So this lecture is part of an online course on commutative algebra or possibly homological algebra and will be about the X groups where A and B are modules over a ring R and we're going to define these groups. So they're sort of analogues of the tents of the tall groups. So you know we can take a tent of products of two modules over a ring and from this tent of products we defined various tall groups of A and B by taking a resolution of B and tensioning with A and then taking homology of it. And instead of looking at the tents of product of A and B we can also look at the group of homomorphisms from A to B which will be an R module as far as commutative and these will give rise to groups, the X groups in a similar way. So how do we do this? Well we're going to be very lazy. I'm just going to go through how we defined tall groups and then wave my hands and say we do a similar thing to get X groups. So let's have a slightly closer look at how we define tall groups. In fact you notice the definition of tall group works for any right exact functor. So let's take a right exact functor f of A. So what does this mean? Well it means that if we've got an exact sequence nought goes to A, goes to B, goes to C, goes to nought then f of A goes to f of B, goes to f of C, goes to nought as exact. However this map here need not be onto so it's not left exact. And a typical example of this is f of A equals m tensor A for some fixed module m. So tensor products are right exact. And we can define the so-called derived functors or left derived functors li of f as follows. So these are going to be a generalization of tall. So first of all let's take derived functors of A. We take a resolution of A by three modules. So we have A and r to the n nought and r to the n1 and so on. And then instead of taking a tensor product with B, which is what we did to define tall, we just apply the functor f.  So we sort of delete A, f r to the n2 and so on. And we notice that this is not an exact sequence in general. There's no reason why it should be because f doesn't preserve exactness, it only preserves right exactness. And thirdly we take homology. And the homology of these would be called the left derived functors of f. So the homology of here will be l nought of A and the homology, sorry, l nought f of A. And this will be l1f of A. I mean the homology of this will be l1f of A and so on. And you can see in the special case when f is m tensor A, this is just the definition of the tall groups we had. And there are some basic properties of these derived functors.  Secondly, li is well-defined. So for that we just copy the proof that tau is well-defined. You remember there was something about a homotopy between maps of chain complexes and so on, and that all works for any functor. Thirdly, it's functorial. Again, this is very easy to check. And fourthly, we get a long exact sequence, essentially by copying the proof of the long exact sequence that we had for tau. Now we notice that for this we do not need free modules r to the n i. We can use projective modules. In other words, we can take a resolution of A by projective modules p nought p1 and p2 and so on. And the reason for this is the only property of free modules we ever used was the fact they're projective. So we repeatedly used the fact that if we've got a map from b to c that's onto and if we've got a free module that maps to c, we can lift it to b. And that was the only property of free modules we used. Well, that's just the definition of projective modules. So we may as well use projective modules. Well, that doesn't really gain us anything because we may as well just use free modules. What's the point of switching to projectives? Well, there's a dual notion of projective module called injective module which says that if we've got a submodule c of a module b and we've got a map from c to i, then we can always lift this to a map from b to i. So this is the definition of an injective module.  So what's this good for? Well, suppose you've got a left exact functor. So this means that if nought goes to a, goes to b, goes to c, goes to 0, is exact.  And there's an obvious example of this. Let's fix a module x. And let's put f of a to be hom over r from x to a. Then f is left exact. And you can see this, if nought goes to a, goes to b, goes to c, goes to 0, is exact. And so as nought goes to hom, x to a, goes to hom, x to b, goes to hom, x to c. And you can ask, is this always onto? Well, not necessarily. For example, we can take our standard countere example, nought goes to z, goes to z, goes to z, over 2z. And we take x to be z over 2z. And then if we look at nought, goes to hom, z over 2z to z. This maps to hom, z over 2z to z. This maps to hom, z over 2 to z over 2z. And this group is 0. And this group is 0. And this group is not 0. So this map here is not onto. So hom from x to x to something is left exact, but it's not generally right exact. And now we can define derived functors of this. So we're going to define the right derived functors of f as follows. So if we want to define this on a, we take an injective resolution of a. So we take nought goes to a, goes to i nought, goes to i1, goes to i2, and so on. Then we apply f and forget about a.  