Thank you for the introduction. Ladies and gentlemen, my wife.  I work at Woodville for Image in a Box, which is a digital marketing agency in Huntsville. Yeah. You can catch me on Twitter at JacobMCDev. I actually just tweeted out some slides. If anybody wants those, there's also a link at the end. But who wants to write down links? So real quick, how many people know what Gutenberg is? No one's like, what's this? Okay, most, kind of half and half, how many people have actually used it via demo or on your site? Okay, just a couple. Awesome. So this is a 101 class fundamentals basics, the gist. I actually really kind of missed an opportunity when I named the talk. I realized after I'd submitted it and it got accepted, there was a much better name. So we're going to print it in like this never happened. And the real name is Gutenberg, right? Yeah? All right. So welcome to Gutenberg. We're going to be going over the Gutenberg user interface, some very basic stuff. Look at what it looks like. Some of the current features, including blocks and some other things. We'll look at a little bit of Gutenberg in the wild, not a whole lot, but some things that are being done with it and some things you can already utilize. And then we'll talk about how to prepare your website for Gutenberg. Very cursory stuff, don't get too excited. It's all just pretty basic. So the first thing we'll look at is the Gutenberg UI, title slides, right? And the WordPress we all know and love. This is what is now being dubbed, if you're not aware, as the classic editor because post-Gutenberg, we have two things now, the classic editor and the Gutenberg editor. This is the WordPress editor everybody knows. You should all be familiar with it, I would assume. It's really basic. We'll go over it just for the sake of comparison. You've got your post title field at the top, you enter stuff in, that's the title of your post, really advanced stuff. Below that you've got your content where you'll type in your content. You have your little buttons for formatting, your add media button to add images and stuff. Over on the right, you've got your published box, categories, tags, featured image. And if you have plugins enabled, which you most likely do, you will most likely have other boxes floating around somewhere. It's all basic. We're all familiar with it for the most part. And this is where we've been and this is where we're going. This is the Gutenberg editor. It looks kind of sleek, you know, a little fancy, it looks like there's a lot less going on. There's actually a lot more going on, it just kind of is packaged better. We'll start, you know, in the top left again. You've got this little, it might be hard to see, I don't know, can everybody see it pretty well in detail? All right, you've got this plus button where you can add blocks and we'll talk about that later and talk about blocks so don't get too excited, hold on. We've got these fancy little arrows which are actually undo and redo buttons.  And those didn't really work that great a few versions ago, but now they work awesome, so don't be too scared. Then we've got this little I icon for document information. We'll talk about that pretty in-depth. In the middle, you've got your add title button, or add title field, which is the exact same thing as the post title field. It's not different at all. It just looks a little different, so don't get too scared. Below that, you'll see this little prompt to write your story. This is where you'll add blocks and whatnot, you can add paragraphs and all kinds of stuff, and we'll talk about that more. Don't get too excited, calm down. Over on the right, way over here, I need a pointer. I don't have one, but it's kind of great, so it's kind of hard to see. We've got preview and published buttons in the top, so it's a little different than the classic editor. We've got a little gearbox, all that does is toggle the settings pane. It's not really that important. And then some dots that lead to another menu where you can look at code or copy all the content, which are kind of cool, but not really pertinent to what we're talking about. And then we've got the settings pane, which looks really different, but is really similar. At the top, you've got your status and visibility, just like you do in your published box. You have a little stick to the front page, which just makes it a sticky post, in case vernacular is confusing. Pending review just sets it as a pending review status. It's really not difficult. We've got categories, we've got tags, featured image. It's all there. It just looks a little different. All these things are also collapsible, much like they are in the current UI. A lot of people don't know that those are collapsible, but they are. And it's all pretty standard stuff. Nothing's too crazy. So we're gonna talk about now what everybody wants to talk about. Those are blocks and features. Because these are the things that are really different. They don't just look different, they're very much different. Gutenberg blocks, this is what a post will look like that has blocks in it already. These are just a sample post thing. We've got a cover image, which is new. It's not something you really did before. We've got headings, which are typical paragraphs. Image is below that. My screen was too small to get a whole lot of them in there, but there it is. And this is kind of what Gutenberg starts to look like. Wordpress now is more visual, whereas