that did have some repercussion you could say perhaps in that secondary kind of way that we saw with the pickup At the time Through june july with those some belt regions, but even that in itself now started to become a little bit more controlled But nonetheless, we kind of peeked up on the the the june print But since then the number has been Decelerating and so the 1.8 million we saw last time out is expected to decrease Once again One thing that bloomberg were noting which I thought was was fairly interesting Was that any gains in august are likely to be partially offset by declines in education jobs In particular If you think about it in america and we saw this actually we use zoom here Amplifiers as much as many other companies now in the pandemic era But zoom crashed a few weeks ago And that was when all the kids in the u.s. Went back to school because a lot of them are doing it virtually via zoom And what virtual learning means then from a school Point of view is you know, think about it. There's fewer bus drivers Taking the kids to and from school There's fewer canteen cafeteria worker staff There's less teachers than are needed Because you know you can deliver virtually to perhaps a larger group of individuals and so on and so forth so You know all of these things Will contribute to to potentially offsetting Some parts of this but another thing to be aware of on the upside figure could be boosted today due to census bureau hiring this is kind of a A one-time anomaly it happens from time to time And really it's important to x out that number of jobs to get a more true underlying sense of employment conditions the figure Boosted by census bureau's hiring is said to be around 240 000 Temporary workers. So if anything that 135 should be more like 1.1 million for example What's going to happen here is that data collection It takes quite a long time for the for the census to be conducted and actually it will probably impact the next reading as well It's not scheduled to end till september 30th Meaning then that october will be when it shows a decline in actual actual government employment in this temporary nature of being then removed from the data set the other thing as well to be aware of is Kind of furloughed workers getting rehired as businesses reopen That will prop up payroll gains But those reopenings depend on the control of the corona virus and this is quite an important thing I think I just want to quickly touch upon This is just a selection of some real household Companies in america and a lot of these job job cuts have come after the reference period So it wouldn't be necessarily included here This report we're going to see today So you're looking at kind of late august to early september these these job cuts are not Huge but obviously the The accumulation of all of these job cuts is quite telling and and I think more for the sentiment then about employing employing Our employer conditions It's quite telling now one of the things here is that on capitol hill obviously They're still struggling to really come to grips with what are they going to do in terms of the form of the next level of stimulus You know, I know trump managed to do various different executive orders by the end of the day You know, what's going to happen with the number of these? Artificial and fixed period measures that have helped prop up a lot of people who are still furloughed And so this is going to be quite key going forward and actually One thing that I was sharing yesterday was them the chaps at ing put together Their latest research report and it was looking at the types of shape of recovery. So if I quickly show you It was kind of looking at this type of scenario Which is where we're at at the moment a sharp recovery and then a gradual recovery there on You've got a recovery that sees then the potential for a where we are at the moment to deteriorate slightly before then picking up into was the second half of next year and then the scenario where we see a big Pullback essentially from where we're at at the moment. I'm talking about general economic activity and economic performance by real gdp here So the varying degree of pace of recoveries and what is this contingent on? Well, it's really contingent on two main things the development of kovat 19 as a virus particularly as we go into the seasonal part of the year when weather conditions will be Will be dropping in terms of temperatures and people generally start to get seasonal flu and so on but also then the speed and success Of trying to get a vaccine to market and then also its Distribution and so when overlaying those two things they've built basically a scenario model Where you've got kind of a being with kovat, you know, are we going to experience from the most? Loose which would be localized type lockdowns to deal with then the economy acting in the most efficient manner Just dealing with locking it down in very small parts As to not impede then economic progress all the way down to Community transmission remember The difference being that an outbreak in an abattoir is a much better situation than a community Transmission outbreak which would be something like what we've seen in the uk in the northwest for example So one factory place where it's contained is very different from an entire community Which is more consistent and widespread in its transmission And what could that mean? Well community transmission as people Actually become more indoors more cost of claustrophobic spaces That could then lead to national lockdowns. That would be by far the worst case scenario And then when it comes to vaccine development You know, there's obviously in multiple different trials that are happening from different pharmaceutical companies at the moment So could we end up in the best positive case? Which is multiple options Kind of late approval coming the end of this year to roll out then before the summer of 2021 And then social distancing can start to be unwound in the second half of 2021 Economy can then start to regain some momentum The opposite would be delays disappointing phase 3m trials No actual vaccine emerges until well into 2021 That means social distancing needs to remain in place that impedes then this mobility aspect for people to actually Start to resume normal activities and therefore economy and the economic response is very slow accordingly. So Yeah, I just thought this was super interesting. I did share these on my my