Hi everybody, I'm Anne-Marie Slaughter, the president of New America, and I'm going to be interviewing Emily Parker, the digital diplomacy advisor to New America, former New America fellow, and I have to add, my former colleague at policy planning in the State Department. So Emily and I have worked together for some time. So you didn't realize that Tuesday afternoon at four is the new happy hour if you have not helped yourself on this snowy day to liquid libation, please go ahead and welcome to all of those of you online, you'll have to get your own liquid libation, but we're going to talk for about a half an hour and then open it up to questions. And this is a wonderful new book. Now I know who my comrades are, Voices from the Internet Underground. So first of all, Emily, I have to ask you about the title, now I know who my comrades are. Tell us about why that's the title of the book and the significance in the case of the book. Sure. So thank you all for coming and thank you, Anne-Marie, for bringing my pleasure. So this book actually took root in 2004 when I was working at the Wall Street Journal in Hong Kong. We were starting a series of columns about China and the Internet. It was called Virtual Possibilities China and the Internet. And as I was thinking about this column, I was trying to understand why does the Internet matter in China? Because, well first of all, back then it seemed like a niche sort of small story compared to now. It's hard to believe, but it wasn't clear that the Internet was going to be a game changer in China. The other thing was that it was clear that even as Chinese netizens were finding ways to get around barriers to information and were finding ways to break through censorship, it was clear that the government was really giving them a run for their money. So every time some small piece of information appeared online that was contrary to what the state wanted, it would disappear quite quickly. So this raised the obvious question, which is why does the Internet matter in China? What I came to realize is that the Internet matters in China not as a mechanism for promoting freedom of speech or freedom of information, but because it can advance freedom of assembly. And China is a government that has zero tolerance for any physical gathering of a group that it distrusts. So given that reality, it's no small event that you can have hundreds, thousands, millions of like-minded individuals gathering every day. So the title of this book comes from a conversation that I had with an Internet dissident in Beijing and I asked him point-blank, why does the Internet matter in China? And he said, because now I know who my comrades are. The idea was that before the Internet, I thought I was alone. I thought I was the only one who had questions about the system or questions about the government, but now I know who my comrades are. Now I know that there are other people like me. So I want to ask you a bigger question about that, but this particular dissident is Michael Antie, right? Is his name Antie? That's a pseudonym, I assume. He's Chinese. Yeah, so his name is Xiaojing, but yeah, he uses Antie, which comes from Lord Antie, you know, to a post. Right, OK, I see. I just finally made that connection. So this idea of now I know who my comrades are. There's a, when I was dean at the Woodrow Wilson School, one of our faculty members, a psychologist, did some fascinating work on public opinion where he showed that people who hold a particular view are likely to believe that they are in the minority, even when they're in the majority. So you're opposed to the government or you think that a particular policy is a bad idea. You will believe that if you're asked, you'll say, well, only 20% people believe that, even when in fact 50 or 60% might be that. So it's a very, and when you tell people that, of course, they are empowered. So this idea that, you know, now I know who my comrades are. I'm not alone. I have support, hugely important. But the government knows who his comrades are, too. So how did, one essential question here is, yes, this empowers the opposition, but it empowered, doesn't it empower the government even more? Something the United States knows quite a bit about. Sure. So you mean empowering the government by knowing who the citizens are? Yes. In other words, they now can see the electronic traces of a virtual assembly in the same way they can see it if everybody comes to the square, although in some ways even more, because if you gather in a square and you run away, they can't prove you were there, but on the internet, they can't. Yeah. So that's absolutely true. And I don't want to downplay the efforts of authoritarian governments to crack down on online dissent, right?  It's not like I'm saying those things don't exist or that they're not formidable obstacles. But the thing to remember about the power of virtual assembly is that it's much easier for a government to crack down on a few isolated individuals, right? And so this is, in part, it is the power of numbers, right? So this book, in the Russia section of this book, I talk about attending a protest in Russia. And protest in Russia, and my friend here was, who was that, that protest with me? Protest here, protest in Russia tend to be, you know, before very recently, it was, you know, a couple dozen activists would come out and they'd protest and the police would descend upon them with a vengeance and, you know, kind of drag them away into buses. And it was completely pointless and it was completely dangerous.  Not only will it not accomplish anything, but I'm going to get arrested, right? Now, you look at the Russian protests that happened around 2011, 2012, and these were the largest protests since the fall of the Soviet Union. And social media played an important role, not just in organizing them, but in showing the Russian opposition that they were not alone, right? So they would go on Facebook and they'd see that 10,000 people were registered for some sort of protest. And that's going to, the power of numbers is going to be a powerful weapon against governments that are trying to crack down on individuals. Something else I remember you told me, I think this was back when I was in the state, and when we were both in the State Department, that it also means that it has a life beyond the arrests. In other words, that we tend to look at it as a protest and the government shut it down and that's that. But that it keeps on going below the surface and kind of erupts protests by protest. So I mean, so if you look at Russia now, I would say independent of Ukraine, we were looking at, you know, there were the Pussy Riot protests, there's Alex Navalny whom you write about and you can tell us about, but I would have said right now it looks like the government has the upper hand. Is that the way you see it as well, or is that just a temporary lull? So authoritarian governments always look all powerful until suddenly they're not, right?  But in reality, it's not quite true. I mean, take Egypt, for example. Egypt's a really good example. You know, this was the Mubarak government had been in power for 30 years, right? And if you had said to somebody, the Internet is going to shake things up in Egypt, you would have been laughed out of the room. It seems completely impossible. But somewhere beneath the surface, you had all these people gathering online. They were joining forces. They were signing online petitions. They were doing online campaigns. And so when that revolutionary moment came, they were ready. And I think Russia is quite similar. You know, I've talked to some of these Russian opposition activists recently, and it's true that Russia is quite dark right now and that it doesn't look like the opposition movement is thriving. However, they're getting ready. You know, and you look at something like Navalny's mayoral campaign, that was a very effective use of online and offline activism. You know, this is Alexei Navalny, who's the most prominent opposition activist and blogger in Russia. And he, you know, he's been completely barred from state media. He's not on the television. He's not in newspapers. And yet he managed to give a pretty good showing in the mayoral election. And that really shows that the Russian online opposition has been really working to kind of organize and be effective. And so they're just kind of waiting. And that they've said as much. You know, they're sort of waiting for the tide to turn. And it could be an external factor. It could be an economic crisis. It could be a political crisis. But they're building up the skills in online organizations so that they're ready to capitalize on that crisis when it occurs. But Navalny himself is shut down, right? For now, yeah.   He's under house arrest with this very interesting condition that not only is he under house arrest, he's not allowed to use the internet, which I don't even know if that, I don't think that's in the Russian criminal code. I think that's just something that they gave to him. So like an ankle bracelet that, you know, yeah, he actually has a bracelet. He has a bracelet, and now the bracelet has its own Twitter feed, you know. But there's always, that's the thing, you know. So Navalny is off the internet, but then his wife takes over the Twitter feed. And also, I mean, one of the things that Navalny has effectively done in Russia is that it's not entirely dependent on him. I mean, he has a lot of followers and he has a little bit of an infrastructure for promoting his message and promoting his causes, even if he physically can't be there. So are there, if you look at the internet opposition or activist communities in China and Russia, are there more similarities or dissimilarities? I mean, how culturally contingent is this? They're different more because the way that the governments control the internet is quite different in all three countries. So China by far has the most extensive censorship mechanism. You know, you have the notorious great firewall of China.  Like three million people, right? It depends on who you ask.  It's people who work at web companies. It's people at all levels of society. Some of them are quite normal people who are actually, you know, doing the censorship.  