I feel like I should sit down and listen to the guy who was described by Mark and John. Thank you all very much for being here. I think as Mark conveyed I have to feel very very lucky to be here. I say, I mean people don't ever ask me this, but I often tell them anyway that I have the best job in the world. This is the best job in the world and I feel very lucky to be here and I'm very flattered that all of you are here. I'm gonna, I'm gonna do a little bit of philosophy, a little bit of philosophy of language and the title and the abstract promised you that I would talk about why the world is safe for metaphysics and other clueless theorizers. And it's gonna sound to you as I go through the talk that what I'm actually discussing is classical philology and geography and things that in general don't sound or seem like metaphysics to those who have any passing, any passing familiarity with the subject. But I assure you that the whole time I am talking about metaphysics and that's because I'm gonna be talking about the conditions under which some bit of terminology picks out delineates a certain subject matter, refers to it. And the problem that I'm going to address concerns what it is that makes it the case that some little bit of language picks out or refers to a particular subject matter. So the question is what makes that the case or in virtue of what is that the case? And it turns out the answer might seem straightforward but it's not altogether straightforward. So here is a dispute that you might have in classical philology. So one classicist might claim that Homer was an actual historical figure. Another that Homer is a completely mythical character made up by somebody along the way. A third that Homer is some amalgam of several historical individuals with a thick layer of myth overlaid. And as you're listening to this dispute it might strike you that the disputants appear to have no common conception of what the nature of the thing that they're talking about is. So if you were to ask for instance somebody who held the first view that Homer was an actual historical figure, who are you talking about? Who's this Homer guy you're talking about? Then if they were particularly honest and conscientious they might give you a speech like this. Well I think that this is an actual historical figure who wrote certain poems. But to tell the truth the historical evidence is not dispositive. So maybe Homer is not a he but an it, amalgam, a group of individuals. Or maybe it's nothing at all. And after he gives you this speech you might think to yourself actually there's a certain sense in which this person doesn't really know what he's talking about. We have radical ignorance concerning the nature of the very thing that's whose features are under dispute. Okay, here's what this has to do with metaphysics rather than classical philology. This is the usual case in metaphysics. So Mark mentioned that I am writing a book on grounding. Grounding is the notion that is expressed by that in virtue of what expression right there at the beginning of that question. So I work on that idea. And as a matter of fact if you start to say the first interesting thing about the features that that idea has you get an academic food fight almost immediately. Nobody can agree, there's no universal agreement at least, among the experts on what the nature of whatever it is we're talking about when we're talking about what it is in virtue of which something is the case is. All right. So we don't have any very good conception of what it is we're talking about. And in a certain sense we don't know what we're talking about. How then could we be talking about something? What makes it the case that we're nevertheless talking about a particular individual? Along comes the causal historical theory of reference and I can't resist quoting this passage from its progenitor Saul Kripke in Naming Necessity in 1970. So Kripke is trying to give expression to a similar question. How is it that people who use the word Richard Feynman but don't really know who he is? How do they manage to talk about him rather than say Murray Gellman or some other relatively moderately famous physicist? And here's his answer. Someone's born, his parents call him by a certain name, they talk about him to their friends, other people meet him. Through various sorts of talk the name is spread from link to link as if by a chain. A certain passage of communication reaching ultimately to the man himself Richard Feynman does reach the speaker. The speaker is then referring to Feynman rather than Gellman even though the speaker can't identify Feynman uniquely. So what's the idea here? The idea here is that by and large we have uses of expressions that are derived from other people's uses of expressions. So now if you didn't before you have the use of the name Richard Feynman. You have no very particular particularly good idea of who Feynman is, right? Your use is derived. How do you manage to talk about that guy? Well you manage to talk about him because although you don't know who he is, your use is derived via a use, a chain of use transmission from somebody's use and that person knows who he is. So you are borrowing that person's ability to isolate a reference for his name Richard Feynman and deploying that. So the original uses are going to refer in virtue of the users associating conditions that single out the reference. So in the case of Feynman the original users are presumably his parent and if somebody asks the parent who is Richard Feynman at the beginning then they would probably say this child here. Okay that doesn't much help metaphysics because it relies on the idea that the original users of the expression know what the heck they're talking about and I am here to tell you that in metaphysics this is often not the case. Even original users of the expression make claims about the nature of the subject matter that's under discussion that are hotly disputed and if the person is reasonable they will recognize that although they have an opinion on the matter that opinion doesn't amount to anything like firm knowledge. Okay fortunately or perhaps unfortunately the same thing happens for ordinary proper nouns in ordinary language and this is known as the Madagascar problem. So here's a little bit of etymological and geographical history. So in the case of Madagascar says Isaac Taylor a hearsay report of Malay or Arab sailors misunderstood by Marco Polo has had the effect of transferring a corrupt form of the name of a portion of the African mainland Madagascar the word is etymologically related it turns out to Mogadishu and the word that Marco Polo picked up was a word for that area of the world emphatically not an island right okay. So it transferred a corrupt form of the name of a portion of the African mainland to the great African island so that's what happened. So I'm going to indulge in a little bit of historical make-believe because the actual story is quite complicated but you're to imagine there's Marco Polo in the middle and he had a source according to Taylor and Arab or Malay sailor who said something about Madagascar using the word Madagascar right okay and Marco Polo picks that up makes a mistake messes up and then inscribes in his diary Madagascar is the great island off the coast of East Africa that formed the source for a memoir that was published in Renaissance Italy right and now C is you you're using the word Madagascar and you might be instructing for instance a child in geography you might be telling that Madagascar is the great island off the coast of East Africa okay here's some facts first your use there up see that's derived by a historical chain of use transmission from a down there right and when you say that Madagascar when you tell the child Madagascar is the great island off the coast of East Africa what you say is true and it's true because when you say Madagascar you're talking about an island not about Mogadishu right um but a does not refer to that island so it looks like we've got a use that's derived from an original use but the referent of the original use is not the referent of the derived use so that looks like a problem for the causal historical theory of reference and now for the sake of exposition I'm gonna make a supposition this is a harmless supposition and as I said there's reason to believe it's historically false but I'm gonna suppose just for the sake of illustration that when Marco Polo wrote Madagascar it's a great island off the coast of East Africa in his diary that was the first use in the history of the universe of a word Madagascar that refers to the island so Marco Polo is the first user of Madagascar to refer to the island right okay so let's look at some extant proposals for how to deal with this kind of case case of reference switch so what I'm gonna do this is a plan is I'm gonna talk about the Madagascar problem then I'm gonna talk about a different problem one so the Madagascar problem's been discussed a lot in the literature so the problem has been discussed almost not at all so I'm gonna talk about a different problem and I'm going to emphasize certain similarities between the two problems that's strategic because I think both problems have the same solution and then if I have time I'll look at objections all right so first proposal Marco Polo initiated a use of Madagascar for the island okay he wasn't trying to he was trying to convey information he messed up though so how come his use refers to the island answer his use refers to the island in virtue of him having associated the condition being the great island so if you asked if you asked Marco Polo hey Marco Polo what's this Madagascar thing you're talking about he would say hey it's the great island off the coast of East Africa right okay there's a problem and the problem is that he also associated different a different condition and a condition which singles out something else that is he also associated the condition being whatever my sources were talking about when they said Madagascar right and in fact this is a condition where he defers to his sources this deferential condition go ask my sources presumably characterizes his dispositions to identify a reference so let's suppose just after Marco Polo as inscribed Madagascar is the great island off the east coast of Africa he'd gotten a postcard from God or an angel whispering in his ear saying by the way Marco Polo nobody in the history of the universe has ever used Madagascar to refer to an island in fact your sources used whatever word they used Mogadishu maybe to refer to an an area on the African mainland right then presumably if he were a reasonable fellow he might have been disposed at that point to say oops I screwed up scratch out his entry and say Madagascar's not an island so he might be disposed to withdraw the claim that Madagascar is an island when he finds out that the island is not what his sources were talking about okay so we've got these two conditions both associated by Marco Polo with his use of the name Madagascar his inaugural use of the name Madagascar one of them singles out an island and the other singles out an area