used to, we don't have a good example of it. But our posts were really just text and short codes. And maybe some images, but mostly just text and short codes. And it was really hard to move stuff around. Because you'd have to cut and paste text, or you'd have to move images. And images are really hard to align if you're not used to it. You might have to add special classes. So you might need to know some HTML or some CSS. But with Gutenberg, we don't really need all that. We just drag blocks over. They have settings. We set them. If you don't like where it is, you just move it. It's fine. You don't have to copy. You don't have to paste. It's really, what's the word, intuitive. It's very simplistic. It works the way you would expect it to most of the time. There's still bugs. It is new technology. But for the most part, it's pretty polished. And for what most people use it for, it's good out the gate. And what's really important in Gutenberg, I think two things that really make a difference. No more short codes, right? Has anybody ever typed a really long short code? I'm been confused. I don't know what happened. Yeah. So what do we do? Short. Well, it really depends. It really depends. Right. And we might talk more about that towards the end. And we kind of go for that a little bit. Maybe. Nathan. Right. Oh, is that what I had for you? Well, we blinked and it killed the projector. Who blinked? Which one of you blinked? Did the tower of vision hear? Yeah. I think I'm back. It's a little darker, but I'm back. We'll get there. Look at this. Technology. So anyways, yeah, no more short codes. I don't know if you've ever written a really long short code with a bunch of parameters and stuff. They're a pain. You don't want to do it. So now you don't have to, most of the time. And we also can now do things like save blocks. You can take some of these settings. And so you have a call to action kind of thing, like you've got a nice little text with a button. And it looks all pretty. And you don't want to have to remake that on every single page. You can save it and just reuse it. It's awesome. Moving on. These are all the blocks that are available. There's a lot. And we're not going to talk about all of them. I'm sorry. But there's a lot. Some of the ones to note.  Some ones I want to point out is the classic block, which is just the classic editor in block form. And this is mainly just used as far as the only use case I can really find for it. When you load a post that's not Gutenberg into Gutenberg, it puts everything in the classic block. So it's there for you still. And it's still compatible. And then you can take stuff out of that and move it around. But it's basically just the old WordPress editor in a block. We have tables now, which are really cool, because I don't know if you ever tried to put a table on a page in WordPress. It's not fun at all. I have done it. Don't recommend it. We have columns, which are in beta, and they have been forever. And I don't know if they'll ever come out of beta. I'd assume they will. But all that does is you can put blocks next to each other. Right now, it's hit or miss. Text works fine. Images work fine. You put some of the more advanced blocks in there. I don't know what's going to happen. And then we have page breaks, which are really neat if you've ever written a really long blog post that is just really taxing to read. You can break it into pages. And you have a little pagination thing at the bottom. You can have page one, two, whatever. One caveat about the page break. The way it does, I think I unplugged something now. We're good. OK. We're going to get through this. The way it does the page breaks is it'll do your post URL slash two. So honestly, I don't know how that affects SEO or if it affects SEO. So you might want to talk to somebody who knows more about SEO than I do about that. But that's how it works. But it is still neat. We have a thing for short codes. I didn't put it up there. I think I forgot it. There is a block for short codes if you still want to use your short code or if you're using a plugin that hasn't developed an actual block yet. They will still work in Gutenberg. And then we have embeds. Because WordPress supports a whole lot of embeds. I think there's like 34 or something. And now we have a block for each and every one, which you could just put in an HTML block and it'll still work. But we have a block for each and every one. So these are all the blocks. They're all pretty neat. I haven't used any that have had a lot of problems other than the columns one. But yeah, there's all the blocks. We're going to look at block settings. And each of these is a different setting for a different block. But this is what the settings will look like. Some blocks will have settings on the actual block. But for the most part, for the vast majority of the settings are going to be on the sidebar. Or at least the same settings that are on the block are also on the sidebar. And we've got some interesting things. If you look at the paragraph, which is the first one here on the left, we've got text settings. You can automatically set your text size to what has been dubbed small, medium, large, or extra large. Or you can toggle it with a slider. Or you can enter it in by hand if you'd like to. We have this drop cap option that's weird. And you can make your first letter be bigger than the rest of your line. I don't understand it. But people like it. We've