So China has this very extensive method of censorship and I guess they feel that they need it because by some statistics there's something like, I don't know, 600 million web users in China. So there's like crazy number. You have Cuba, which doesn't have that same level of censorship because there's so few people in Cuba that are online. You know, Cuba you have by some estimates, you know, 5% of people actually using the World Wide Web. They have an internet. You know, official figures are higher, but you have this tiny number of people and the Cubans, Cuban government regulates the internet by making it prohibitively expensive to use it and making it just inaccessible. And then in Russia. But they don't censor. So like if you can get on, you can talk, but nobody can talk to you because they can't get on. There definitely have been stories of, you know, blocked websites and censorship. It's not that it doesn't exist, but it's less important there because it's less part of the story because it's just so hard to get online and so few Cubans get online. And then in Russia, until fairly recently, the Russian authorities didn't really mess with the internet at all. We're seeing that change now. But what I write about in the book is that one of the main things that kept Russians from organizing online was not government action. It was apathy. It was a sense that, you know, there's no point. And the internet simply reflects the reality on the ground. And in Russia, for most of the time when I was writing this book, 85% of Russians said that they felt they had no impact on their political process. So that's, the internet is not gonna change that. So there was a sense that you could find anything on the internet and you could write anything on the internet, but what would be the point? So the three cultures are very different because the three atmospheres are very different. So I wanna ask you a little more about Cuba and then I'll come back to all three. How did you, first place, how did you research this in Cuba if you can talk about that? With difficulty? Is there any Cuban official here now? No. With difficulty. So Cuba was by far the most difficult. It's funny because I think China and Russia probably have more sophisticated surveillance, but in Cuba, the surveillance is more in your face. So, you know, in Cuba, for example, you'd be sitting at a cafe and you would have the sense that like, the guy at the next table is watching you and that was something you heard over and over in Cuba which is, you never know who is who. You know, an informer could be at the next table and an informer could be in a store and an informer could be in a cafe. People are always asking you questions, you never really know why. So, you know, when I went to Cuba, I kept a really low profile. The lowest possible profile. I didn't write anything about Cuba, you know, during the years that I was researching the book. I didn't even tweet about Cuba. I just tried to stay as below the radar as possible. But you were able to go. I went several times, yeah. And so, I guess then if you're in Cuba and you know only 5% of people are online, what's the point? Yeah, so Cuba, it's funny because that is a very natural question to ask and yet in researching this book, I found that the internet probably had the most transformative effect on the citizens that I met in Cuba who were using it. So, there's basically two things at play here. First, it's true. They're not really using the internet to communicate with other Cubans, but they are using it to communicate to the outside world, right? And that's a big deal because this is a country in which the official media really decides how the story of Cuba is told. You know, this is a nation with great healthcare and great education and a lot of happy campers and high literacy rates, you know? So, for Cuban bloggers, this was their one opportunity to tell Cuba as they saw it and as they felt it. You know, it was also very important for kind of reconnecting the Cuban exile community with the Cubans inside of Cuba. This is kind of a, this is the population that had been really split apart by politics. But most important, I think the internet in Cuba was psychologically transformative because Cuba is the most paranoid society I've personally ever been in. And you know, it's a place where it's ruled by citizen informers. You have to be constantly careful about what you say, who you say it to, how you say it. And I was personally affected by this. You know, it's very debilitating, constantly being worried about how you present yourself. And so, you have these people who never really spoke with their own voice, right? They never really spoke honestly. And then they go online and they start their blog and they're kind of, for once they kind of find their voice and they find themselves and it's really empowering to them. And so, I felt when I was in Cuba there was this real difference. There were kind of the bloggers and then there was everybody else, you know? And the bloggers were, you know, walking through the streets and they're like, okay, is this guy taking a picture of me? I don't care. Is this person trying to secretly follow me? I don't care because I, you know, is one Cuban said to me, I comport myself like a free man. You know, he comported himself completely differently. And this changes the way they behave because fear, in many ways, is the most potent weapon of the Cuban regime, right? It's what keeps the Cubans in control because it's people, it's not, Cuba's not a police state in the sense, not in the way Russia is where you see these riot police everywhere and you're terrified to confront them. In Cuba, fear rules, right? So there's people who just don't, they're scared of their neighbors, they're scared of speaking out, they're scared of banding together and the bloggers that I met had overcome that fear. Because they were speaking and they'd found their voice. Yeah, and it has offline repercussions. You know, for example, I write about one blogger who gets arrested and all these other bloggers rally to her defense. You know, they swarm the state security building and then they all get arrested too. But it totally changes the way they behave. And so my impression of Cuba was, if these individuals reached a critical mass, Cuba would be a radically different place. And I think the government is aware of that and I think that's why they keep internet penetration as low as it is. You know, it does raise just fascinating questions about speech and dignity. And you know, if I put on my law professor hat, sort of the theories of the First Amendment, but it is the First Amendment. It's the first one, right? That Congress shall make no law bridging the freedom of speech, that ability to speak in an authentic voice as affirming your own identity as a human being separate from others is just such a fundamental psychological thing. I mean, it's interesting as you talk about China, I am one and I have comrades. When you talk about Cuba, it's more I have found myself. There are others like me, but that's very interesting. Yeah, I mean, I'm mostly interested in this psychological element. So I spent about 10 years on the ground in these three countries. I spent a lot of time with these people and it really is changing the psychology of citizens in these countries, the internet is. And one of the framework of this book, I talk about how most authoritarian regimes would not survive a sustained mass uprising, right? It's not all about this military power and all these, they wouldn't survive it. And in many cases, these sustained mass uprisings don't come to pass largely because of psychological elements, right? And the three that I identify in this book are isolation, fear, and apathy, right? Isolation, fear, and apathy keep authoritarian regimes in power, right? I mean, it's very simple. Isolation, authoritarian regimes try to isolate critics from one another. So if I go outside and scream at the top of my lungs, whatever I want about the government, it's not gonna do anything. But if a thousand people do that, it's a threat, right? And so authoritarian governments understand this very well. Second one is fear, as we saw in Cuba, that fear, if citizens are too scared to speak out, if they're too scared to join forces with their neighbors, if they're too scared to take action, they never will. And the third one is apathy, which I think was most relevant in Russia. And apathy in many ways is an authoritarian government's best defense, right? Apathy is kind of what happens, what once fear and isolation have done their damage. It's this feeling that, you know, what's the point? It's not gonna change anything anyway. And that's something that we saw a lot of in Russia over the past few years. This feeling that, yeah, I could go outside in protest, but like, I'm just gonna get arrested and it's not gonna change anything. It's not gonna give us a new leader. It's not gonna give us democratic elections. So I'm just gonna stay home, you know? And so what I've observed in these three countries is that the internet in various ways is helping people transcend these psychological obstacles. So in China, people are overcoming their isolation by finding their comrades online. In Cuba, people are overcoming their fear by speaking the truth and kind of, you know, having the support of the international community. Yeah. And in Russia, people are overcoming apathy simply because success is the best antidote to apathy. And so this is where Navalny was so successful. He started out, and there's a great line from him a few years ago when I first met him and he said, I propose to people the comfortable way of struggle, right? He understood very, very well that Russians were not about to get on the street and do a mass demonstration. That they just were not interested in doing that. But what he might be able to get them to do is fill out an online form, right? So he would give them a template. He'd say, okay, here's a form. We're protesting corruption at this bank. And all you need to do is sign your name and we'll send it off to the Ministry of the Interior. It took people two seconds. They could do it from their living room. And then they started getting results. You know, suddenly officials were getting fired, bad deals were getting reversed. And I think once Russians saw that they could make a difference, they became more motivated to act. So it's funny, because we talk about slactivism here.  