of the African mainland and now we need to know from the proponent of proposal one what makes the island one the right one right we've got two conditions they don't agree how come we should go with the island one rather than the mainland one okay second proposal second proposal is this that Marco Polo Marco Polo's use refers to the island in virtue of the islands being the dominant causal source of his associated body of information so he could be highly mistaken about the nature of the thing that he's responding to but as long as that island is the thing that he's responding to is the thing that is sending information his way that he's organizing into his diary under an entry Madagascar as long as that's the causal source then he's referring to the island he's referring to that dominant causal source okay so this is a little bit of historical make believe again because Madagascar was not the dominant causal source of his associated body of information but let's just assume for the sake of argument that this is right right okay this faces a problem so here's the problem let's suppose that I have been deputized by the FBI to find whitey bulger who is public enemy number one and they've received a tip that bulger is going to be at a philosophy talk at UVM on a certain date right and so following that tip I come to the talk and I find somebody who resembles whitey bulger in the mugshot and I say aha I found him and I start texting back to the FBI dozens of messages right whitey bulger is sitting with his right law right leg crossed over his left he's got a brown jacket and a red sweater on and so on and so forth right I'm extremely verbose and I keep this up for as long as it takes somewhere between a few seconds and a few decades for this person to be the dominant causal source of the information that I associate with the name whitey bulger right okay the problem is this okay so here's Johnny Depp right if you're familiar with philosophy examples and you may have already guessed that this is not whitey bulger right this is a lookalike a dead ringer right okay so this man right here is the dominant causal source of my associated body of information but my reports are false I've been taken in bulgers not here he's not the reference of my uses of whitey bulger despite the fact that I'm getting the bulk of my information from right here and in fact there seems to be no number of encounters with this man right here after which my reports will stop being false reports concerning the whereabouts and features of whitey bulger and start being true reports concerning the whereabouts and features of Johnny Depp so these are problems I think these problems with the extent of proposals are actually a symptom of a deeper problems I want to consider in detail that counterfactual case that I just described so Marco Polo inscribed the Madagascar sentence into his diary and then he's better informed about the nature of his situation and he withdraws right here seem to me some plausible claims first that his withdrawal would have been correct in that case he would have been correct to say nope Madagascar is not an island and on our assumptions that means that he would have been in a whitey bulger style case so what is this deeper problem with these proposals is actually every other proposal that I know of other than the nutty thing that I'm going to say here in just a moment you've got in each case you've got in this case an actual case in which that inscription of Madagascar refers to an island and a counterfactual case in which that inscription of Madagascar does not refer to an island and according to the associated conditions view Marco Polo is wrong in the actual case he's got this disposition to withdraw the claim when he is apprised of the nature of a situation according to the causal links view Marco Polo is wrong in the counterfactual case but both views seem wrong it seems like actually we've assumed this for the sake of argument Marco Polo is the very first person ever to use the word Madagascar to refer to an island but also in the counterfactual case he was correct when he said Madagascar is not an island was correct to withdraw his claim so it looks like we need a view which distinguishes between the actual and the counterfactual case so here's another problem and this is going to get us much closer to metaphysics okay so this I think well this is made up history but this is this plausibly tracks the causal history of our use of the name morning star right okay so we're to imagine somebody probably somebody Greek or Babylonian somebody like that let's use morning star to refer to the brightest star visible in the morning sky and then this person goes on to say and this is the first use of the word morning star the morning star is brighter than serious so let's suppose that that use B of morning star is the original use of the name right okay that refers to Venus that's where our one of our words for Venus came from right but serious not Venus satisfies the condition the brightest star visible in the morning sky why is that Venus isn't a star it's brighter it's not a star right okay so this it seems is what I'm going to call a bad dubbing a dubbing is an introduction of a name so a bad dubbing is a situation which somebody introduces a name I'm going to use the word morning star to refer to a certain thing and says what they're going to refer to they use a description the brightest star visible in the morning sky to say what it is they're talking about and they succeed halfway so they succeed in referring to something they managed to do it but this is what makes it bad what it refers to doesn't satisfy the description so Venus the reference of this original use of morning star is not the brightest star in the morning sky all right so how common is this how common is this phenomenon how often do we encounter bad dubbings well they seem to happen sometimes so here's some examples I won't go through the examples Morgan Stern is just a German name for morning star if you do metaphysics then you're constantly encountering them you're constantly encountering people who introduce technical terminology but they kind of misintroduce it they screw up the introduction in one way or the other right so again we have a problem the problem is this what makes it the case that original uses of expressions introduced in bad dubbings refer to particular individual rather than some other individual say the individual actually satisfies the description right okay two extant proposals these are going to seem very familiar to you first first proposal is Dubber dispositions so the proposal is this the Dubber the person who originates the name is disposed to identify the planet Venus as the reference to be when given the relevant astronomical information and there's a generalization to all cases which I won't worry about right okay there's an obvious problem with this which is this why do you think that's true right so you can imagine that morning star was introduced for religious purposes and it was very important for this religious observance that they single out the brightest star in the morning sky right if if that were the case and the person got a little postcard from God telling them the nature of their situation then it's plausible to think that they would be disposed to withdraw the assertion that the morning star is brighter than serious and where they would conclude the morning star is serious so we actually don't know what the Dubbers dispositions are and importantly we don't care we credit the Dubber with introducing a use of the term for Venus anyway right okay second proposal involves causal links so when the person says the morning star is brighter than Venus and their use of the morning star is responding to the planet not to any star that's what makes it the case that they're referring to the planet not to any star there are a variety of problems with this proposal but again there's a deeper problem again this diagram should look very familiar to you at this point so let's assume that the behavior on the right in the counterfactual case with a Dubber of the of the name morning star is told that's not a star and she withdraws as a result right then in the actual case we have an inaugural use of morning star and it refers to Venus in the counterfactual case the very same use does not refer to Venus refers instead to serious or so it seems according to the dispositions view the Dubbers wrong in the actual case according to the causal links view the Dubber is wrong in the counterfactual case and again it looks as if both views are wrong okay so time to say why the world is safe for metaphysics we've got some similarities here so we've got an actual case in which you have an original use of the name which refers to one thing and you've got a counterfactual case in which the original user is apprised of the nature of her situation and in the counterfactual case we're imagining that the user would have as a result of reasonable dispositions withdrawn the original claim the very claim that they actually made it seems as if the person would have been correct to do so and that in that situation their utterance would not have originated a use of the name for thing one maybe it would have originated a use of the name for thing to the area on the mainland or serious in the two cases maybe not so it looks as if in the counterfactual case the original use would not have referred to what it actually does so what we need is an explanation of the actual case that doesn't also apply to the counterfactual case to yield the conclusion the result the false result that say in the counterfactual case Marco Polos withdrawal of the claim that Madagascar is an island was mistaken he should have stuck to his guns or at least he would have spoke the truth if he stuck to his guns right okay so what's the difference between these two cases the answer I think is that in the two cases the very same act by the very same person puts them in a different social position so here's the case of the bad dubbing of Venus with the name the mooring star so in the actual case the originator stands at the beginning of a certain social practice a collective activity in which you have unwittingly participated so the originator stands at the beginning of the practice of using morning star to refer to Venus you're part of that practice by virtue of having a use of morning star derived from her use in the counterfactual case well she's not the beginning of any such practice okay so this is a philosophy talk we must have new terminology a referential practice is roughly a body of social practices associated with the use of a particular name like morning star Madagascar or some other expression to refer to something to eliminate some subject matter so in the actual case in the bad dubbing case the dumber inaugurates a referential practice that I'm going to call the morning star Venus referential practice it's the practice of using the word morning star to refer to Venus that's why I call it that it's got all the marks of a social institution so there are offices in this social institution you hold an office