And thirdly, we take the homology of this. And the homology groups of this complex are going to be the right derived functors f of a. And everything works just like the left derived functors we defined earlier. So we just want to reverse every single arrow. So instead of looking at right exact functors, we look at left exact functors, which are the same except you've reversed all the arrows and so on. And this is the usual properties. It's well defined. And our nought f is just the same as f. And it's functorial. And we get a long exact sequence, which I'll show some examples of in a moment. So we can do this for, if f a is hom x to a, then these right derived functors are i of f of a is just the functor x of i over r of x and a by definition. So it should be hom over r. So let's work out some examples of this to get familiar with it. So we'll do the simplest example, which is we just take r to be z. And we need to know what are the injective modules for r? Well, we'll study injective modules in a little more detail in a later lecture. So let's just comment for the moment that injective modules are the divisible ones. This is true over any principle ideal domain, but not true in general. But it's fine for the integers. So let's calculate x of z over nz with an nz. So this is going to be x i over the integers. So we need an injective resolution of z. So we need to map z into a divisible group. So we get nought goes to z, goes to, let's, we can just take the rationals. Rationals are divisible. And so the rational is modulo z. So we get a very short resolution. So these are the injective ones. And now in order to work out x, we apply hom of z over nz to whatever this is. And we delete the z. We forget about the z. So we get nought goes to hom z over nz to q, goes to hom z over nz to q over z, goes to zero. And now we need to take the homology of this sequence. I should call it the cohomology. So there are no homomorphisms from z over nz to q. So we get nought here. And the homomorphism from z over nz to q over z is just isomorphism from z over nz. So the homology of this bit is zero. So this is x0 of z over nz to z, which is the same as hom of z over nz to z. And this is now x1 of z over nz and z. Well, we should really work out this group for all. I think I should have said here that I'm taking n to be greater than zero, because if n is equal to zero, you get something a bit different. Then this isn't right. You see, this is only zero if n is not equal to zero. So now let's work out x of z and z. So this would be the case when n was equal to zero in the previous one. Well, again, we take the resolution nought goes to z, goes to q, goes to q over z, goes to nought. And this time, if we apply hom z to something, we get q goes to q over z, goes to nought. And now if we take the homology of this, well, we just take the kernel of q, goes to q over z, which is just z. And this map is on toast of the homology of zero. So this is x1 of z with z. And this is x0 of z and z. And we should finish off just by calculating x to z over nz, z over mz. So now we need to take an injective resolution of z over mz. And we can do this by taking nought goes to z over mz, goes to q over z, which is divisible so injective, goes to q over z, goes to zero. And this is just multiplication by m. So here's our injective resolution. And now we apply hom z over nz to everything except for this, which we forget about as usual. So we get nought goes to, well, hom z over nz to q over z is just z over nz. So we get z over nz goes to z over nz. And this is multiplication by m. Now we take the homology of this. And the homology of this is easy to work out. Here we get z over the highest common factor of m and n. And we get the same here. So this is x0 of z over nz, z over mz. And this is x1 of z over nz, z over mz. And the x2 and x3 and some always vanish. In fact, they do for any modules over the integers, as we will see in a moment. The next question is, why is x called x? Well, this is short for extensions. And the reason is that x1 classifies extensions. More precisely, x1 of c and a classifies extensions nought goes to a, goes to b, goes to c, goes to zero. These are extensions of c by a or possibly extensions of a by c. I can never remember which round they go. And two extensions are called, you can divide extensions into equivalence classes. So two extensions are considered to be the same if there's an isomorphism from this b to this b. And it turns out that the isomorphism classes of extensions of a and c correspond to this group here. Let's see why this is so. Well, first of all, suppose we've got an extension, nought goes to a, goes to b, goes to c, goes to zero. Well, then we can look at the long exact sequence of x groups. So we get nought goes to home c to a, goes to home c to b, goes to home c to c, goes to x1 c to a, goes to something where we don't care what happens after that. And now home c to c has an obvious element. It has the identity map as a sort of canonical element in it. And we can just take the image in here. So we get an element of x1 c and a. So x is just d. I'm not sure we really call this boundary map d. But it's the image of the identity map under this map from home c to c to x1 c to a. So that gives us a way to get from extensions to elements of the x group. Conversely, suppose we've got an element x in x1 c a. Well, we can get an extension out of it as follows. What we do is we take a resolution of a by injective modules. And now the x groups are the homology of home c to i0 goes to home c to i1 goes to home c to i2. So x is in this group here. That means x gives us a map from c to i1 whose image in i2 is zero. Now what we can do is we construct an extension of a by c just by taking the product of i0 and c over i1. In other words, we take the elements of the product i0 and c for the same image in i0. And now you can check that here we have an extension of c by a. So this shows how to get from extensions to elements of the x group and how to get from elements of the x group to extensions. And you can check that these two correspondences are inverses of each other. So this really does classify extensions. We can just see a very quick example of this. Suppose we take a equals c equals z over 2z. Then we calculated x1 of c and a was isomorphic to z over 2z. So it has two elements. It has an element zero and an element one. And these correspond to the following two extensions. We can take nought goes to z over 2z goes to z over 2z times z over 2z goes to z over 2z goes to zero. So this is the split extension. It's just a product of these two. And the split extension is the one that corresponds to the zero element. This is easy to check. And there's a second extension. We