It's like, oh, you know, this is just internet activism. It's not real life activism. But what Navalny did was he showed people that internet activism could make a difference. And once they realized that, they were just more likely to be active, period. You know, online and offline. So that's my last question actually. It's exactly the online offline connection. So I remember in Tina Rosenberg's book, Join the Club, she talks about the impact that the group that responsible for the Serbian Revolution had in working with groups of revolutionaries in Egypt, in Tunisia, in other countries. And essentially her argument was, you know, you can have 70,000 people as part of a Facebook group, as you did with, we are all well-lead. Yeah, Khaled Said in Egypt. But that does not, 70,000 people on Facebook does not readily translate into a sustained demonstration on the ground. That you need a whole nother set of teaching you how to organize, teaching you how to make it cool to organize, to come out. So I wanted to ask you, what's the link there? Where, you know, the internet activism overcomes fear and apathy and isolation. But do you need something more to translate that into the kind of crowds and sustained demonstrations that actually make a revolution? Yeah, that's a great question. So yes, you do need something more. And the argument that I make is the internet just reflects the reality on the ground. So if that society is ready for offline protests, if it's ready for revolution, then the internet will play that role. If it's not, the internet is not gonna create it. So I would never argue that the internet on its own causes revolution. I think that's a crazy argument. I don't know that many serious people who are arguing that. It's a convenient straw man for anyone who wants to take down. Yeah, it's a straw man. And you know, Egypt, so you look at Egypt, right? So Egypt, for years, you had all these people gathering online, doing these kind of online campaigns and building up online skills. But the conventional wisdom on Egypt, and actually there was an article three months before the Egyptian Revolution with the title, Egypt's online descent has yet to move to the streets or something like that. That was the conventional wisdom. It's like, this is all virtual. Forget it, it's all virtual. And we saw that wasn't true. However, in Egypt's case, there needs to be a spark, right? It's what Lennon called a revolutionary situation. There needs to be a revolutionary situation for a revolution to happen. In Egypt's case, I think that spark was Tunisia. I mean, there were obviously a lot of things happening, but Tunisia was this neighboring country that also had an authoritarian government that also looked like it would last forever. All of a sudden was gone. And I think Egyptians saw that and they thought, well, maybe we have a shot. And so suddenly this kind of online energy became translated into offline. And Russia's another good example. I mean, for Russia, for years, there were kind of interesting things happening on the internet, but it seemed only virtual. And then all of a sudden, there were 100,000 people in the streets. In that case, they were riled up by election fraud. They said, we've had enough. There's always something. There's always something that pushes people there. But in the years that people spend online, kind of finding their comrades, overcoming these psychological barriers, it means that when the revolutionary situation comes, they're ready for it. So that is a great note on which to turn to the audience. Is there a microphone? Yes. So Leonie, we'll bring you the microphone and we'll just raise your hand to Kirsten, brother. Sorry, Kirsten. Oh, sorry. Hi there. First of all, thanks so much for being here today. My name's Adam Fivinson. I'm a graduating second year at the MS and Foreign Service Program at Georgetown. Just wanted to ask you a kind of a policy-related question. So for those of us who are going into the policy, development, diplomatic world, whatever it might be, what are the recommendations that you have, assuming our goal is to support the growth of an internet that does support freedom, open speech, and democracy in other countries around the world. What recommendations do you have? Yes. No, no. So, both of us have worked specifically in this area. I think the first thing for the US obviously is to keep its own house in order, right? It's pretty hard for the US to go around the world talking about internet freedom when other countries are saying, well, you guys are not really doing a good job of that. So I think that's number one. I think that Secretary of State Hillary Clinton did a good thing in kind of putting internet freedom on the agenda. I think that's important. She said this is a foreign policy priority. The good thing about that is it also turns a lot of eyes on the US because once you say, once you come out there and you say, I wanna promote internet freedom abroad, of course everyone is gonna kind of say, well, what about this and what about that? You're not doing this, you're not doing that. I think that's a good thing, right? I mean, I think that's really what the US can do first and foremost. There's, I think like State Department funding for kind of online dissidents, you don't have to be careful with that. I think sometimes it's more effective than other times. I mean, I think what a lot of these governments try to do is paint anybody who's an online dissident as kind of a mercenary of the CIA. So I think you have to be careful with the US government and who you support because it could actually make those people less effective. I also think that there's a big role to be played by our corporations, by Twitter, Facebook, Google. I mean, some of these places are like the best brand ambassadors for the US. And there's things that these companies can do to make their technologies more accessible in countries like China. Great, yes, here in the front. Hi, my name is Ananda Leek. I'm the founder of Digital Sisterhood Network. And I'm really interested to know while you were in Cuba, did you have a chance to talk to Yoni Sanchez? And can you tell us a little bit about your experience with her? Sure, so yes, they did meet Yoni Sanchez. And actually one of the main figures in my book is not Yoni, but her husband, Ronaldo. Yoni, I didn't focus on as much only because Yoni at this point is not your typical Cuban blogger. She's practically like a diplomat at this point. So Yoni, for those of you who don't know, no, Yoni is by far Cuba's most famous blogger. And she's a real example of what the internet can do for an individual. I mean, she's someone who started blogging under her own name and kind of writing about the things she saw in Cuba. In Spanish, I assume. In Spanish, but now her blog is translated into many different languages. And she actually, even at one point, she wrote questions to President Obama and he answered. I mean, that's kind of the weight that she has. And she is almost like, I don't know, a digital diplomat or even a non-digital diplomat in the sense that she, recently, Cuba made it easier for Cubans to travel outside of Cuba. So Yoni did this kind of world tour and she went to Miami and she was kind of arguing about lifting the embargo. So it's kind of amazing. I mean, this is just this ordinary person who is a total critic of the Cuban government who via the internet has been transformed into actually someone who might be able to change policy. Wow. And then her husband. So her husband is amazing. I mean, he's less famous than Yoni, but he's fast and he's actually the one who said I comport myself like a free man. He's a really, he's much older than her and he used to work for the Cuban state media, which is kind of interesting. He used to be a Cuban official journalist and then he slowly kind of moved from the official media to foreign media to blogging and now he's kind of writes with his own voice and he's an amazing person. He's really interesting. He's kind of walks through the streets of Havana with no fear at all and he's quite inspiring actually. Oh, no. Great, thank you. Other questions? Yes, David. Thanks. David Sturman, a research assistant here. My question regards in countries like Cuba where there's a economic, either an underlying economic reason why people don't have access to the internet or government imposed false economic reason for why there's a digital divide. Do you see internet advocacy producing groups that are more tied to outside the country and are therefore either in actuality not really representative democratic movements or even if they're not, even if they're not are more easily portrayed that way by the government to their detriment. That's a good question. So I think the government will definitely portray people like that that way. But in Cuba, for example, the people that I profile in my book, interestingly enough, are not even really dissidents. They're Cubans who were just fed up with the system. They felt that they weren't making enough money to live and they found a way to express themselves. So, for example, one of the Cubans I profile in my book is a judge, she was a judge. She worked for the Cuban judicial system and she found herself kind of having to deliver these verdicts that she thought were completely unfair. She had to go to the family and say, your relative is convicted of stealing a bottle of oil because he was starving. I mean, she found that completely unfair and so she wanted to find a place to vent her frustration and so she started writing her blog and then in doing that, she was labeled a dissident and she became a dissident. So actually, most of the Cubans I've met in Ioanni might be a little bit different. Ioanni is more of your classic dissident blogger. Some of them are just normal people who had had enough and then they become dissidents almost by default. I'm Susan Morgan. I run the Global Network Initiative. I had a question about your book and I'm making an assumption here which might be incorrect. I'm presuming you just focus on China, Cuba and Russia. That's true. And I was just wondering why you chose those three countries and just as a little aside, when I was in Cuba a few years ago, one of the things that really struck me in the streets was in the residential streets, there was almost always a little sign that said committee for the defense of the revolution. So it was just a little visible thing that said someone somewhere is watching. Yeah, so the CDRs and that's basically what those are citizens, our networks of citizen informers that were put in place by Fidel Castro himself. It's the idea that like somebody on every block is like keeping files on residents and that's still in place. I mean, there are still signs for that and can you imagine living under something like that where you know that there are people, you don't know who they are, you don't know who is who, you don't know who's informing to them, but yeah, that's a very effective mechanism of control in Cuba. Why did I pick these three countries? So China was obvious because I kind of come from a China background and I was looking for other case studies where you had a government that was trying to control information and assembly and you also had a vibrant online opposition. So actually, originally, Egypt was one of the case studies in this book and what's funny about that is literally in 2009, 2010, if you went to people, if you went to publishers or think tanks or whatever and you said Egypt's gonna be a game, the internet could be a game changer in Egypt, you were like laughed out of the room.  And I wasn't even saying there's gonna be a revolution.  