the office of participant user of the name so there are participants their experts their newbies their teachers and so on there are norms that characterize what participants are supposed to do with their proper role is so participants by and large for instance take assertive morning star utterances like the morning star is brighter than serious to be true or false depending on the features of Venus rather than say serious so that means that it's a norm of the practice that what I say the morning star is brighter than serious what I say is true rather than laughably false right there are various moves that you can make in this practice sometimes unwittingly so here's one move becoming a newbie by being initiated into the practice you might attempt to seize the role of expert by teaching somebody else by offering them an explanation of what it is that the morning star means or refers to and so on so it looks like it looks like we have all the marks of an informal unadmittedly informal social institution and we are all doing our part in the social institution in the counterfactual case we might be doing lots of stuff but we're not doing any of that right okay so here's my proposal that the dover's use of the word morning star is an original use of a name for Venus in virtue of the fact that it inaugurates a referential practice that use refers to Venus in virtue of the fact that the referential practice it inaugurates is a practice of using morning star to refer to Venus that's the view these two explanations carry over straightforwardly to the Madagascar case so Marco Polo didn't intend to initiate a social practice but he did and you participate okay now this paper has been around since 2013 and I have not tried to publish it and the reason I have not tried to publish it is in part because the view seems objectionable in a variety of ways so I want to talk about two objections do I have time to talk about two objections I do have time to talk about two objections the first objection is basically just an objection which is a very natural objection and which I'm going to discuss just to fill out the view just a little bit and the second objection really makes me worry okay so so here's the objection so we've got the original user person says the morning stars brighter than serious we say correct that's arrived uses you say the morning stars brighter than serious correct right okay so on the proposed view it looks like the original use refers to Venus because the subsequent uses do right and the subsequent uses refer to Venus because the original use does and that's a terrible explanation right we never explained what it is in virtue of which we refer to Venus we've got a circle right okay so the objection is basically on the right track it shows that we've been working with the wrong distinction so in the causal in the original statement of the causal historical theory that this distinction between derived uses of say Homer and original uses where in the best cases the originator knows what the heck they're talking about when they use the name that's the wrong distinction we're talking about social practices and we need to distinguish between derived uses and what I'm going to call authoritative uses so these are part of the norms that constitute the social practice of using morning star to refer to Venus right okay so which uses are the authoritative uses there are those uses whose associated conditions and causal ties to the putative subject matter do the reference fixing they determine what it is that the practice is a practice of referring to okay so now you're going to ask me okay I know in general what an authoritative use is supposed to be which uses are authoritative this is a little bit like asking who is the authority on the band kisses first three albums right this is an informal social institution and authority and these kinds of institutions is messy contested vague and so on that's life but the authorities are the ones who play the expert role in the referential practice and if you've ever had a conversation about kisses first three albums then you'll recognize that there are some people who style themselves experts and the rest of us right so in a lot of cases styling yourself an expert will be enough because people will let you fake it until you make it that's also the nature of an informal social institution so I think by the way that original uses are always authoritative because you say here's a new word uses in certain conversational contexts like naming ceremonies I hear by name this ship the Queen Elizabeth those are authoritative I think mapmaking may be an authoritative use so when the mapmaker writes Madagascar on a certain region of the map that has a certain air of authority when you read the textbook and you've got a little blue box around some particular term like you know by Madagascar we geographers refer to the great African island off the east coast great island off the east coast of Africa and as with kiss albums there's a high degree of de facto influence so is a significant element of serendipity in who counts as an authority right so there's the objections just reproduced up there for you the reply is this authoritative uses refer in virtue of associated conditions causal links and ties to whether authoritative uses in the same referential path practice so you can think of reference for the whole practice is determined by a vote so each use backs a candidate for a particular reference and each use has a weight the weight records the degree to which the use is authoritative authorities a matter of degree to right so Terrence is fairly authoritative on kiss albums and there are higher authorities okay who's the reference the reference is the winner of the weighted vote right okay so this allows us to give a very natural explanation of what's going on in that whitey bulger case so you've got the FBI FBI says this is whitey bulger they give you the mugshot okay so the mugshot in codes is basically a causal link to a particular man the man the shot is a mugshot of right okay the FBI's authority with respect to the whitey bulger practice is very high there's my use of whitey bulger my authority in the whitey bulger practice is nil and that's why no matter how many times I come into causal contact with Johnny Depp I can't switch the reference of what whitey bulger from the Boston criminal to the actor right my votes don't count okay again in the morning star we've got the originator so the originator says the morning star is brighter than Venus and they annunciate the condition being the brightest star visible in the morning sky and that happens to be serious that's a picture of serious over there supposed to be right the originator's authority is high right and then you've got astronomers and textbooks and so on and they're all saying the morning star is brighter than serious and they associate conditions with their use of the morning star that singles out Venus that's a picture of Venus over there in the morning sky their authority however is higher and this is one of the norms I don't have a lot of time to talk about this but one of the norms is this that you can acquire authority in a naming practice by being in a superior epistemic position so the astronomers and the textbook writers and so on know better whereof they speak than does the originator and that's why their authority is higher okay so that's the first objection and that just helps me limb the view here's the second objection is the one that worries me so this is a general claim to temporal inter internalism so basically what it says is this if you want to know what a speaker refers to on a given occasion want to know for instance what Marco Polo refers to on a given occasion when he inscribes in his diary Madagascar is the great island off the coast of East Africa then you got to look at conditions that obtain at the time or earlier right so the reference of a use of a term is determined by facts that are either contemporaneous with the occasion of use or in its past that's temporal internalism and it's highly plausible right the view that I've discussed I guess I haven't said this explicitly yet but the actual in the counterfactual cases with respect to Marco Polo's originating use of Madagascar they're exactly the same up to and through the occasion of use the inscription of Madagascar they differ only in what happens afterward so the view that I've just shopped for you is a view on which what happens at a future time makes it the case that Marco Polo's inscription right now refers to Madagascar rather than say to Mogadishu right okay so the objection is this temporal internalism is very plausible how can you bring about the past proposal at hand implies that temporal internalism is false so it inherits the implausibility okay this bugs me I have a couple of replies I do think temporal internalism is very plausible in the abstracts one of those general principles it sounds pretty good when you first hear it it's also true in most cases so the Madagascar case is a pretty weird case most cases are like the Feynman case right where the parents referent looks like it's more authoritative the parents use looks like it's more authoritative than subsequent uses of Richard Feynman with the possible exception of Feynman himself so so in that case you don't need to reach for review which violates temporal internalism but it does seem to me that there are clear counter examples they will strike you as weird but that's because it's temporal internalism is true in most cases all right so here's here's a linguistic stipulation I'm actually doing this right now so I want to introduce a new word Newman to it's called Newman to because this is modeled on an example of David Kaplan's where the introduced name was Newman one okay so I hereby stipulate that Newman to in my mouth is to refer to the first person born in the 22nd century according to whatever timekeeping conventions are enforced as of 11 p.m. 31st December 2099 okay so here's a little bit of history did you know that the island nation of Samoa moved the international dateline in 2012 so they basically leap forward two days at midnight okay so they're now more or less the very first highly populated place where the 22nd century begins okay so I'm pretty confident they're not going to switch back like they switched because they wanted to be on Australian and New Zealand time didn't want to be on American time basically right so I'm pretty confident they're not going to be switched back so here's my conjecture Newman to the first baby to be born in the 22nd century according to whatever timekeeping conventions are in place on at midnight on 31 December 2099 that Newman to will be born in Samoa that's my conjecture it's either right or wrong right and it'll be right or wrong depending on who Newman to is right however who Newman to is may depend on future timekeeping conventions so for instance if Samoa switches back and decides to go with American Samoa again then Newman to will not be born in Samoa right okay so that is the failure of temporal internalism because whether my conjecture the conjecture that I just enunciated is correct depends on what happens in 2099 what happens in the future okay so temporal and internalism highly plausible but in weird cases it fails and it just so happens that an awful lot of metaphysics may turn out to be a highly weird case so let me end with a coda how should you interpret somebody who offers a new bit of terminology in the course of theorizing happens more or less constantly right so here's what they say let in refer to the Phi where Phi you just fill in the blank with some condition or other right like the first child to be born in the 22nd century according to the timekeeping conventions in place on 31st December midnight 31st December 2099 okay that's what they say very very simple this is actually on the view that I've said that I've sketched here this is actually a very complicated move in a certain social practice so here's what they're saying here's a new bit of vocabulary and I think we'll find it useful I think it refers to the Phi I'm gonna go on to use this new term do it us to ascribe to its reference some features that I think it has please join me in reasoning for now on the presumption that in as a reference and that the reference is uniquely Phi let's investigate together information may come to light that that presumption leads to trouble of some sort if so then some revision may be called for look this is metaphysics this investigation is going to go on for a while I may not be there to help you decide how to revise the presumption I am after all mortal please do the best you can thank you very much