got background colors and text colors. And if you don't like what's there, you can select a custom color. We've got advanced tabs where you can add CSS classes. And I chose these blocks to use, because they have a good sampling of all the fields that are available in settings. Under headings on the heading block is the next one. You've got the H1 and H2 through H6. Those actually change the HTML element of the heading as you would expect. There's alignment buttons. You can set an HTML anchor so you can put an ID on that tag and actually navigate to it with a link. And then we've got cover images where you can change the fixed background. It's like a parallax type thing if you're familiar with that. You can change the opacity with the background in this. And then over on the images you can set alternate text and whatnot in different sizes. It's all really kind of standard to what we already do in WordPress, but it's a lot more difficult to do. And there are custom settings you can set where your theme can actually manipulate. Some of the options are available to make your content editing experience stay within the bounds of your brand, or at least your theme settings. But those are block settings. And it's really standard. Oh, and I didn't mention in the beginning I meant to. They're in the same pane where the document settings were before. You'll notice at the top the document and the block. You can toggle between the individual block settings or the entire post settings. So this is what block settings kind of look like. When you click it, it happens. You don't have to like garner the days of changing something, preview post. Changing something, preview post. Changing something, preview post. You can see it right there, which is kind of neat. I have a lot of gifts, by the way. None of them are funny, but they're all there. And they can be helpful, I think. Moving on, this is how you add blocks in Gutenberg. We've got the plus symbol up there that I told you about earlier. It drops down. And you have all these blocks available. And you can search for blocks, which is really neat and really quick. Block developers can actually add tags to their blocks. So when you type in title, heading pops up. You don't have to search for the exact block name, because you may not know the block name, but you just type what you want. And there it is, assuming it's tagged correctly. Another way to add blocks is inline. You can actually put it in the content. Instead of going up to that, I mean, you can do it where you're editing. I mean, it works the exact same way. It's just in a different spot. Moving blocks is really easy. We have these fun arrow buttons that slide the blocks through the content real smooth. And the content moves around your block, so you're always looking where you're editing. It's not going off the screen. You have to scroll. And it works really nice. I didn't really like it when we first got it, when I first looked at Gutenberg. It has really been polished and works much, much better. There is another way to move blocks by dragging. You have to kind of grab in a really obscure area that's not obvious and jiggle it around to get it to go in the right place sometimes, because it can be difficult. But they're working on it. See, I was having trouble. So moving the content around, and you kind of start to see how Gutenberg really makes things easier from a layout standpoint. This is the document outline I mentioned before. It has some handy information that we've never really had in WordPress, such as word count, heading count, paragraph count, and for whatever reason block count if you really want to know how many blocks you've used. But more importantly, it has this document outline which you can actually use to navigate between your post. If you're editing something, you want to go, oh, I want to edit the other section, you just open the document outline and jump to it, especially if it's a longer post that can be helpful. Which that kind of leads us into some other features of Gutenberg that I'm really excited about, and I think are really helpful, and kind of where the WordPress community has been going a lot, especially the past couple of years, and that's accessibility. The document outline, if you notice, this H4, it's all red and stands out, because that's not how you should nest your headings. They need to be one, two, three in a correct order for SEO purposes, but more importantly, for accessibility, because a lot of people who use assistive devices to navigate your website, they do so via the headings. That's how text readers do it. That's how other assistive devices do it. They go through the heading. So it's really confusing. The technology can get confused whenever you put an H4 under an H2 without an H3 between, because it really doesn't understand what you're trying to do there. And Gutenberg will tell you, you're like, hey, that's not right. You should fix it, because people use your site, and you want them to be able to find things. Over here, we also have, in the color selector, if you select two colors that don't have a very high contrast, as you noticed, I selected the red and the orange. Those are terrible together. It'll tell you that the color combination can be hard for people to read. I appreciate that, because I'm color blind, and I really hate when people put blue text on red backgrounds, but it happens more often than you would think. But so with Gutenberg, we start