They said, here's Mubarak, he's all powerful and the internet is only virtual. So Russia, I got a fellowship to go to Russia and had a chance to live there for a period and so I did Russia instead. I think the easiest answer to your question, it's maybe not the most sophisticated answer is, I really wanted to write about countries where I had access. I wanted to write about countries where I could spend a lot of time there, where I had contacts, where I had some language ability and sometimes people ask me, why isn't Iran in this book, for example? And actually, I would love for Iran to be in this book but it wasn't feasible for me to go and make six trips to Iran hanging out with internet dissidents there. I mean, maybe someone else could have done it. I didn't see it as feasible and I really didn't want one of those books where I'm sitting at my desk in Washington or my desk in New York just kind of like opining about how internet dissidents must feel. I really wanted places where I could go back and live there and kind of experience it. So that was sort of how this came about. It's just kind of figuring out where those places were. Here up here in the front, I have another question too, but I'm gonna. Hi, George Peeler, I'm an attorney and I write for Forbes.com. I wonder if you could draw some analogies between your work as in the book and some things going on in the United States, not just the Snowden revelators that have met people somewhat paranoia about their own communications but actual use of the techniques you're describing and affecting policy.  The Rand Paul filibuster on drone strikes against Americans, that type of thing. So yeah, it's a good question. The United States, look, the United States is not perfect in this regard, right? I think that in terms of freedom of information in the United States, I would never say that the US has like a perfect record on this and you do see movements in the US that the same kind of things that you see in these countries, you see online collective action, some of the examples you gave, to reverse what people see as restrictive US policy. I guess the one thing I would say about the US is that even though the US does not have a perfect record in this area, at least we are having an active debate about US freedom of expression and I think that's the one thing that sets the US apart from some of the countries in this book and that regardless of what you think about Snowden, at least we're talking about it, at least the media's investigating it, at least there's like a variety of opinions about it and that at this point is not something that could happen in China, Cuba or Russia. I don't know, Emery, I'm sure you have. No, so I think that's right, but I take your question a little differently which is particularly with Pippa and Sofa, I saw that actually as an interest, you could argue that's a case where internet activism overcame a kind of apathy here that many of us would have thought, come on, what are we gonna do? This is Hollywood, so you know, backing, this is an intellectual property, this is really hard, but what you saw, at least I mean, I experienced it almost entirely through Twitter, but was suddenly this thing just taking off and you realize by putting the Pippa or Sofa sign on your avatar, you could join this movement. So I think there's some examples, at least that's one where you could say even in a country where we are free to assemble and we are certainly unrestricted in many ways on the internet, if it's apathy, more than fear, I think, or isolation, it's been important here too, it'd be interesting to carry that out. I was thinking more of those times was actually getting information out into a critical map of the person. Right, and a much wider group. We'll never get out that way. Exactly, exactly, exactly, exactly. Yes, they're on the side. Right in front of us, the other way.  Hi, Maria Rapnikova, Georgetown University. Thank you so much. Hi, it's been nice to finally meet you, we were in Tash, I don't know, years ago, I guess, in Russia, so I have a couple of quick points. First about the actual process of research, what kind of challenges did you encounter the most, say in Russia, China versus Cuba? You talked about Cuba a little bit, but how do the other countries differ in that regard in terms of access and kind of just open dialogue? How welcoming were people to your questions? And also, what surprised you the most in this process? You spent a long time there, did something surprise you at the beginning or more towards the end? Like, what was the process like? And did you interview any nationalistic bloggers or those who are pro-regime and how do those conversations differ from the dissidents? Thank you. Thank you so much. Okay, let me make sure I get all those questions. So the challenges of Russia and China versus Cuba. So yeah, there were challenges in all three, in terms of my own personal safety, I mean, you're always thinking about that, but the difference between writing a book and writing a series of articles is you have the option of being very low profile for a long time. So I was very quiet about this project in order to get it done. I wasn't constantly blogging from these countries or tweeting from these countries, and that helped a lot. I think one of the big challenges in all three countries is building up trust, right? In China in particular, why would we talk to you? Like why is it in their interest to talk to you? And I think some of that just came with time. When you first show up, it's like, oh, here's some other American just coming to talk to us, and why would we? But when you come back year after year after year, they're more inclined to open up to you. And also when they see that you're, it is all about kind of connectivity when they see who else you're talking to and who your friends are, they're more likely to open up to you. So those were challenges, I think, building up trust. And of course, yeah, I mean, in Cuba, in Russia I didn't feel this so much, but in Cuba definitely, I mean, just feeling that surveillance was everywhere and you had to be very careful and you couldn't really operate the way you normally would. Normally you come into a country, you wanna have as many meetings as possible, you're kind of running from one place to another. In Cuba you can't really do that or you're gonna attract too much attention to yourself. And in the book I talk about an academic who was in Cuba right before me also doing research on the Cuban internet who was confronted at the airport and told to leave. He told never to come back. And that was because he just met so many people and he actually, what was interesting is he met people from all across the political divide. He met pro-government Cuban bloggers and anti-government Cuban bloggers and I think that was in a way even more threatening because I think somebody perceived that he was trying to like connect people or something. But yeah, it was just all about keeping a really low profile. Your next question was, oh, being surprised. So I think I was most surprised by Russia of all three countries. So when I came to Russia, I didn't even really have a way, kind of a means of analysis. I was coming from China, the Great Firewolf, China, and netizens writing in code to evade government censors and Russia, that paradigm didn't exist at all. So Russia for the most part had a perfectly free internet which is really interesting and the academic research on Russia at that time, this was a few years ago, was Russia, the web that failed. It was this idea that, okay, so you have all these people that are unhappy with the government and you have this internet where you can get all sorts of information and you can organize however you want and nobody's using it. Highly literate people, too. Yeah, highly literate people. Yeah, and significant internet penetration. I mean, that wasn't the issue and yet nobody was really using the internet for activism. So I found that sort of surprising. But then I also, what I saw in Russia is how just there are different versions kind of a possibility underneath the reality, right? So the first Russia I saw was this like apathetic Russia. You could find everything online, but nobody cared. You know, the second Russia was like late 2011 where all of a sudden there's 100,000 people in the streets and then the third Russia is like Russia now where it's kind of like more, it's gone back to this dark authoritarian feeling. So I think Russia has consistently surprised me and that's why I think Russia is really worth watching because I think Russia is very, very able to defy our expectations. Like I just don't think we can predict what's gonna happen there at all. And the last question. Negative bloggers. Negative bloggers. Yeah, so I've definitely, you know. Nationalists, sorry. Yeah, no, I've definitely. Sort of the same view, the same thing in my view. Yeah, actually, so Emery actually asked me a question about this on the podcast right before this. And yes, I have met nationalist bloggers, especially in China.  And I think one thing I'm not saying in this book, that everybody who gathers online is gathering as a force for good or as a force for democracy. There are definitely people who are, have less, I don't know, admirable views, right? I mean, maybe they're promoting violence, maybe they're promoting nationalism. So yeah, I have met them. I mean, nationalism in China is quite interesting because the conventional wisdom on online national in China has always been that the government promotes it and that they're stirring it up. But I found that often online nationalism exceeds what the government wanted. And so they tend to be victims of censorship as well.  Right? Yeah. Hello, my name is Oliver Krimmer. I'm a correspondent for an Austrian newspaper here in DC. Did you find that revolutionary conditions in any of the three countries you mentioned? And in its absence, what good is the internet? Anyway, doesn't it simply sort of put people who would be isolated, apathetic, and feared in the real world into that same condition in a virtual world? Thank you. So no, do I think that these countries are in a revolutionary situation at this second? Not necessarily. Do I think that could change quite quickly? Yes. However, I definitely disagree with the statement that if there's not a revolutionary situation, the internet isn't doing anything. You know, I think that's, you know, we were just talking about this also. I think there has been this kind of binary equation that people talk about here. It's like, well, is the internet creating revolution or not? And if the internet's not creating revolution, it's not doing anything. And I firmly disagree with that. You know, like you take China, for example, and I think the internet has already transformed China. I think it's really hard to argue with that. You know, I mean, first of all, the government can no longer completely control the flow of information. They try really hard, and they can control a lot of it, but not all of it. Information always gets through. It always, always gets through. Second, you know, people are using internet tool, online tools to demand accountability, to demand transparency, to demand a response from officials, and often they're getting it. And so the internet has created a world of possibilities in China that just didn't exist before. So no, is there a Chinese revolution? No, but that doesn't mean that China is not a radically different place. Also, I think the important thing to remember is to sort of be careful what you wish for, right? A lot of people here say, well, where's the revolution in China? But what would a revolution in China look like, right? I think that China's not really there at this moment. And I think if you went and asked the average Chinese person, do you want a revolution? They wouldn't say yes, right now. They wouldn't, because who would run China? You know, I mean, it's not there. And I think, you know, with the Arab Spring, everyone was kind of cheering on the Egyptian revolutionaries. And then once revolution happened, everyone was like, oh, these guys, they never, they don't have a plan.  