Hello and welcome to this presentation of the STM32 flash memory interface. It covers all the new features of the STM32-F7 flash memory. STM32-F7 microcontrollers embed up to two megabytes of flash memory. The flash memory interface manages all memory access, read, programming and erasing as well as memory protection and option bytes. Applications using this flash memory interface benefit from its high performance together with low power access. It supports read while write and allows dual bank booting for devices supporting the dual bank feature. Flash memory has several key features. It has the adaptive real-time or ART memory accelerator with a unified instruction and data cache. Also, a pre-fetch buffer allows a linear performance in relation to frequency while executing from the tightly coupled memories for instructions or ITCM interface. The STM32-F7 series has up to two megabytes of flash memory configurable in single or dual bank mode with a read while write capability that can program or erase one bank while executing code from the other bank. Note that the dual bank feature is only available on STM32-F7 6x-F7 7x devices. The flash memory interface has three main buses, a 64-bit ITCM interface, a 64-bit AXI-AHB interface and a 32-bit AHB register interface. The flash memory is mapped on the ITCM interface of the Cortex-M7 processor and used for code execution and data read accesses to the flash memory. Write accesses to the flash memory are not possible through this interface. The flash memory remains accessible to the Cortex-M7 processor and other masters such as DMA controllers through the AHB interface. The 32-bit AHB register interface is used for control and status register accesses during user flash memory and option byte programming. The flash memory is divided into two functional blocks, a main memory block containing up to two megabytes for user code and data. The default configuration for the main flash memory is in single bank mode. For devices supporting dual bank mode, the flash memory organization can be changed via the ND bank user option bit. The second flash memory functional block is an information block which contains three parts. The first part is the system memory which is reserved for use by ST Microelectronics and contains the bootloader. When selected, the device boots in system memory to execute the bootloader. The second part is a one kilobyte one-time programmable area. The OTP area cannot be erased and can be written to only once. The last part contains the option bytes for configuring user options. This slide shows an example of the flash memory map. This example is from STM32-F76X-F77X devices with two megabytes of flash memory. In single bank mode, there are four sectors of 32 kilobytes starting from sector 0 to 3, one sector of 128 kilobytes which is sector 4 and seven sectors of 256 kilobytes starting from sector 5 to sector 11. The sector number is used in the software procedure to erase a sector or to protect it. In dual bank mode for STM32-F76X-F77X MCUs, the two megabytes of flash memory is organized as 12 sectors for each flash memory bank. Where each bank has four sectors of 16 kilobytes, one sector of 64 kilobytes and seven sectors of 128 kilobytes. Bank two sectors start from sector 12 to sector 23. Dual bank mode is also available for STM32-F76X-F77X MCUs that are one megabyte devices. This slide describes the memory organization in this configuration. The embedded flash memory is a dual bank memory with read while write and dual bank boot capability able to boot either from bank one or bank two. The ND boot option in the user option bytes is used to select the dual bank boot mode. When the ND boot option is reset, the device boots either from bank two or from bank one depending on the boot ADD address and valid bank content. When dual bank boot is enabled, the device boots from system memory to execute bank swap when valid bank is detected. With a dual bank memory, it is possible to read from one bank while programming or erasing the other bank. Code execution is not stopped when the flash memory is being programmed. When programming or erasing data in the same bank, the fetch is stalled as long as the flash memory controller is busy. Using the FV mode bit in the system configuration memory remap register, the two flash memory bank addresses can be swapped. When this bit is cleared, bank one is mapped at address 0x0800000 for AHB access and 0x0020000 for ITCM access. When this bit is set, bank two is mapped at address 0x0800000 for AHB access and 0x00200000 for ITCM access, which allows the device to boot into bank two. The dual bank boot allows a safe firmware upgrade as the previous firmware version is still present in the other memory bank. The dual bank boot is managed by the bootloader. The device boots in bank two using the ND boot option bit programmed in the flash option bytes. The boot pin and boot option are configured for booting in flash memory. If the ND boot option bit is set, the device boots according to address in boot ADD option bytes. If the ND boot option bit is cleared, the device boots in the system flash memory. The bootloader checks the bank's content indexed by boot ADD address as it must read there the stack pointer at that location. If bank two's boot ADD address content is a valid SRAM address, the bootloader swaps the banks to remap the bank two at address 0x00200000 and jumps into bank two. If it is not valid, the bootloader swaps the banks to remap bank one at address 0x00200000 and jumps into bank one. Note that the bootloader uses resources in SRAM for dual boot. The content of SRAM is not preserved after reset. The flash memory interface implements program and erase operations. The programming granularity depends on the MCU supply range. Word parallelism is only available in the 2.7 to 3.6 volt range. Half-word parallelism is available starting from 2.1 to 2.7 volts, while byte programming is available on the full VDD range. To enhance programming parallelism, it is possible to use double-word programming. This is only available starting from 2.7 volts while applying an external VPP while programming. Erase granularity can be down to sector size. Also, it is possible to erase the complete flash memory using the mass erase option. Bank erase is only available with devices supporting dual bank mode. As the flash memory has a long read access time, in order to read the flash memory, software is required to configure the number of weight states to be inserted in a read access, depending on the clock frequency. The number of weight states also depends on the MCU supply voltage range. For the 2.7 to 3.6 volt range, the flash memory can be accessed at up to 216 MHz with 7 weight states. It can be accessed with 0 weight states at up to 30 MHz. For the 2.4 to 2.7 volt range, code execution at up to 216 MHz is possible with 8 weight states. For the 2.1 to 2.4 volt range, code execution at up to 216 MHz is possible with 9 weight states. When VDD is below 2.1 volts, flash memory can be accessed with 0 weight states at up to 20 MHz. Thanks to the Cortex-M7-L1 cache and to the Adaptive Real-Time Memory Accelerator, the Art Accelerator on the ITCM interface, the program can be executed with 0 weight states independent of the clock frequency. This provides an almost linear performance in relation to frequency. The Art Accelerator is available only on the Cortex-M7-ITCM interface. It brings outstanding performance and reduces cache usage while executing code from flash memory. It consists of 64 unified cache lines with a 256 or 128 bit width depending on the flash memory interface with configuration. For each Cortex-M7 fetch, a lookup in current buffer, pre-fetch buffer and art cache is performed. If there is a hit, the data is immediately returned to the CPU. In case of a miss, the Cortex-M7 processor takes the data directly from the flash memory. In parallel, the flash memory line is copied into the current buffer enabled and art cache if enabled. If pre-fetch is enabled, another flash memory access is performed to fill the pre-fetch buffer with sequential data. Several option bytes are available in the flash memory information block to configure certain specific features of the device. The option bytes are saved in the flash memory after programming. The user option bytes are loaded in two cases, either after a power or a brownout reset when exiting from standby mode or after option byte programming. Three option bits are used to configure the brownout reset threshold. Two options are available to prohibit or allow the stop or standby low power modes. Four options are available to configure whether the watchdogs are enabled by hardware or after a software configuration and if the independent watchdog is frozen or not in stop and standby modes. Two options are used to enable dual bank boot mode and dual bank boot in devices supporting dual bank mode. The boot ADD options are used together with the boot zero pin to configure the memory address used for booting. The RDP option byte is used to set the memory readout protection level. Several flash memory protection options can be configured using the option bytes. The readout protection is configured using the RDP option byte. The readout protection prohibits any access to the flash memory or the backup registers by the debug interface or when booting from SRAM or when the bootloader is selected. The write protection is configured using the WRP option byte. This option protects specific code areas from unwanted write accesses. The write protected area can be defined with sector granularity for single bank flash memory devices while in dual bank mode. One bit protects two consecutive sectors. Three interrupts can be generated by the flash memory. The end of operation interrupt is triggered when one or more flash program or erase operations is completed successfully. The programming error interrupt is triggered when a flash memory program or erase operation failed. The write protection error interrupt is triggered when a write access is attempted to a write protected area of the flash memory. This shows a summary of the flash memory interface states in each of the power modes. The performance of the flash memory is almost linear with the frequency using the ART accelerator or Cortex-M7 processor using the instruction and data cache. The core mark score is 1082 at 216 megahertz which corresponds to roughly 5 core mark per megahertz when executing from the ITCM interface with the instruction cache, data cache and prefetch buffer enabled or when executing over the AXEM interface with the Cortex-M7L1 cache enabled. This is a list of peripherals related to the flash memory interface. Please refer to these trainings for more information if needed. For more details please refer to these application notes about the STM32F7 microcontroller execution performances, dual bank feature and system memory boot mode.