twitter yesterday So check it out But I think this is why my endpoint is to link this to payrolls is why I don't think payrolls now is that important the Fed have kind of set their stall We know what the deal is there. They're not going to make any changes regardless of what this figure comes out as What's going to be important now is what does this figure look like? In oct in october in november, you know when a lot of these fiscal measures start to be unwound And that leads us on to then really Our man rishi in the uk the uk chancellor Reading quite a lot about this this morning and that argument I've just been talking about about the This kind of the uk in particular globally is going to be one of the first countries in europe To start unwinding these emergency measures post-covid as I said yesterday this kind of phrasing of there's no such thing as a free lunch And you know, ultimately you're going to have to pay in taxes and it's very crude and simplistic But the idea here is that his job here is he needs to manage an economy But he also needs to be somewhat fiscally prudent in order to think about then managing it appropriately going forward So it's a balancing act But there's a couple of things here that I thought were quite interesting and for one politically Boris Johnson's government's under quite a lot of pressure at the moment Sunak has warned that rebels in the uk conservative party will vote against the autumn budget if the government proceeds with tax increases, so it's been a particularly divisive Kind of idea here about raising taxes as we start to wrap up and unwind and withdraw the emergency measures post-covid This could obviously have some real tangible impact on the actual lives of people in britain on the on the street if you like So not only have you had number of policy u-turns which have hurt the conservative party's reputation and therefore their performance in the polls Things could get a lot worse if you start to see internal fragmentation within the Tory party party in itself So all of this I think it's quite interesting because obviously there's quite a few other things coming up like brexit as well to deal with So just to give you a flavor Sunak this week ended a subsidy program aimed at helping the restaurant industry A ban on evicting residential tenants in rent arrears is set to expire on the 20th of this month furlough ends in october Think tanks estimate that as many as three million people Will still be on furlough at the point of which it ends three million Soft deadline for brexit obviously coming at the beginning of october According to the times newspaper this morning Senior officials close to borrower see only a 30 to 40 chance of a trade agreement with the eu as it stands right now Another point many mortgage borrowers will face the end of their payment holiday Remember a lot of people were given the choice then to just just park their mortgage payments. Yes, they got to pay it Later at some point but while they're under pressure perhaps being furloughed on No real job security to just alleviate some of that short-term pain of those payments going out They're on a holiday, but that's got to come to an end as well So there's a number of things happening here and I know the The pound obviously has been you know on resurgent Resurgency of the pound over the last couple of months, but for me that's predominantly being based on the dollar Not really the pound and a lot of these things I think will start to build up With the uk over the period really I think of the coming weeks and what actors are growing fundamental weight on sterling And particularly we've had a bit of a bounce of late in the the Dixie, of course If that gets sustained to any degree, you know, I think there's there's different room for a bit of a pullback there in cable What's my overall take here? Well, germany as a as a comparison have pledged to extend its furlough program into next year As the crisis goes on but obviously from a fiscal situation germany is quite different to the uk, but politically I just wonder whether Given all of the risks associated with so many things expiring all at the same time and compounded by brexit uncertainty I don't see much in the way of choice that sunak has other than to start pushing these things out and extending them for a little longer Before we have a little bit more clarity on the economic situation and the bracelet situation So I wouldn't be surprised if He has to swallow a little bit of a further worsening in the uk's debt situation in order to just offset a lot of this Because there's also a lot of political pressure as well to manage The potential threat of mass job loss. So yeah, just thought I'd cover these things This is obviously one of the main threats then This is a little snippet from the ft of why this is so critical Is that here are some numbers associated with with these types of programs that are due to basically end And the fear here is that many banks are worried about a potential wave of defaults Meaning that without any type of state guarantee They could be pursuing thousands of struggling companies through the courts And that's because the government's lending scheme has provided only 53 billion pounds to some 1.2 million companies through three programs bounce back loans guarantees for small business loans and the coronavirus business interruption loan scheme And if all of these start coming to an end, you know, can these companies legitimately survive Particularly these small medium-sized firms which do make up the proportion of the main types of companies that employ people in britain So push comes to shove I think he rolls it over for another month or two at a minimum But we'll shall see until that comes to fruition I think it will add a growing potential negative weight to sterling's outlook in the short term All right, final few things This is looking at crude oil crude obviously came off quite sharply yesterday in tandem with equities Looking at the chart here technically with oil Again, I think perspective is probably warranted I know we've been generally fairly bullish with oil and I don't really Think that I would turn bearish at this point just given what's happened over the last two days I do think that As we come down here, we're in close proximity on the low We printed yesterday to the 40 dollar handle and the futures just around 20 cents above that I do think though that the around 38 50, which