I think you have to be really careful just, yeah, what you wish for. And so a lot of the activists in my book are not actually working for revolution. They're working for evolution, right? They're working to make their countries better. And they're working to like give more citizens a voice and kind of make their governments more open and transparent. And in many ways, they're succeeding, so. Certainly when I lived in Shanghai from 2007 to 2008, which is still relatively early, I was stunned by the role of the netizens. So admittedly, I'm reading the China Daily and I'm reading it in English and I'm aware that I'm reading in part what lots of people in the Chinese government want me to read. Even so, the sense that the netizens were a whole part of the population who had to be responded to, even if it was to shift cultures kabuki a little bit. But that was very surprising to me. I just, you know, this notion there was a Chinese internet public. And I would not have said that before. There definitely is. And you know, so there's a handful of people in this book who are more like your traditional dissident types, right? You know, the guys who talk about freedom and equality and whatever. But there's a lot of other quieter Chinese internet stories that are less known here, you know? And one of the bloggers in this book made a really interesting statement. He said, Chinese people don't care about freedom, but they do care about justice, right? So what does that mean? It's an interesting statement, it's somewhat controversial. What he was saying is that if you ask a lot of people in China about these kind of abstract goals of freedom of speech or freedom of information, it's not gonna resonate, right? But people want justice. And if they see that a local official is messing with their family, if they see that somebody is taking their house or their land without compensation, if somebody's getting beaten for no reason, they're gonna want justice and they're gonna wanna write about that online. And if somebody is censoring that, they're gonna fight back, right? So there's all these little battles that are being fought on the internet in China that are not like, they're not about this abstract idea, like we just wanna have Western freedom of speech. It's like we wanna use the internet to get the justice that we deserve. Thank you, yep, in the back there. Hi, Anna Swanson. I'm a student at SICE and also a researcher here at New America. I was wondering, you were kind of alluding to this with your last conversation, but your thoughts on the success of authoritarian governments in policing the internet. I mean, the internet in China, for example, the internet has definitely transformed society, but at the same time, I would say, I mean, I think that the government may have been more successful in policing the internet and squashing dissent than some people in the West thought it would be five or 10 years ago. But I'm curious to hear what you think of that. So I think that's true. You know, I think again, this is one of these false binary situations, right? You know, if you write a book about the power of the internet, people say, well, but what about the governments? And it's like, yeah, I know. I mean, I'm not denying that at all. Like, again, if you read this book, you will see a lot of harrowing stories about how the governments are trying to intimidate these bloggers. And I will not for a second deny that that's happening and I will not deny that the governments are effective. But I think in many cases, these governments are on the defensive.  The internet was so kind of limpid and nobody was using it. And now you're starting to see all these kind of blacklists and anti-extremism laws. And it's because the Russian authorities woke up and they said, this is a threat. You know, so often in my experience, the, as the government ramps up control of the internet, it's usually a testament to the internet's power, right? Why would they do that otherwise? Why would they exert so much time and so much effort and so much hardware into controlling the internet if the internet wasn't a threat, right? So the two kind of go hand in hand. And it is a cat and mouse game. It's a cliche, but it's true.  There's always going to be a struggle. But I have not seen any case in which, you know, any of these governments have been able to completely control the flow of information. And in that sense, they are not winning, right? I mean, because they could have done that more effectively before. Well, and this is where, for me, reading your book, it comes back to the psychological effect. Because even if you shut down the actual protests, the actual links, the psychology, the sense that I'm not alone, the sense that I act like a free man, the sense that I can have, I can make a difference, those things can't be put back in the bottle. Yeah, you just said that much better than, yeah. Yeah, that's exactly right. And that's kind of the whole point of the book, right? Is that we tend to talk about this kind of internet versus authoritarianism in terms of like how many words were censored, how many websites were shut down, how many bloggers were put in jail. And that's all relevant, and that's important. But the psychological elements, which are much more abstract, are impossible to erase, right? And when you kind of have this realization online, I'm not alone, I am not afraid, and I'm ready to take action, like censoring a website is not going to reverse that, right? And you have that realization online, but it's still an offline realization. It changes who you are, and it changes what you're ready to do, you know? One last question. Becky wants to ask a question. Kind of going off of that. I'm curious, oh, Becky Schaffer, New America. I'm curious, you talk about the importance of a revolutionary spark, and the apathy, the isolation, the fear, but how important is a narrative in creating the revolution? And with social media like Twitter, where there's such a limited space to tell someone's story or to share a narrative, how do you think that the role of narrative is changing in revolutions? I love that question, that's a great question. I come from a literature background, so I love questions about narrative. So yeah, I actually, you know, I'm really interested in the idea of narrative, and I'm really interested in the idea of, you know, who tells the story of a country, right? And so, you know, there's all sorts of work on this, and I'm sure you're familiar with this idea of like national narratives, right?  And when you have these countries where, you know, the past, present, and future of the nation is entirely dictated by the state, right? They're telling the story of that country. And then you have these bloggers who are kind of reclaiming that narrative, right? And I find that really fascinating and really empowering. And you see that in Cuba, you know, bloggers basically express that in so many words. They said, I'm gonna tell the story of my country, like as I see it and as I feel it. And they're kind of reclaiming the national narrative. I mean, your point about Twitter, that's well taken, but, you know, a lot of this is also happening kind of in blog form. And some of it is just, you know, it's just about different versions of the narrative, you know? And I think for people in countries where it's all the authoritarian narrative, that's very debilitating. And when they can kind of take it back and tell the story as they see it, it's an empowering thing. Well, so I have to, we'll end on that note, but the one, maybe this is a final question, I'm reading John Meacham's biography of Thomas Jefferson. For those of you who've come to many of these events, you can see how long it's gonna take me to get through it, because I've been saying this for a month or so. But, you know, it is so striking, all the documents that we have, right? I mean, that he wrote, you know, wrote constant letters that, and we have that, our national narrative is so shaped by our conception of the revolution and the way that gave rise to it. And so one thing I do wonder is, how on earth are we gonna tell this story looking back? I mean, you know, I'm not sure that even a million tweets adds up to the Declaration of Independence or a letter or the Federalist Papers or any of that. I mean, how, good note to end on, you are somebody who has a literary background, if you haven't seen it, there's this just wonderful book review of Emily's book by Mario Vargas Llosa, who sort of fuses the two, the politics and the great, great novelists. But, you know, what will happen to these narratives if they're all tweets and blog posts? Yeah, it's a real problem. I mean, even the kind of effort I had to make just to kind of get these things in this book. And it's funny because there is a certain blogger personality where they're just not interested in what happened the year before, you know?  Like, I don't even use that tool anymore. But yeah, no, it's really, I mean, just trying to kind of track these things down and kind of create this story was challenging. Yeah, I mean, because these websites, they disappear. And not simply because of censorship, but yeah, it's a fleeting story and I tried to document it as best as I could. So we not only have a new profession of digital curator, we're gonna need digital historians and they're gonna be able to rescue floppy disks and things, you know, what we think about it.  So anyway, so hang around a little bit. Emily's gonna sign some books and we have refreshments and drinks and it's wonderful to see all of you and give Emily and her wonderful book now I know who my comrades are, a round of applause. Thank you.