Let me recall what we did last time. So we consider it so-called perfectoid fields. And let me recall that a perfectoid field is a topological field k whose topology is induced by a non-discreet valuation of rank one. Such that the residue field is finite of characteristic p. And such that the Frobenius is surjective on k naught mod pi. So let me recall that k naught was the power-bonnet elements, which is a set of all x, absolute value at most one, using this rank one valuation. And in there we have the maximal ideal, which is also a set of topologically new potent elements. And it's also the set of all x of absolute value less than one. And last time we constructed a function called the tilting function from any such perfectoid field of characteristic p. And note that in characteristic p, the condition is basically that this complete non-archimedean field be perfect. And so some of the theory is much simpler over this characteristic p field. And we are trying to compare the two somehow. And so let me recall the description of k flat. So it's inverse limit over the piece power map r, for example, of k. And somehow if you consider truncated versions of the power-bonnet elements, then they become isomorphic. So pi and pi flat were chosen to be somewhat corresponding uniformizes. And so this was summarized somewhat in the following picture. So here's the spec k0 with its generic point and its special point and some infinitesimal sickening, which is the spectrum of this reduced thing. And then we have the tilt, which somehow starts the same. And then somehow goes on differently in the spectrum of k flat not. And last time I stated the following theorem that there is the tilting function l maps to a flat induces an equivalence between the finite etal extensions of k and of k flat. So that, in particular, the absolute gather groups of two fields will be canonically isomorphic. And implicit here is the statement that any finite extension of k will again be perfectoid. And the plan of proof for this was to use some almost mathematics. So recall that we defined the category of almost k0 modules as k0 a modules, which was the category of k0 modules, where you divide out by the six subcategory of m torsion objects. And this was possible because we are in a non-discrete valuation and hence the square of the maximal ideal is equal to itself. And then we defined notions of k0 a algebra and so on and so forth. And the refined version of the theorem is then that we have the following chain of equivalence of categories that somehow these finite etal extensions extend almost to the integral level. And some of them there's a lifting theorem stating that one can lift finite etal extensions over such, over a new potent elements. And then we can go to the other side because of this isomorphism of the truncated rings. And so part of this long equivalence of categories will generalize to a more general setup, which is the setup of perfectoid algebras. And these I will introduce now. So first over k, so a perfectoid k algebra is a Banach k algebra R. And then the first condition is of a topological nature saying that the ring of power bounded elements in R, it's always open but the condition is that it's bounded. And then comes sort of the perfectoid condition which says that reducing mod pi is surjective. As the Frobenius mod pi is surjective. And so let me remark that sort of in the classical setup of which a geometry will also consider Banach algebras. And this condition that the suffering of power bounded elements is open and bounded is precisely true for reduced algebras. But this condition will basically never be the case. So the second is then over the almost integral level. So there it's a flat and pyridically complete k not a algebra. A such that, again, while you look pi, the Frobenius should be surjective. But now we additionally require that the kernel is generated by the piece root of pi. So it was possible to somewhat choose these uniformisers to be equipped with a canonical system of p to the n roots. And the last point, also the reduced version, perfectoid k not a mod pi algebra is a flat k not a mod pi algebra a bar. And then we, again, put this condition about Frobenius, such that Frobenius uses such a nice morphism. OK? And then we have the following series of equivalents of categories. So let me denote by k perv, the category of perfectoid k algebras. And then we have k perv is equivalent to k not a perv. Then it's equivalent to k not a mod pi perv. And once we have this, then this will be just the same as on the other side again. And then we can live to the other side. So the proof splits into two parts. So first we have to prove this equivalent and then this. And at the first step, let me explain what some of the conditions imposed on such a perfectoid k not a algebra mean. So we put the condition as flat and pi at t complete. And so let me explain what this means in classical commutative algebra. So let m be a k not module, almost module. And recall that then we had to find this module m lower star, which was the homomorphisms in this almost category from k not a to m. And so this is a k not module. In fact, if you take the associate with almost module, we get back m. So the first part describes the condition when m is flat. So m is flat if and only if m lower star is flat, if and only if m lower star has no pi torsion. And let me recall that this thing here is called the module of almost elements of m. And for the following reasons, it's a good name. So assume m is the almost module associated to n. And n is flat, so flat k not module. Then the almost elements of m are described in the following It are the elements x and n revert pi, such that for all epsilon and m, the epsilon multiple really less than n. So some of it are not really elements of n itself, but as soon as you multiply them by something, in the maximum ideal, they become elements. So in this sense, almost elements. And so this basically explains the flatness condition. Now we want to explain what it means to be periodically complete. And there we have the following. So assume that m is flat. Then for all x and k not, one can compute what x m times lower star is. And it's just x and lower star, so that's OK. And secondly, one can wonder what m or x m lower star is. And there it will turn out that it does contain this thing. But in general, there are more elements. And I will give an example in a second. But it's not so bad, because for all epsilon and m, the image of m mod x epsilon m in m mod x m lower star is just this sub-module. And the last point is that, again, if m is flat, then m is periodically complete, if and only if m lower star is periodically complete. So in particular, the condition that this k not, almost algebra a, is flat and periodically complete can be rephrased as saying that a lower star, which is an actual k not module is periodically complete and flat. OK? And so let me first give you a remark about point three. So recall that, for example, if you want to compute k not mod pi lower star, then there was a general formula saying k not a mod pi. Lower star is a general formula saying that this is a set of morphisms from the maximal ideal into k not mod pi. So more generally, if you have any module, take its associated almost module and its almost elements, then this is computed by this formula. It takes more morphisms from the maximal ideal into there. And so this contains, as already mentioned last time, elements of the form sum of pi to the 1 over p to the i x i, where the x i are any elements in k not mod pi. Because as soon as you multiply by some small epsilon, this will just be a finite sum, which will then make sense. And so the problem is that these elements do not lie in k not mod pi, which is sub set of this in general. Cannot lift them to elements of k not. OK. But also we see that somehow these elements will not lift any further, because somehow there is this accumulation point of the exponents here. And they will not lift any further. And that's some of the reasons that here in 3 we get this last statement.  So for 1, first we check that if m lower star is flat, then m is flat. That's easy, because we call that m is flat means, by definition, that for all x k not modules and i bigger than 0, that's the tau i of x and m lower star is almost 0. But of course, if m lower star is flat, then this thing will be 0. That sort of this implication is clear. Then in the second implication is that if m is flat, then m lower star has no pi torsion. How do we do this? So first note that, again, using this formula for the almost elements, it's a set of homomorphisms from m into m lower star. And this description implies that this thing has no almost 0 elements, because m squared is equal to m again. So then we use that as a particular case of this condition, we use that the tau 1 of m lower star was k not mod pi, that this thing is almost 0. And this just translates into the condition that the kernel of multiplication by pi on m lower star is almost 0. But again, the kernel can have no almost 0 elements, because of this, and hence it follows that this thing is 0. And so which means that m lower star has no pi torsion. And then there is the last step saying that m lower star having no pi torsion implies m lower star flat. But of course, this is just a statement in standard commutative algebra and standard effect. So now for the second statement, so again, we use this formula that m lower star can be computed as the set of homomorphisms from the maximal ideal into n. And because n is flat, this embeds into homomorphisms from k into n invert pi. And how do we describe these homomorphisms that lie in the image? So homomorphisms are given by the element of 1, which is an element in n invert pi. And the condition is that as soon as you multiply by some epsilon in m, you have to get an element of n. But that's just what we claimed in the third point, if m is flat. Exactly, so we write x m lower star. We can write it as the set of elements in y in m lower star n invert pi, such that for all epsilon in m, some epsilon x times m lower star. And this can be rewritten as the set of all x times the set of all y in m lower star pi inverse such that for epsilon m epsilon times y lies in m lower star. But using the dot we proved in part two for the special case of n equal to m lower star, this is just x times m lower star.  