was that 10th of july low Also on the low on the 30th of july and you got some points before there at around 40 39 50 As well where the price has previously reacted to I still think these are pretty good areas of support One thing that people are looking at is The volume of crude arriving in china, which is the world's largest importer of crude Is set to slow according to data from refinitive after rising for five straight months But generally speaking, I think from some of the chinese data we've had things have been relatively stable I guess again Exactly the same argument that we were looking out for with oil. It all comes down to this Yeah, the the shape of the economic recovery and so therefore de facto how these assets will react And particularly with the consumption of oil the net demand for that product Is intrinsically tied towards the expectation of the economic recovery So the clovered 19 situation how that is dealt with by authorities and The the speed of which a vaccine is forthcoming Will be very telling then for the performance of oil going going forward in the medium term But yeah a bit of a bit of profit taking perhaps. I mean, you know when you look at oil These are looking at weekly performances And you can see there's it's been very seldom the case over the last four months where we've traded negative on the week The last time we did was back in the transition through july into august And that also happened as well in that june to july period as well. So Yeah, it's been flat lining a little bit of late But just reversing probably the last four weeks or so of gains to to be pretty scratched at the moment. So again I'm a little bit reluctant to start putting out new Bullish or bearish views. I think you've just got to wait to see how the price unfolds Through today and then going into the resumption of trade next week before we could see about how sustainable The market kind of reaction was to what happened yesterday more broadly. The other thing finally with the fed Two fed officials spoke yesterday and you remember this is before the blackout period We are now into a blackout period Which means then for anyone new to markets the no fed official can speak now until that fmc meeting on the 15th and 16th of September this is kind of their routine in order to divert any type of leaks and rumors and things ahead of the official unveiling of their latest announcement that september meeting key, of course a little bit more definitive details Perhaps that the market is awaiting in regards to following on from average inflation targeting being adopted as outlined by pal Jackson Hole speech And also we get our latest update in the summary of economic projections What these two chaps said then the final two speakers Were evans and bostic and they played down The chances of updated public guidance on the path of interest rates of their upcoming policy meetings suggesting There would be more clarity on the outlook for the economy first and there were some hints that that might not really come until the spring and so I think that's pretty Standard management of the situation the markets have heard what they need to a little bit extra dovish policy Maneuvering with the inflation side of things from power Jackson Hole. There is no real Way to quantify it accurately. I think about projecting the past of us economic performance of unemployment Even inflation As well as then interest rate predictions over time Because of the degree of uncertainty with the number of things that are happening in q3 q4 of this year, you know, whether that's the The u.s. election, of course What's going to happen with co vid what's going to happen with the vaccine? All of these things mean I think the Fed are right to adopt this kind of more generic blanket statement type um Approach rather than being definitive because if they're definitive they're only going to end up being wrong And they can't really afford the loss of credibility in that sense. So not too much to interpret I think from that I think that's the prudent approach that they've taken And perhaps that could be what the outcome is really to the september meeting that anyone who's looking for more definitive concrete kind of Milestones as to what might indicate in the future the Fed to take some kind of action. I think they could be disappointed to be to be honest All right quick finish off and a look at the calendar A few things to be aware of then This morning Construction PMI in the uk is not really going to be too much of a factor. So really it's all about the afternoon Non-farm payrolls. Don't forget Um, I will be covering non-farm payrolls live with the team On an exclusive webinar conducted via zoom All you need to do is go on to the link in the description of this video and on the comment section You can register for that still some spots left to join us. It is a capped event So yeah, join us for that. We'll do a full thorough more Advanced kind of rundown ahead of that and we'll have all the members on the team on board So that's really the focal point for today Otherwise then the other thing I I guess I'd be very keen to look at are those key technical levels that I mentioned in us indices and Yeah, let's just see how things play out at this point Okay guys Have a good session ahead. Hopefully I'll see you live for payrolls later And if not, have a great weekend and I'll see you on monday. Thanks very much


There's a couple of big comments that we have not pulled yet, especially out of raw yellow mega pack. There's like $2, $10 comments. So we have a destiny hero plasma. That could be money because it's actually playable. Judgment dragon, not sure about that. I mean, all these cards are good. Flamely man, I know his money. Dimensional prison, malicious. Also one of the best cards in the set. Secret rare. I think it's about 15 bucks. Cobalt eagle. Wow. It is mowing outside. All right. We got grandmaster. Flute, Electrum might be okay and rainbow dark dragon. So very nice pulls at that time. That was our best pack for sure. Raw yellow. It feels like maybe a smaller set in raw yellow mega pack. Yeah. Look, another good common test tiger. Only the second one of those, but so pretty good. Hero mask is good. Groth, Dimensional prison, crazy ultra. E-emergency call. Another huge bag. DD crow. If there's a Prismo or a Rio back here, we're going to die. Oh my goodness. Okay. It wasn't that time, but that was still like a $50 pack.