Let us continue with what we are doing in the previous lecture I am trying to prove this theorem that if X is an affine variety that the function field of X is just the quotient field of the affine coordinate ring of X which is the same as the ring of regular functions on X and if Y is a projective variety then the quotient the function field of Y is actually the degree 0 part of the homogenous localization at 0 ok and of course Sy localize at 0 itself will have a being homogenous localization will have a gradation and it is a degree 0 part of that if you take the degree 0 part you get field ok. Now I will explain more about this soon so but this is the point the point is that we define something in geometric ok and then we try to capture it using commutative algebra I mean that is the translation from algebraic geometry commutative algebra and that and what this is what this the theorem tells you what this translation means for a function field ok if it is an affine variety then the function field is the quotient field of the affine coordinate ring which is a projective variety then it is given by this which also produces a field ok. So let us try to prove that and then I can use this to go and show also that there are no non constant regular functions on a projective variety ok. So here is a proof so we start with x affine variety x is irreducible closed in some n and well ax is you have ax is just by definition this is affine coordinate ring of x is just the ring of polynomials in n variables namely the polynomial functions on this affine space restricted to x ok. So this is just this is by definition a of an divided by the ideal of x which is just and a of an is just can be identified polynomials in n variables over k the ring of polynomials in n variables over k these variables x i's are the coordinate functions on the affine space the n coordinate functions and here is your ax and by the way since x is affine this is the same as o x these two are one and the same ok. And what I am trying to show is I am trying to show that kx is the same as the quotient field of ax alright so yeah so you know we already have that o x sits inside kx this is something that we have already seen ok. In fact any regular function is also a rational function after all kx is supposed to be consist is supposed to be represented by regular functions on open sets and o x is regular functions defined on all of x and all of x is also an open subset of x ok therefore o x is a subring of kx ok this is something that we have already seen we have I have mentioned it here o x sits inside any local ring and that is further sits inside kx which is the function field of x and on the other hand o x is the same as ax ok and ax has its quotient field q of ax ok any integral domain sits inside its quotient field which is the quotient field is same as field of fractions any integral domain sits inside its quotient field ok and there is a property which says that you know the quotient field is smallest field in which the integral the quotient field of an integral domain is the smallest field in which the integral domain sits ok that is also called the universal property of the quotient field or field of fractions and therefore since the, since q ax is a quotient field of ax which is the same as o x it is a smallest field which contains o x and since kx is a field which contains o x ok this has to be contained in that therefore you know you get you get a unique ring homomorphism from here into this ok. So this ring homomorphism is a unique ring homomorphism which just expresses the fact that an integral domain sits inside its field of fractions or quotient field and the quotient field is smallest field which contains that integral domain. So if some other field contains that integral domain then this quotient there is an isomorphic copy of this quotient field inside that field ok and that is what is given by the image of this map which is an injective map. And in fact it is pretty easy to write out what this map is see the what is the quotient field of Ax you see Ax is given by polynomials f and g Ax is just O x, O x is regular functions but here the regular functions are just restrictions of polynomials ok. So you take an element of Ax it is a polynomial model of this ideal ok so you can write it as f bar ok. So if I take a polynomial in n variables namely f then there is this quotient which is going to model of the ideal of x and f goes to f bar which is an element here ok and of course f bar is the coset f plus i x ok. So you have this f bar so this quotient field will consist of things like this there will be things of the form f bar by g bar ok. So because of the field of fractions of an integral domain is consist of all fractions ok quotients of two elements of the integral domain with the denominator element not being 0 ok so the g bar cannot be 0. So this is how an element here will look like ok and what is the element to which you will send it after all f bar by g bar if you think of it as after all as a function this is just f by g ok after all f bar if what is f bar in Ax what does it mean it is simply restriction of the polynomial f which is which defines a function on this affine space to the subset x.   So if it intersects x you will get an open subset of x ok so the moral of the story is that this will define this x intersection dg, f by g. So this is I have this element which makes sense here so what you must understand is that you see dg is the basic open set in affine space where the polynomial g does not vanish ok dg is just the complement of z of g which is a zero set of g ok the zero set of g is a zero set set of points where g vanishes ok and mind you that is you know that is a hypersurface ok of course g g need not be irreducible but even if you factor g into its irreducible factors then this z g will be union of those z of those g i's which where the g i's are irreducible factors of g and each z g i will be a hypersurface ok which means a co-dimension 1 sub variety ok that is n-1 dimensional close sub variety of an ok. So the z g will be a union of hypersurface and its complement is dg which is a basic open set mind you this is a basic open set in the affine space and this is itself isomorphic to an affine variety ok by the Rabinovich trick this is isomorphic to a close subset of an plus 1 ok. So this dg is an open subset and x intersection dg is an open subset of x and there f by g makes sense right and so this is how you define this map ok and so that is how this map is defined this is how you will define this map naturally you can check that this here I have defined this map in a very geometric way by thinking of everything as functions but then I told you this map also comes because of the universal property of the field of fractions of an integral domain ok you can check that there also the definition is the same ok. So this map that you get either geometrically or via commutative algebra it is one in the same map right and so all this tells you that the quotient field of Ax is certainly a sub field of kx ok what I am to prove is that they are equal ok. So what I will have to show is that I have to show this map is surjective alright and how do I show that this surjective well take so you know let me call this map as something ok let me call this alpha ok.  So it is the same k here so the constant functions are there and this is a field which contains this smaller field so this is the field extension and this is again another field extension ok kx is a bigger field extension in fact we want to show that these two fields are the same ok why is alpha surjective because you know how does an element in kx look like in any element of kx is of the form well is of the form u, f, u, phi where phi is in O u and u inside x is open non-empty I mean this is how we define kx it is equivalence classes of pairs consisting of an open set and a regular function on that open set ok and this open set has to be taken to be an open set inside x right and but then you see a little bit of fins part will tell you that this is the image of an element from the quotient field of Ax and what is that element is very easy to guess by just concentrating attention at a single point so you know if x belongs small x is a point of u inside x ok then phi looks like f by g right with x belonging to dg I mean this is how a regular function is defined a regular function is something that locally surrounding each point looks like a quotient of polynomials so phi looks like f by g means the function phi from u to k can be identified with the function f by g which is also defined on a open neighbourhood of the point x to k and of course that open neighbourhood should be a neighbourhood where g does not vanish so it has to be in dg ok of course x is in dg intersection x which is an open neighbourhood of small x in capital X right this is just the definition of regular function ok but then you have f bar by g bar this element if you take f bar by g bar that is certainly an element in the quotient field of Ax certainly and alpha of f bar by g bar will be you will get this as I have defined this map alpha it is just going to be x it is going to be x intersection dg, f by g this is what I am going to get ok but then notice that this is the same as well and of course you know but this is the same as w f by g these two are the same where w is the open neighbourhood of small x in x intersection dg where phi is equal to f by g see after all phi looks like f by g locally at every point so I am taking only one point it is enough to take just one point to call that point as small x so there for this point small x I have an open neighbourhood w ok which is in x intersection dg ok and there on the w phi is the same as f by g ok that means you know so f by g make sense on w phi also make sense on w and so this is the same as w, phi ok and that is the same as your u, phi because that is the equation this is w, phi restricted to w so I should write phi restricted to w the restrictions are obvious let me not write it ok so in other words so this implies alpha is this alpha is subject and you have done thus the quotient field of x is the same as well isomorphism is alpha as I have defined it it is the quotient field of the function field of x kx is the same as the quotient field of ax ok which is the same as quotient field of ox because ox is the same as ax ok so it is very simple to get the function field of a affine variety simply take it is co-ordinating and take the quotient field that is it ok that proves the first part ok now you are going to prove the second part which is about the which is the projective case right so how do you prove that so let you start with y the radius will be closed in Pn, Pm ok y is a projective variety and I am trying to now calculate the function field of y alright and well then you know of course the the the projective space is a quotient of the punctured affine space above and you know the homogenous co-ordinate ring of the whole projective space is the same as the affine co-ordinate ring of the affine space above and this is just polynomials in M plus 1 variables and the the homogenous co-ordinate ring of y is just this modulo ideal of y where ideal of y is the is all those homogenous polynomials which vanish on y so this i h I will put a subscript h to just remind you that it is a homogenous ideal it is ideal generated by homogenous elements namely it is generated by all those homogenous polynomials in these M plus 1 variables which vanish throughout the locus y ok. Now we have seen