And so last assertion is left as an exercise. And for the last point, part four, we note first that both functors m maps to m lower star and n mapping to n almost at mid-left adjoins, giving the second case by the funcars that I have not yet introduced, which is m maps to m lower star, which is m lower shriek, which is m tensor m lower star. And hence commute with inverse limits. And so in one direction, we then get that if m is pi equal complete, then m lower star is the inverse limit of m mod pi to the n lower star. This commutes with inverse limit, so it's the inverse limit of m mod pi to the n lower star. And now we want to see that this is also the same as the inverse limit of m lower star mod pi to the n. So a priori would think that this might be only an inclusion because we only have this inclusion, but we require that somehow we have elements in this inverse limit. And so everything that occurs in this inverse limit has to lift to something larger. And then we use this statement here saying that then it's actualized in this sub-thing. And for the other direction, it's even simpler somehow. And so now we can prove some propositions. The first is that if we have a perfectoid k algebra, then phi induces, in fact, an isomorphism between r not mod pi reduced mod pi to the 1 over p, r not mod pi. So remember that we only required this phi to be surjective, but in all other rings, we required that we have such an isomorphism, and here it's automatic. And also, if one considers the almost algebra associated to r not a, then this is a perfectoid k not a algebra. And for the proof, just note that if x is an element of r not such where x is an r and r not such that its p's power is an r not, then it follows that such as this thing is power bounded, then obviously x divided by pi to the 1 over p is also power bounded, which gives the first statement. And for the other note that we know that r not is flat and pyridically complete by the condition that it's open and bounded, and hence by the previous lemma we say that the almost algebra is also flat and pyridically complete, and sort of the last condition also translates into the almost setup. And now we want to, so we can now pass from perfectoid k algebras to perfectoid k not almost algebras, and now we want to go the other way. That's the following lemma, so let a be a perfectoid k not a algebra, and let r be the algebra you get by inverting pi, and we want to see that this gives you a perfectoid k algebra. So first we give it a topology, so with topology making a lower star open and bounded. And then in fact, the power bounded elements of this thing will just be a lower star, and so in particular we see that r, we can reconstruct a by taking the almost algebra associated to r not, which means that the two functions will be inverse. And this also, then, by what's already on the blackboard implies that we get an isomorphism between a lower star mod pi to the 1 over p and a lower star mod pi, which might be surprising at first sight, because a priori we only required it to be an almost isomorphism, but it turns out that it's an isomorphism automatically, and r is perfectoid. And so let's prove this. So first we show that phi is injective. So first we know that phi, this map, is an almost isomorphism by assumption, meaning that the kernel and co-kernel are almost zero. And so the first step is to show that phi is, in fact, not only almost injective, but injective. So what does this mean? So we have some x in a lower star, such that x to the p is pi a lower star. And we want to prove that, in fact, x lies already in pi to the 1 over p a lower star. But almost injectivity says that this is at least true after multiplying by some epsilon. So for all epsilon and m, we see that epsilon times x has to be an element in pi to the 1 over p times a lower star. But this means that this x is almost an element of pi 1 over p times a. So it's in this set of almost elements. But then the last lemma tells us that this is just the same as what we have. So we get to the desired conclusion. And then next we need the following small lemma that if x is some element of r, such that its piece power lies in a lower star, then already x lies in a lower star. And for the proof, there exists some integer k, such that pi to the k over p times x lies in a lower star by definition of r. And if k is at least 1, then in particular we see that y to the p is pi a lower star. And by the injectivity that we have already proved, it follows that y lies in pi to the 1 over p times a lower star, which just says that also pi to the k minus 1 over p times x still lies in a lower star. And then we do induction to get the claim. And so next we prove that the power-bounded elements are just a lower star. And for this note that it's clear that a lower star is contained in the power-bounded elements, because all the powers of an element of a star will lie in a star. And a star is by definition a bounded subset of r. And for the other direction, assume that x be power-bounded. And then for all epsilon in the maximal ideal, epsilon times x is topologically new potent. And in particular, there exists some big n, such that epsilon times x to the p to the n will be in a lower star for n large enough. And this implies that epsilon times x is a lower star by the previous lemma. And hence, x again defines an almost element of a lower star. And hence, x itself, by the description of almost elements, is an a lower star. And so now we have to prove that we have the following equivalence. So in other words, we have to see that the a maps to a bar, which is a tensor for a mod pi. And so we have to see that a bar admits unique deformations. And for this, we will use a theory of the cotangent complex, the foundations of which are laid out in the Caesars of Euler-Z. And let me recall a little bit about this cotangent complex. So first in the classical commutative algebra setup somehow. So to any ring homomorphism, R to S, we associate the so-called cotangent complex, which is an element. You can consider it as an element in the derived category of S modules, whose homologies concentrated in degrees and non-positive degrees. And some properties of this are that first of the H, I guess it's more standard to use homological notation, H0 of this thing is just the differential of S over R. And for any R, S, T, one gets a triangle in the derived category of T modules connecting, now I have to get this right. L S over R pulled back from S to T, goes to the cotangent complex from T over R, goes to the cotangent complex of T over S. And so some of this extending the short exact sequence that one knows, somewhere from just omega 1. And let me say one word about the construction of this thing. So construction runs as follows. We will also some more needses. So first you use the dot-con equivalence saying that this derived category of S modules with this boundedness restriction is equivalent to the category of simplificial S modules, module of weak equivalence. And so instead of this thing in the derived category, we will construct some simplificial S module. And so the way we do this is to choose some simplificial resolution of S by 3R algebra. So meaning just a polynomial algebra and possibly infinitely many variables. And there's even a canonical way of doing this, if you like. And then once some are for such a polynomial algebra, it's easy to define what the associated cotangent complex should be. It should be just omega 1. And so one extrapolates from this case.  And then you're getting this right. Yes.  And so why are we interested in this? It's because of the following theorem. So in deformation theory. So let R be some algebra, some ring.  And so set R0 to be R mod I. And let S be a flat R0 algebra. And we want to lift S to a flat R algebra. And we are asking, what's the obstruction to doing this is? And what are the possible ways of doing this? And that's answered by the following theorem, which says that there is an obstruction class in x2 of the cotangent complex of S0 over R0 with values in the ideal tensored up to S0. And the possible lifts are a torsor under x1 of the very same objects. And the automorphisms of one lift are x0 of the same objects. So in particular, for example, if the cotangent complex vanishes, then there is a unique lift. And it's sort of, there's no ambiguity whatsoever. And the second deformation result that we will need is that if we already have S and S prime to flat R algebras, and we are given a homomorphism from their reductions. So all the reductions to R0. And we are given a homomorphism from S0 to S0 prime, morphism of R0 algebras. And we want to lift this to a homomorphism from S2 S prime. And then there's the following that there is an obstruction class in x1 of the cotangent complex of S0 over R0 with values in S0 prime, tends to R0 i. And the possible lifts form a torsor under the group x0 of the same thing. So again, we see that if the cotangent complex vanishes, then also homomorphisms lift uniquely. And so it will be interesting for us to have a criterion for when the cotangent complex vanishes is the following very nice proposition that also appears. So I learned this from this book of Gabbard and Tramero. And I don't know whether this was observed before. So there's a following very easy criterion. So if R is the perfect Fp algebra, then the cotangent complex of R over Fp vanishes. So we will need a slightly enhanced version of this. So that R2 S be a morphism of Fp algebras. That R lower phi be the R algebra R via the homomorphism via the Frobenius. And in the same way, we define S phi. And we assume that the relative Frobenius 5S over R is an isomorphism even in the derived category between this and this. So this isomorphism in the derived sense just means that without the L, it's an isomorphism and all higher tor terms vanish. And then, again, the cotangent complex vanishes. And so for example, the first statement can be used to say that the ring of fit vectors of R adjusts a unique deformation of R to a flat and periodically complete Zp algebra, which is somehow much closer to how I think about the width vectors than the usual definition. And also, it's clear that the omega 1 of R over Fp is 0. Why is this? So if x and R and x is y to the p, then dx is equal to dy to the p. And that's p times something. And this is 0. And that's somehow the crucial identity behind this proposition here. And so we just have to go for a slightly more enhanced argument. And so this goes as follows. So we choose the simplification resolution. And so we