can take z over 4z and this is non-split. So non-split extensions correspond to non-zero elements of the x group. Well, for Tor, we had this balanced property that Tor A B is isomorphic to Tor B A. This obvious analog of this is certainly not true for x. We can see that x of A B is not equal to x of B A in general. That's not even true for x zero. However, there is a sort of analog of the balanced property. This corresponds to the fact that we can compute Tor by taking a resolution either of A or of B. And similarly for x, we did it by computing a resolution of using a resolution of B. But we can also do it by using a resolution of A. However, you have to be a little bit careful. The problem is, first of all, that HOM of C to A is contravariant in C. So that means it sort of reverses arrows. So if we've got an exact sequence, north goes to A, goes to B, goes to C, goes to zero. Then we get a sequence HOM C to M, HOM B to M, HOM A to M. And the arrows go the wrong way. And this map here is not onto. And this gets a bit confusing because does this mean is this right exact or is it left exact? Because if it was right exact, we should take a projective resolution of C. And if it's left exact, we should take an injective resolution. Well, it turns out that we can calculate x of i C to A by using a projective resolution of C. And the result is the same as using an injective resolution of A. So this is the balanced property of x. It means you can either use a resolution of C or a resolution of A. And the proof that you get the same both time is sort of vaguely similar to the proof that I gave for Tor. So I'm not going to go through it. So we should have one example of x to the i not equal zero for i greater than one. So over the integers, x to the i always vanishes if i is greater than one. And it's not difficult to find examples. We could more or less copy the example we gave for Tor. So we just take r is the ring of polynomials in one variable modulo x squared. And we're going to take the modules A and C to be just k, which is r over x. And we want to calculate x to the i of C and A. So let's take a projective resolution of C where we can do this by taking r goes to k goes to naught and then gets a resolution of this as we had for Tor. And now we apply hom of this to A. And the only thing you've got to be careful about is that arrow is now going the opposite direction. So we get hom r to C goes to hom r to k and so on. And these are all just k. And the maps between them are all just zero. So if we take the homology of this, we just get the groups k everywhere. So we find the x of i of k and k is just isomorphic to k for all i greater than or equal to zero. Okay, so that's the end of the summary of x groups. There's one problem with the x groups that we sort of didn't go into too much detail. We said we used an injective resolution of a module. The problem is we haven't actually shown that a module has an injective resolution. So the problem is, are there enough injective modules over a ring? In other words, for every module, can we find an injective module that it embeds into? And the answer is you can. In fact, there's even an almost canonical injective module that's invented to cause an injective envelope. So we will discuss these next lecture.


Throughout the past four episodes we've discussed over population in general what effects life extension may have on the size of the human population, how population growth can be contained and what science could do to allow us to support a larger population. Fundamentally, I am an optimist. I'm not 100% sure whether life extension therapists will dramatically increase the size of our population or not, but I am more than convinced that science will prepare us for that hypothetical moment well before it actually arrives, if it ever will. Still, for the sake of the argument, suppose that the worst-case scenario came true. Suppose life extension pushed our numbers through the roof and science failed to prepare us for that. Suppose we did become too many people. I know from experience that many people would argue the most ethical way to prevent this from happening is to ban life extension. Never created in the first place, forget about it and let sleeping dogs lie. If you ask me, that's the most unethical way to prevent our population. Welcome to Extend, your one-stop YouTube show for all things life extension. If you're into this topic, click subscribe down below to follow our channel and while you're at it, click the little bell icon too. By doing that and selecting all notifications, YouTube will let you know every time a new episode is out. Before I continue, I want to stress that the views presented in this video are mine and mine alone. Ethics are not absolute, so I can only speak for myself. It's up to you to decide whether you share my views on the subject or not. The overpopulation objection to life extension can be summarized in a very simple way. If we created life extension, we would eventually end up with more people than our planet can sustain. Therefore, we should not create life extension. Of course, this objection hinges entirely on the assumption that life extension would inevitably lead to a population so large that we couldn't no way sustain it. And as we've seen in previous episodes, that's far from certain. But let's say that it is certain and that we only have either of the following two options to prevent overpopulation. More deaths or fewer birds. And as a side note, fewer births doesn't mean forced abortions. It means deciding to make fewer babies in the first place and acting accordingly. Now, which option is more ethical? The more deaths option means that someone must die. Living, breathing people who probably have dreams, ambitions, friends and family who will grieve for them and so on must give everything up and stop existing for the sake of averting a population crisis. The fewer births option means that hypothetical people who are not even in their