in earlier lectures that you know that the projective space has an affine cover consisting of M plus 1 open subsets which are each isomorphic to affine M space ok. So you know we have P M k is union of U i i equal to 0 to M U i is just the complement of the 0 set of X i ok this is 0 set in this is a projective 0 set this is 0 set in P M each X i is homogenous polynomial of degree 1 it is 0 set will define a close subset of projective space it is called the hyper plane projective hyper it is called the projective hyper hyper plane defined by X i vanishing of the X i and it is complement is this U i ok and these U i's and these U i's are all open sets and we have seen that there are isomorphisms phi i from U i to A M ok there isomorphisms like this and you know how these isomorphisms are written U i corresponds to all those points with projective homogenous coordinates with ith coordinate non-zero. So what you do is if you take a coordinate like this which is written as lambda not through lambda M with colons to indicate that it is a common that it can be scaled by a common ratio these are the homogenous coordinates you simply send it to lambda not by lambda i lambda M by lambda i where you omit lambda i by lambda i this is the map this is the map that is defined theoretically and we proved that this map is an isomorphism of varieties and under the isomorphism of varieties this translates to this is a geometric map and it translates to commutative algebra to an isomorphism of the affine coordinate rings namely A of A and K so this is my pullback is pullback regular functions and give me a regular function here you compose with this you get a regular function here that is a map like this the opposite direction and this is the K algebra isomorphism this is of course isomorphism varieties okay and here I will get O of U i and we proved that what this O of U i is actually it is the coordinate ring of Pn Pm localized at Xi and then you take degree 0 part okay so we proved this so we proved this in an earlier lecture right and now this is the picture with the whole projective space now you rewrite the picture that corresponds to this closed subset Y okay so what you will get is if you intersect with Y you will get Y also as a union of I equal to 0 to M Y intersection U i okay so Y is covered by this Y intersection U i and each Y intersection U i is an irreducible closed subset of U i and under the isomorphism of P i it will give you an irreducible closed subset of affine space so it will correspond to an affine variety. So this just expresses the projective variety as a union of at most M plus 1 affine varieties so you know we use this to show that any projective variety is a union of affine varieties and we later on proved that any quasi affine variety is also a union of affine varieties and therefore we proved that any variety is a finite union of affine varieties so the affine varieties are the building blocks of all varieties anyway so you have this and what happens is that what happens to this isomorphism this Y intersection U i you will have an isomorphism P i the P i will give you an isomorphism with P i of Y intersection U i which is a variety there and what will happen is that the and if you this if we will translate if you go to combative algebra this will translate to A of affine coordinate ring of P i of Y intersection U i which is an affine sub variety P i of Y intersection U i is a irreducible closed sub variety of A m ok and again this is P i star which is pull back of regular functions by P i ok this is also a K algebra isomorphism and what you are going to get here is the ring of regular functions of Y on Y intersection U i and you know we proved that this is this can in fact we identified with S of Y you localize at X i and then take the degree 0 part ok. So it is this thing when you intersect with Y you get this alright and now here is a point the point is well you know K Y is the same as K of Y intersection U i if Y intersection U i is not 0 is not empty ok because I told you that the function field is not going to change if you go to a non-empty open subset Y intersection U i is non-empty open subset of Y I mean of course I am choosing an i such that Y intersection U i is non-empty open subset of Y if some Y intersection U j is empty I do not choose that j ok that is certainly at least one i such that Y intersects U i ok and you take that Y intersection U i that is an open subset of Y non-empty open subset of Y it has the same function field but then K of Y intersection U i is the same as K of Phi i of Y intersection U i this is equal to this because Y intersection U i and Phi of Y intersection U i are isomorphic because Phi i is an isomorphic but then so this is the function field of Phi i of Y intersection U i Phi i of Y intersection U i is Phi i of Y intersection U i is a sub variety of A m ok and so it is affine it is an affine variety and we have just now proved we have just now proved here the first part that if you have an affine variety then the function field is given by taking the quotient field of the field of a quotient field of the ring of regular functions in the affine coordinate ring so this is the same as quotient field of the affine coordinate ring of Phi i of Y intersection U i ok and of course here you know when I put equality maybe I should put here isomorphism because I am using the fact that if two varieties isomorphic then the function fields are isomorphic so I have this isomorphism between Y intersection U i and Phi i of Y intersection U i so these two are isomorphic varieties isomorphic varieties have isomorphic function fields ok so that is the fact that is pretty easy to understand I did not state it but let me try to write that down here suppose you have X so let me put you some other notation so if X1 is a variety and X2 is a variety and Phi is an isomorphism of varieties then I have Phi star the pull back of functions will induce isomorphism of K X2 with K X1 ok so if two varieties are isomorphic then their function fields are of course isomorphic the idea is very simple because you see if you give me an element of K X2 it is going to be a rational it is going to be it is a rational function so it is a regular function and open subset of X2 and if you pull it back you will get a regular function on the inverse image of that open subset in X1 and that is going to define an element of K X1 and you can check that this is an isomorphism it is a trivial check that if two varieties are isomorphic then their function fields are isomorphic ok so an isomorphism of varieties induces an isomorphism of function fields in the opposite direction always the isomorphism at the level of regular functions or at the level of local rings or at the level of function fields ok all isomorphisms that involve functions either locally or globally they are all induced by pull back that is something that you should not forget ok. So isomorphic varieties have isomorphic function fields and since y intersection ui is isomorphic to phi i of y intersection ui they have isomorphic function fields but the quotient field of phi i of y intersection ui is the quotient field of its affine co-ordinate ring ok that is what we have seen in the first part and then but this is the same as well you know so this is the same as quotient field of this isomorphic to the quotient field of o of y intersection ui because again if two rings are isomorphic if two integral domains are isomorphic then their quotient field isomorphic so the quotient field of this isisomorphic to the quotient field of this because these two integral domain is isomorphic but I already but I have already identified o of y intersection ui with sy localize it, so this is just the portion field of sy localize at xi take this degree 0 part. And now I want you to check it is a very simple check in combative algebra that this is the same as simply sy localize at 0 and take the degree 0 part ok. So it is the last equality that has to be checked and it is kind of it is very easy to check ok because you know so this is the only thing that needs to be checked, so you must understand what I am doing here, here I am using the fact that if two varieties isomorphic their function field isomorphic and here I am using the fact that if two rings are isomorphic two integral domains are isomorphic then the quotient fields are isomorphic and then I come to this ok. And yeah so how does one prove this last equality that is also little bit of combative algebra, so let me rub this part so that I can continue. So what is sy localize at xi is just elements of the form f by xi to the power of r equivalence classes like this where f is in f is a polynomial in these m plus 1 variables and r is greater than equal to 0. I mean this is what it is and this square bracket is equivalence classes ok this is what the localization is sy localize at xy is just invert xi ok invert xi means you are allowing denominators with positive integral powers of xi right and what is sy xy xy xi localize at 0 is well it is such things which with homogeneous degree 0 which means that you know the numerator should also be homogenous denominator should also be homogenous and they should be homogenous of the same degree so that when you take the difference of degrees it is 0 ok. So this is this will consist of things of the form well sums of finite sums finite sums and of course also finite products for that matter well I could simply write it instead of saying all that I could simply write it as f by xi to the power of r such that f is homogenous of degree r this is a degree 0 part right and of course the degree 0 part sits inside this ok and what about this so that is and you know if I take the quotient field of this ok this is an integral domain alright and I take its quotient field so if I take quotient field of sy xy at 0 this is what this will sit inside and this will consist of quotients of things like this so it will be something like f by xi power r divided by some g by xi power s it will be quotients like this where f, g homogeneous of degree r and s respectively this is what it will be alright if I take the quotient field of that right. And well and what you can understand is that you know I can identify I can identify an element like this with the element f times xi power s by g times xi power r ok this if you just divide these two you know very nively this is what you will get ok and the numerator will have degree r plus s denominator will have degree r plus s so this is the quotient of homogeneous polynomials of the same degree ok so what you are going to get is you are going to get an element here ok so see this belongs to sy localize at 0 degree 0 part I will tell you what this means see sy localize at 0 is equal to is gotten by inverting all nonzero homogeneous polynomials ok this is the notation is ring localize at a prime ideal ok so the point what I want you to notice is that this is homogenous localization see normally