also have sort of the relative Frobenius over R, which is a map from R phi tensor R S dot to S dot phi. And the assumption just says that this is a weak equivalent, or quasi-Somorphism, or whatever you call it. And so we have two resolutions of S phi by a free, simplificial S phi R phi algebra. Some of both sides of this give you such a resolution. And so it follows that they both compute the same cotangent complex, which means that R phi tensor R of the cotangent complex of S dot over R goes isomorphically to the cotangent complex of S dot S phi over R phi. But on the other hand, we can explicitly compute what does this thing here do on the differentials. So somehow if one part of this simplificial algebra is just some infinite polynomial algebra in general, then phi S k over R on differentials maps dx i to dx i to the p, which is again 0. So it follows that this very same map, which is a quasi-isomorphism, is also the 0 map. And in particular, both sides of this isomorphism have to vanish. And this is exactly what we wanted to prove. But that's somehow the same thing just considered from a different point of view as the cotangent complex of S over R. Now we have to use this vanishing of the cotangent complex. But first we have to sort of adapt the theory to the almost context. And this was also done in the book of Gabber and Tramiro. So they show the following that if R and S are k0 algebras, then if you consider the cotangent complex of S over R, but consider it in the almost way, then again, this is an object of derived category of S almost modules. And they show that this depends only on the morphism from R a to S a of k0 almost algebras. And so you get La a over b in the derived category of a modules for any morphism a to b of k0 almost algebras in this way. And in fact, Gabber and Tramiro define a slightly refined version of this cotangent complex, which actually is an actual module, somewhere over an actual ring. But we will only need this almost version. And then the previous theorems state true, theorems about deformations state true in the almost world. And we will then need sort of the following corollary of this vanishing of the cotangent complex, which is that if a bar is a perfectoid k0 a1 pi algebra, then the cotangent complex of a bar over this thing is 0. And one way to reduce it to the case that was already handled is sort of it's enough to represent a bar by a flat k0 pi algebra R such that Frobenius is an isomorphism from R pi to 1 over p was R. And again, because of these strange elements which appear in the, so someone could try this, but this does not work. But in this book, there's a different way of representing a normal string by an actual ring. So it's what they call a double lower shriek, which is you take a lower shriek, which was, I recall, maximal ideal tensor a1 star. This will, in general, be flat already, so that's better. So this thing might not even be flat. But it's not an algebra anymore of a k0 because it does not contain the identity. So we artificially sort of have to add the identity and sort of mod out m. And so this works. Don't want to get into the details here. And I guess one could also just do the same proof somehow. Or is it possible? Which is not that? Yes? OK. Well, somehow, if it's not a zero thing, then it will be facefully flat here somehow, so I guess. Well, maybe I should check again, but somehow. OK. And so as the corollary, we now get that the categories of k0a algebras, which provides us correctly last time, the perfectoid k0a algebras and the perfectoid k0a mod pi, algebras are equivalent. So that's for the proof. So we need to prove inductively that a bar lifts uniquely to a flat k0a mod pi to the n algebra, whatever I want to say, a bar n. And then a will just be the inverse limit of the a bar n. And to see this, we have to see that somehow, at each step, the cotangent complex will vanish. I want to add the almost thing here. And so we have an exact sequence. What does this go? Some easy verification, somehow that this lifts l a, a bar n over a mod pi to the n. And so this is 0, this is 0, and so this is also 0. OK. So finally, we get some of the equivalents of all of these categories. In particular, we get from perfectoid k algebras to perfectoid k flat algebras. But some of the way we did it was completely inexplicit. And it's nice to know what this equivalence looks like after all. So we want to compare this with this front-end construction. So let R be a perfectoid k algebra. And A be the associated k0almost algebra. And then we define A flat to be the inverse limit over Fabinius of A mod pi. And this is a k0, k flat, not A algebra, via k flat not A as the inverse limit over Fabinius of k not mod pi.  And then we have the following proposition, that R flat is really the tilt of R. And we have some more properties somehow. R flat can also be described just as a set, as the inverse limit over the piece power map on R. And then also the ring of power bounded elements in the same way, A r star. In particular, one sees that the construction is independent of pi because the way we proved the equivalence we somehow implicitly chose some pi. But for example, one can see from this description that it doesn't matter. And somehow, again, we have the property that after reducing mod, the uniformizer, the two things come the same. And in particular, we get a continuous multiplicative map from R flat to R, which again is noted x maps to x sharp. And so the key point is just that somehow we have the series of equivalence of categories. k flat mod A mod pi to 12. And then we again lift to the other direction. And somehow, if we start with this R, then this maps to A and this maps to A bar, which is A mod pi. So that's all easy. And some of the key point is that this lifting here on characteristic p from the stuff mod pi flat to the stuff on the integral level, which we somehow did with this vanishing of the co-changing complex and which was completely inexplicit, can somehow, in this case, be made explicit just by taking the inverse limit of the fulvinous. Reasonably easy. Well, all the other verifications are basically standard. So maybe I should say how to get this map from R flat to R. So it's easy to see this. And then we have the projection map again here from the inverse limit over the piece power map. And in the same way that this was proved for fields, one checks that again there's an isomorphism here. And this then gives the desired map. It extends somewhere to the rational level. So one might wonder whether that's some or some or no, we have described the explicit function going from characteristic 0 to characteristic p. And one might wonder whether it's possible to go the other way and then you did this. So by some, the commerce function is given the following way. So some commerce function S maps to S sharp is S sharp. So it takes the ring of fit vectors of the power bounded elements in your characteristic p-ring. This becomes an algebra over the rate of fit vectors of k flat-node. And then in PAD code theory, you define this map theta from this ring of fit vectors to k. And then you tensile along this map theta. And so when we're drawing a picture, this would mean that we recall this picture somehow. And so we somehow went this way. And we showed that one can sort of invert both arrows. But this explicit function now goes somehow over this direction. This is the explicit function. So it does not go with a special point, but there is this whole surface somewhere in between which is some of the ring of fit vectors or a spectrum of the ring of fit vectors of this characteristic p-ring. And so you first extend over this whole surface and then specialize to a different point that somehow the explicit functions go as a circle. And it's possible to give a direct proof without using any almost mathematics that one gets an equivalence between these categories of perfected algebra, which is done by Kedler and here. So some of the independently came up with pretty much related ideas. And after I told them that there is this equivalence, I sort of saw that it's immediate to deduce it from their results. So let me give an example now. We have not seen any perfected algebra until now. So maybe that's one can, for example, take the following algebra. You take convergent power series in a certain number of variables, but you join all of the p-power roots of the variables as well. So by this ring, I mean you take the polynomial algebra over the ring of integers, complete it periodically and then invert pi. So this is perfectoid. And it still looks just the same. So still all flat. Very easy proof of this is to observe that first of the power-bounded elements are really just the PAD completion of this thing. And that phi is subjective on R0 mod pi. So we see that it's indeed a perfectoid algebra. And to see that the two are tilt of each other, it's enough to see that R0 mod pi is the same as R flat mod pi flat. So by how we proved the tilting equivalence, it's enough to show this isomorphism that is clear. Because it's just the same thing. And of course, you could sort of do the same example for any toric, for any monoids or for toric varieties. And finally, we want to use all of the stuff to finally prove this equivalence of Galois groups on the two sides. So we need to prove something about fields again. And that's the following lemma. So that if R is a perfectoid, k-algebra was tilt R flat. And somehow R flat is a topological algebra. And it makes sense to ask whether it's a perfectoid field. And that's precisely the case when R is a perfectoid field. And so proof runs as follows. So the rank voting valuation is necessarily given by equal to the spectral norm, which in our case can be defined as XR as the supremum overall, no the infimum, overall T inverse, where T is some unit such that T times X is an R0. And it's easy to see that spectral norm on the tilted side can be computed by taking the spectral norm of this sharp representative. And it's also easy to see that R is a perfectoid field if and only if this is multiplicative and R is a field. And because R flat is the inverse limit of R, it follows that if R is a field, then R flat is a field. And somehow it's also true that if this is multiplicative, then by this formula, this thing is multiplicative. So one gets one direction, and the other direction is also not hard. But so I leave it as an exercise. And so now we need to discuss the lifting of finite detail covers. So somehow how do finite detail covers? You want to show that if you have any perfectoid algebra in any of these settings and something finite at all over it, then again, it is perfectoid. And the starting point is the following proposition. Again, it's very nice everything that I need is already in the book, that if A bar is a perfectoid, K naught A mod pi algebra, and B bar is finite detail over A, then B bar again is a perfectoid. And for the proof, so that B bar flat over K naught A mod pi is clear, because it's flat over A bar and A bar is flat over this. So the hard part is to show that the following is true. So what's the following theorem? That we have this, and we have the Frobenius map here, and we have this thing, and this is a pullback diagram. So some other classical commutative algebra that might be well known, but some also extends to the almost setting. And once one has this, one easily sees that indeed the Frobenius does the expected thing. And so we get the following diagram. So we have some R corresponding to A corresponding to A bar, which is then the same as A bar flat.  