mother's wombs yet will not be born. This, by the way, happens all the time. The number of people who could be born but aren't is astronomical and it completely dwarfs the number of all people past, present and future. The more deaths option means some of us and eventually all of us will lose everything themselves included. It means we'll all lose our one shot at existing and the people who care for us will lose us. The fewer births option doesn't involve anyone's death nor any losses of any kind on anybody's part. Some people may be unhappy about not getting to have as many children as they'd like, but frankly, I think that asking Mary and John to have fewer children than they'd like is more acceptable than asking Bob to die so that Mary and John can have more children. Besides, even if things got so bad that births had to be cut down to zero, this would not last forever. It would only last the time it takes to develop the science and technology to manage a larger population and as we've seen in the last episode, several solutions are in the works already now. Mary and John would just have to wait a little bit and when the length of your lifespan is indefinite, you can afford waiting. The assumption that managing a much larger population in a sustainable way will be forever beyond what we can do is simply not one that I'm willing to make because frankly, it's as dumb as a box of rocks. Maybe you think I cheated when I introduced the more deaths option because it kinda sounded like the deaths I was talking about weren't old people. After all, I said living, breathing people who probably have dreams, ambitions, friends and family who will grieve for them and so on must give everything up and stop existing for the sake of averting a population crisis. That doesn't really sound like I'm talking about an old person, right? It sounds like I was talking about a young person and no one, to my knowledge anyway, says that we should prevent overpopulation by letting young people die. Only old ones. Simply by not creating life extension and letting aging do its thing. Yeah, that's wrong. On several levels and for different reasons. The first reason is the hidden assumption that old people have no dreams, ambitions, purpose or anything to contribute left to do or possibly want.  I don't think I could say it better than Varda Joran, a 90-year-old woman, did on the Huffington Post during the COVID-19 pandemic. I am not disposable and I'm saddened that there are people who think age dictates whether a human life is worth saving. I can tell you that I and my loved ones want me to live for many years to come. I want to attend my grandson's high school graduation and see which college he'll attend. I want to see my older grandson, who is married, become a father. I want to continue my joyful life. I am unable to travel as extensively as I once did, but I want to visit Israel again. Just because I'm 90 doesn't mean I don't have things to learn and skills to hone. The second reason is that even if the first assumption was universally true, it would be so because of aging itself. For example, as Varda said, she's no longer able to travel as extensively as she used to. It's fully understandable that if your health keeps worsening, the list of things that you have the energy for shrinks, but the point of life extension is precisely that of preserving the exact same vigor, energy and health you had as a young person. The third reason is yet another assumption, namely that if a choice had to be made between existing old people and imaginary unborn ones, the existing old people would get the shorter end of the stick by default. Not only are they disposable and should die for the sake of potential people that are presently scattered around the world in the form of brainless chemicals, they also get to spend their last couple of decades in a state of worsening health, which by the way is precisely what leads them to eventually dying.  That's another point that I will never tire of repeating. Life extension or rejuvenation or what you will call it are just fancy names for medicine. Future medicine able to do something that present-day medicine can't, preventing or curing the diseases of aging by attacking their root causes. The root causes are several, grouped under the umbrella term aging. If life extension must not be a thing so that aging can keep the population from growing too much, it means that some of the diseases of aging need to stay around and kill people. That's easily done. We only need to pick which age-related diseases we want people to die of and make sure we never attack their causes. Personally, I'm not comfortable making that choice, especially not on behalf of others. To sum this all up, if the problem is that we don't want to have more people than we can afford having, we should not make more people than we can afford having. It's as simple as that, and I think it's far more fair and selfless than telling old people, look, we kind of need you to die, you know, we'd rather have more children. As a side note, one of the papers I consulted during the making of this episode series says that during the 1994 UN population conference in Cairo, several approaches to slowing population growth were considered, all of which we discussed in the third-over population episode. Unsurprisingly, they were all aimed at reducing births, not at increasing deaths. No one proposed to drop medical research against the diseases of aging as a solution to population growth. I've heard more than once that wanting to extend your life is supposedly selfish, and I don't really see why. There is nothing wrong with not wanting to die. I guess this belief stems from another one, that is that there is a right length of life, a more or less set number of years that somehow is the right number of years we should live, and that if you want to live past that, you want more than your fair share. This belief is entirely groundless. There is no criterion