when you localize a ring at a prime ideal you are supposed to invert everything outside that prime ideal ok but here it is homogenous localization what you are doing is you are not you are inverting every nonzero polynomial but you are inverting only nonzero homogeneous polynomials the reason why you are inverting only homogenous polynomials is because only then this localization will get a gradation ok if I mind you this sits inside sy localized at 0 so you know this when I write like this this is the usual localization 0 is a prime ideal and ring localize at a prime ideal is invert everything outside that prime ideal so this is actually the quotient field of sy this is quotient field of sy this is very big this is very big because this consists of quotients where in the denominator you have put any polynomial which is not 0 in fact you are inverting even nonzero homogeneous polynomials even now I mean sorry you are in fact you are inverting even non homogeneous polynomials alright but this is smaller you are inverting only homogeneous nonzero polynomials only those homogeneous polynomials which are nonzero you are inverting them so this gives you a subring of this which is the quotient field ok but this is graded because the gradation can be defined as degree of the numerator minus degree of the denominator because denominator consists only a homogeneous polynomial which is nonzero because you are here you are inverting only the nonzero homogeneous polynomials and in this if you take the degree 0 part that is what this is so this consists of exactly elements like this ok this consists of exactly elements which are given by numerator homogeneous polynomial denominator also homogeneous polynomial they are both of the same degree ok and therefore you can see that this is certainly sitting inside this homogeneous localization degree 0 part to which this belongs and that is a map you send this guy to this guy alright and you can check that this map is injective alright what we are trying to say is that this is also surjective that is what we want to prove we want to show that this there is an isomorphism between this and this ok I have written equality but in most cases the equality is just isomorphism with proper identification an isomorphism can be treated as an equality I mean if you think of this function as the same as this ok as a function then you get equality but if you prefer to write it like this with all these weird brackets here and another weird bracket here if you want ok then you can you can think of it as a formal isomorphism ok but if you just think of them as functions they is just equality right. So this is sitting inside this and conversely you know you can check that if you give me any element here it comes from an element here ok so the only fact that needs to be proved is that this is surjective alright and how does one show that it is very very simple conversely you know see if you want let me call this map as alpha sub i let me call this map as alpha sub i so I just want to say that alpha sub i is well an injective kalgebra homomorphism of course you know see the fact is that this is a non-zero homomorphism from a field and you know a non-zero homomorphism from a field is always injective. So you do not have to worry about verifying the injectivity because the sources are field alright being the quotient field of this integral domain alright I have to only worry about the surjectivity so how do you get surjectivity surjectivity is also pretty easy well take an element here how does an element here look like it will look like a quotient of 2 homogenous polynomials of the same degree alright. So take f by g take a quotient in this which means that f and g are homogeneous of the same degree and of course g is non-zero because whatever you put in the denominator whatever you invert is non-zero so any element here looks like this and I want to tell you that this element actually comes from an element here ok and that is perhaps pretty easy you can easily see that I just have to divide by the same power of xi as the degree of f which is the same as degree of g so it is obvious so you can see that so it is surjective obviously so clearly this element f by xi degree f degree of f divided by g by xi degree g this element this fellow is in sy localize at xi degree 0 and its image under alpha i is f by g ok. So that implies that alpha i is an isomorphism surjective hence an isomorphism ok as functions if you consider everything as functions then these two are not just isomorphic they can even treat them as equal ok so this proves the other fact that ky is nothing but take the homogenous coordinate ring of y you localize at you take homogenous localization at 0 and then take the degree 0 part this is an isomorphism ok so that proves the theorem ok. So it is very clear that you can you know how to write commutative algebraically what the function field is for an affine variety for alpha projective variety it helps you with calculations ok and there is another fact that that comes with there is another fact that comes with that comes with in connection with these two diagrams the fact is the following the fact is that of course the function field is an extension of the base field small k ok and the fact is that the transcendence degree of the function field O of a variety over small k will give you the dimension of the variety ok so that is also true. So you know so here is a so that is a very well corollary corollary is if x is affine dimension of x this is topological dimension of x is a dimension cruel of ax this is also the dimension cruel of ax at mx by x of the local ring this is also equal to a transcendence degree over small k of the quotient field of ax which is the same as transcendence degree over small k of the function field of x and mind you mind you that this guy is the same as this isomorphic to O x small x and this guy is isomorphic to O x so this is one corollary so you know you have so many formulas for the dimension of an affine variety it is either the topological dimension of x or it is the cruel dimension of ax if the ring of polynomials recited to x ok or it is also the cruel dimension of the ring O x which is a k algebra ok it is also the cruel dimension of ax localized at mx because you know ax localized at mx is the local ring of capital X at small x ok and this is the same as the transcendence degree of the quotient field of this and that is the same as the transcendence degree over k of 4 small k of capital Kx so this is any this is one important corollary and if you have and you know the point is that if it is projective also all these statements are correct except that you do not bring in the global regular functions because as I told you we are going to prove that for a projective variety all the global regular functions are all the only global regular functions are all constants ok so if if if y is projective then you know I will get I will have to forget this ok this is the only thing that I have to throw out but everything else everything else can be written so let me write that out if y is projective then also you will see that dimension y topological dimension of y is the same as the cruel dimension of the the local ring of y at this point small y and there is also the transcendence degree over small k of the function field of this is what will happen ok and so why is this a corollary of all that we have seen I have to explain that and the see we have already proved the topological dimension is the same as cruel dimension of the affine coordinate ring for an affine variety and but you know but then you know if you take the cruel dimension of the local ring ok then you see the cruel dimension of the local ring is just the cruel dimension of this ok but the cruel dimension of this is actually the same as the height of this maximal ideal ok see because when you localize at a prime ideal in the local ring your only the only prime ideals that survive are the prime ideals which are contained in the in the localized prime ideal in the in the prime ideal that you are localizing ok. So what is one thing in community algebra that you should remember about localization if you take a ring if you take a commutative ring A and if P is a prime ideal if you go to AP what are the prime ideals in AP the prime ideals in AP are precisely the prime ideals in A which are contained in P by going to AP you are you are throwing out you are removing all the prime ideals which are outside P which have elements outside P. So by going to AP you are actually only focusing attention on prime ideals which contain P so you know the cruel dimension of this thing this is a local ring so the cruel dimension will be it has only one maximal ideal and you know cruel dimension of a ring is just supremum of the heights of all its prime ideals so it will be the height it will be here the biggest ideal is this maximal ideal so the cruel dimension of this ring is the same as the cruel dimension of the height of this maximal ideal. But then this local ring model of this maximal ideal is the is just the base field K ok this Ax localize at Mx modulo Mx is just K ok and dimension formula will tell you the dimension of K is 0 therefore the dimension of Ax mod therefore the dimension of Ax localize at Mx is the same as the height of Mx ok and but but then that height will be equal to the cruel dimension of Ax ok and that is the reason why you get this equality ok and this equality comes because any integral domain and it is that if you have a so this equality comes because of the fact that you know so this also comes from a formula that I have stated earlier if you have a finitely generated K algebra which is an integral domain then the transcendence degree of the quotient field of that integral domain is the same as the cruel dimension of that of that algebra ok so so so if you apply that to Ax you will get that this is equal to this ok because Ax is a finitely generated K algebra which is an integral domain ok and it is so it is transcendence degree over K has to be equal to the cruel dimension of Ax ok so that gives this equality alright so you do not get this is equal to this rather you get this is equal to this and therefore this is equal to this because these two are equal ok and of course we have proved Q of Ax is same as Kx so this is also equal to the transcendence degree over K of Kx ok and the same kind of argument will give you will yield these equalities in the affine case ok and you have to only remember that when you if you want to do this for y it is enough to do it for yi where yi is an affine piece of y which is gotten by intersecting with ui ok and if you do it with a instead of if to do this computation it is enough to do this computation with y replaced by yi intersection ui which is yi if you want ok and then that case is already covered here therefore this follows ok therefore that is the proof of this statement ok so I will stop with that.