So let these be fixed somehow. And then we have all of these equivalents of categories. And so in the middle, we have some A bar finite at all, A bar flat finite at all. And we have just seen that there are these functions. And recall that last time I stated the theorem to the effect that one can uniquely lift finite at all covers over such algebras. So this is somewhere from last time. So in particular, we also get this function here. And so on also on the other side. So we have A finite at all. You'll have this function. And then these, of course, map to by inverting the uniformizer to finite at all algebras on the generic fiber. The first observation is that this diagram automatically implies the proposition that the outer upper arrows are fully faithful, because this here is an equivalence. And we get the same algebras here. So we have these inclusions. And later we will prove that both are equivalences. So this is known as faulting's almost purity theorem. I mean, so if I have B1, B2 in A finite at all say, then I'm interested in the set of homomorphisms A from B1 to B2. And so that's by the equivalence of categories. Yes, fully faithful. Yes. So if I have two such algebras which are finite at all over A, and I know that then they are also perfectoid. And so because for perfectoid algebras, I know that going to the generic fiber is fully faithful. I know that we have this equality. And this is just what we wanted to prove. And what we will do first is to prove this almost purity theorem somehow in characteristic p, which is much easier than in characteristic 0. So that's the following proposition. And again, it's already in the book of Gabel and Ramero. So that k be of characteristic p. So in order not to write flat everywhere, I sort of assume that from the start I'm in characteristic p and let R be a perfectoid k algebra and S finite at all over R. Then because S is a finite projective R module, it has a canonical topology. And then it makes sense to say that S is perfectoid. And on the almost integral level, it's still finite at all. So this means that this function here is fully faithful, essentially subjective. And there's the following extra thing that is useful, is that in fact, this thing is a uniformly finite projective R0A module. So in particular, there's a good notion of the rank of this extension. And so there's a good notion of speaking of the degree of some finite exhaled extension here. And so let me give the proof. So S, obviously, is a perfect k-banach algebra. But it does not yet mean that it's perfectoid because we have to show that the set of power-bounded elements is bounded. And so how do we do this? So let S0 in S be a finitely finite R0 subalgebra. So it's possible to choose one such that let S0 bot in S be the set of all x in S such that trace S over R x, say, is S0, S y is an R0. So here we use the trace form of S over R, which is a map from S to R. Last time introduced in the context of almost algebra, and of course, it exists also in usual commutative algebra. And so this is a perfect pairing. And then it's easy to see that S0 and S0 bot are open and bounded. And we claim that the power-bounded elements sit somewhere in between. So we let, what was my name for it, y is the integral closure of R0 in S. And then first of the trace form pairing on y takes values in R0. And it's also clear because this is a finite algebra that all elements in there are integral. And by this property, it follows that it's somewhere squeezed in between S0 and S0 bot. And so it follows that y is open and bounded. And but somehow using the arguments from the beginning, it's easy to see that the integral closure of y is almost the same as the power-bounded elements because if you have a power-bounded element and multiply it with some small epsilon, then it becomes topologically new potent, intense integral over this thing. And so it follows that S0 is open and bounded. So we know that this algebra S is perfectoid now. And now we have to see that there exists some n such that for all epsilon and m, there are maps to R0 to the n. And the other way around was composite equal multiplication by epsilon. For this, we'll imply then that S0 is a uniformly finite projective R0 module, S0A, which is part of being some of the assertion. And how do we construct this? So now it becomes a crucial argument. So let E and S10 or RS is the idempotent, showing that S over R is unremifed. And then there exists some big n such that pi to the n times E is an S0, 10 to the r0, S0. But because for being this projective, it follows that for all epsilon and m, in fact, epsilon times E is in this thing. So somehow, a priori, we only know that it's true that up to some power that it is unremifed, but then we can lift it almost to the integral level because for being this projective, it's really the crucial argument. And we write epsilon times E, some of the sum of xi tends to yi, where these xi and yi are in S0. And no, no, S0. So I'm not passing to almost stuff. So we have these. And if you think about it for a second, then you see that you can choose this n somehow uniform for all epsilon, somehow by taking one expression here and then taking just the piece roots out of it. And then we get maps from S0 to r0 to the n sending S to the tuple y1. And we get a map in the other direction sending r1 up to rn to the sum of ari, ri. And it's easy to see that they're composite multiplication by epsilon. So this way we see that S0a is such a nice module over r0a. And what's left to see is that S0a over r0a is unremifed, but we have already seen that E defines the required almost item potent somehow. So here we are. Yes, sorry. I shouldn't change notation from my notes. And so it follows that the diagram here extends here to an equivalent. And in particular it follows that there's also this arrow here. Or we have just proved that they are all perfectoid. And also another to extract at this step because to have the Frobenius. So somehow, once you have the Frobenius, somehow really not more p, but really on the ring itself, sort of you can really make this argument that if something is true up to some pi power, then it's really almost true already. This argument is much more difficult to make a perfectoid stuff in characteristic 0. And usually one has to work much harder to prove the same theorems. But somehow amazingly they all stay true. Yes, you can sort of, it's easy to make this explicit what it means. And I guess it's also a little bit stronger because you didn't assume that it's perfect. But I'll just say that it's complete. Good. It also follows that somehow in this picture all finite etal algebras are uniformly finite projectives in this picture. So it makes all settings sense to talk about the degree of some finite etal extension. And so as a final theorem for today, I want to finish the proof of this equivalence of Galois groups, which is to say that the only step left is to show that this very leftmost arrow there is an equivalent. And maybe first I collect something. So I want to collect sort of what we proved for perfectoid algebras in characteristics 0. So now assume that then there is a fully faceful function from the finite etal algebras of what's tilt to finite etal algebras over itself, which is inverse to the tilting function. And the essential image are those S and R finite etal such that S is perfectoid and S0A in this extents to the almost integral level. And somewhere in this case, if this is true, it's also true that S0A is uniformly finite projective over R0A. And then the final theorem is that it's a perfectoid field, then this function is in fact an equivalent. And for the proof, so we can assume that R is k. So we can just rename our k as this perfectoid field R. And so it has a tilt k flat. And so the idea is now that if k is algebraically closed, then we already proved this last time. And so we reduced to this algebraically closed case. So consider m, which is the algebraic closure of k flat hat. And then m is a perfectoid field. So it's perfect and complete. That's enough to see. And so in particular, it's a perfectoid k flat algebra. And so we can untilt it. We get an untilt m-sharp, which lives over k. And this is again a perfectoid field by the proposition that somehow tilting identifies the perfectoid fields. And it's algebraically closed by what we proved last time. And so for any finite extension L of k flat, so L is contained somewhat in an m, we can go in the other direction, get something over k. And we let n be the union over all L of these L-sharp things. So it's contained in m-sharp. And it's easy to see that this is a dense subfield. So because you know that modulo pi, some of the tilt, and this ring itself are the same. So modulo pi, you know that these are basically the same, and then it follows that they are dense. And then if some of such a field has algebraically closed completion, then it follows by Krasnar's lemma, that n itself is already algebraically closed. So in this setup, some are algebraically and separately closed are the same. And so what we get is that any finite extension F of k is contained in L-sharp, where in some L-sharp, where we can assume that L over k-sharp is some finite Galois extension. But tilting preserves automorphisms and degrees. So that's somehow the reason that I paid attention to this uniformly stuff, so to see that it really preserves the degrees. So it follows that L-sharp is again Galois. And so this F contains some L-sharp, this F contained in L-sharp corresponds to some subgroup of the Galois group of L-sharp over k, which is the same as the Galois group of L over k-flat. And so this corresponds to some F-flat over k-flat. And then we really have that this thing will give rise to some because we have the equivalent of categories. OK, and here we are. So we have proved this almost purity theorem for fields. And of course, I mean, it was a little bit, it would be easy, one could get easier proofs for this, but this somehow comes naturally also out of the theory. OK, that's it for today. Yes, so somehow, because you have this equivalent of categories, it's true that F-flat-sharp will be fixed by the subgroup. So it's contained in F, which are the invariants. And because they have the same degree again, it follows that they are equal. Yes, yes, yes, yes. So somehow you get this finite subgroup. This also exists on the other side because you have this equivalent of categories.  And if you untilt it, and by the equivalent of categories, the elements in there will be fixed under the subgroup. So it must be contained in F, but they have the same degree. And hence they are equal. Yes, yes, I need that to feel the sensilium, but it is, so somehow, a union of complete extensions.