scientific or otherwise to decide whether others have lived long enough or not. In my opinion, there is only one person in the entire universe who gets to decide if and when you've lived long enough, regardless of everything. That person is you. Others don't get a say, and this kills both ways. Nobody can force you to die, nobody can force you to live. Life extension is not selfish, and it wouldn't be even if it meant making fewer babies for a while. We're not talking about hurting or killing others for the sake of saving yourself. If we banned life extension for the sake of making more children while avoiding overpopulation, now that would be selfish. It would indirectly force older people to die, so that other people can fulfill their own wishes to have children without having to worry about any consequences. Now, the choice to make children is not selfish in general, but in a hypothetical situation where making children and life extension were mutually exclusive, it very much would be. People don't see it that way because in reality there is no identifiable victim. Saying that older people need to make room for the young is easy. It would be much less easy if you knew for a fact that an old person would die specifically to make room for the baby you were planning to have. Let's make a thought experiment. Forget life extension and say that you lived in a country so overpopulated that the government enforced a one birth one death policy. You get to make all the babies you wanted, but for each of them, the government would select an older person according to certain criteria and euthanize them. You wouldn't get to know who, but you would know for a fact that as a result of your choice, someone would die. Do you find this more ethical than not having babies? Probably not, and yet it's not very different from a situation where we banned life extension for the sake of baby making. The only differences are methodological.  Also, no old person would die to make room for any specific baby, but all of them would for the sake of making room for some babies anyway. If we decided against life extension for this reason, we would all be guilty of countless age-related deaths, not only those of our own generation, but also those of all the following ones for whose sake we decided against life extension in the first place. Look at this chart. It shows the total death count in 2017 broken into age brackets. The peak on the left is children under five years. Over 5.3 million of them died that year. It's an average of about 15,000 children per day, which our world in data rightfully calls an everyday tragedy of enormous scale that rarely makes the headlines. Now look on the right at the deaths that occurred at age 70 and above. They add up to over 27.7 million dead that year for an average of about 74,500 people per day. Five times more older people died in 2017 than children and that's been the case in any given year for a while now. Yet, while 5.3 million dead children in a single year is a huge tragedy that rarely makes the headlines, 27.7 million dead elderly in a single year is what? Oh, it's just the circle of life, the natural order of things, and it never makes the headlines because we don't see it as a tragedy. We don't think it's even worth mentioning. At best, we see it as a way out of population growth. It's a crystal clear sign that our society sees older people as second-class lives that don't matter as much as younger lives and I really don't think we should see that way, let alone if we are talking about ethics. In a way, I think this whole debate is pointless and I already told you why. We're never going to have to choose between having children and life extension. Life extension will not happen overnight and it will not suddenly triple our population. It will take a long time for life extension therapies to be effective and widespread enough to have measurable demographic effects. On the one hand, that's really sad. It means many people will die because we won't be able to save them. On the other hand, it will give science all the time it needs to do what it does best, finding a way for us to have our cake and eat it too, making both parties happy in the process. This episode concludes Extents Over Population series. I hope you enjoyed this as well as the other episodes and if you didn't see them, check out the playlist up here. I also hope the views I expressed in this episode didn't piss too many of you off, but especially if they did, let me know in the comments below. As long as we keep it civil, discussion is always welcome and, you know, aren't lights live long enough to see life extension happen, so please don't bite my head off because of ethical disagreements, will you? If you liked this episode, consider giving it a like and sharing it with your friends, and maybe subscribe to our channel. If you do, don't forget to click the bell icon and then select all notifications to be always notified whenever we publish a new episode. If you'd like to extend Extents Life, you can do so by becoming a supporter of Lyspyn.io, the organization producing the show. Visit Lyspyn.io slash hero to find out more. Take care and I will see you next time.


This is the Late Round Podcast with your host J.J. Zacharias. The Patriots have the second-lowest pass-to-rush touchdown ratio in football right now. Arizona is the lowest. But the Patriots have run for 12 touchdowns while throwing for eight. Since Tom Brady entered the league, New England's never had a pass-to-rush touchdown ratio below one. They were at one during his career, meaning they threw for as many touchdowns as they ran, but it was the year Tom Brady missed the entire season. So essentially what I'm getting at here is, I would expect this ratio to start to favor the pass a little bit more moving forward. I'm not sure Brady is a buy candidate because buying a quarterback who doesn't run isn't generally a smart idea, but we should expect some positive regression to hit.