to get some features that help us to be more proactive in making sure that our sites are usable by people who may have special considerations that need to be made. So some other features we have are opt-in features, as they like to call them officially, and you gotta attach code for these. I'm sorry, what you do? You may need a developer to help you out. It's really basic though, if you've ever done anything with your functions.php file, you can drop these in, it's not difficult. They've got default block styles, which, for whatever reason, aren't enabled by default. I mean, it makes sense in a post-Gutenberg world, but in the transition, it's kind of really confusing. So basically what that means is, when you load Gutenberg on your site, none of the styling that Gutenberg provides would be provided. It will all be relied upon you or your theme to provide that styling. And if you don't wanna do all of that work, and you don't mind the default styling, you probably wanna enable the default styling. Otherwise, it won't be there. They have wide alignments, which lets you set your content to a little bit wider than the content area or to full width of the content area. Those will need special CSS handlers in your own. I mean, it kinda works out of the box, but there's a good chance you're gonna have to edit CSS and make sure it works well with what you have. Block color palettes. You can have your theme actually set the colors that are available to you to make sure that you don't use a blue that's slightly different than the blue that you use everywhere else on your website. You can have them disable custom colors so you don't have your content person go in and put red on a blue background or whatever. I've not known anyone that did that. That wasn't a shout to anybody who may be in the audience. You can also do block font sizes. So you can set custom font sizes. If your default font size for your site is at 18 pixels rather than 16, which is generally the norm, you can set a special font size to that so that whenever you can click the small, medium, large, extra large, and you always use the same font size. And it's what you've determined and not what Gutenberg thinks you should use. Those are all again kind of advanced, but there is, I did link to in the resources a long post that is the official WordPress documentation for it on how to do all these. And again, it's really simple, really easy, straightforward, especially if you know any CSS or PHP. Moving on, we're gonna look at a few examples of Gutenberg in the wild. First, the atomic blocks theme, which came out recently. And as far as I know is the first theme in the WordPress repository that officially supports Gutenberg 100%. There might be others, but that's the first one that I saw because they do good marketing. So this theme actually enables all the features that you saw before and it's programmed to work really well with everything that Gutenberg provides. And it's a good example to look at if you want to check out what other people are doing to make sure that you kind of do the same thing and can figure out those stuff, you get it. Atomic blocks Gutenberg blocks collection, which is a little mouthful, adds a whole bunch of stuff that's kind of like post builder type blocks, very similar to like what you might find in Beaver Builder or one of the other ones like Divi or something. I don't like Divi, but anyway. Post-grade container, all those things, share icons called actions, really standard stuff. I haven't actually used it, but I've looked into it a little bit and from what I can tell it's really solid. There's a lot of good reviews about it, a lot of good write-ups on it. One that's kind of innovative is the Drop It plugin, which is really neat, I think, and kind of a good example of where Gutenberg can take us. It allows you, if you're familiar with Unsplash.com, which is a website where you can get royalty free photos. They're all high definition and they're free to use on your site, however you would like. It lets you actually search the unsplash.com gallery in your post editor and drop them in there as you would like. So you don't have to go through Unsplash, scroll through everything, it went away again. You don't have to go through Unsplash and scroll through everything like you would normally and then download it, upload it to your media folder and then go that way. It's all right here in the editor, so it's very easy. One moment, I don't think any of you know what we're doing. I thought it was battery-driven, I thought it was battery-driven. You can find some delays, that's my guess. Yes, see the batteries are going there, see the batteries. Uh-oh. Yeah. Has anybody else had this much trouble at all? I don't know, yeah. Someone's got it. Right. So I'll take the bullets nobody else has to. I'll try to talk about it so everybody can hear me. So this is a good example of what Gutenberg can do. Technologically speaking, it opens up a lot of possibilities for us that weren't really possible or at least were much more difficult before. It's not gonna be a lot different now, especially on a front-end standpoint from like what your users will see. But the post editing experience is much different and will eventually, we hope, turn into what we'd like to, you know, like true front-end editing where you can go to the front of your website, open the WordPress customizer, and it works like this. Kind of like what a lot of page builders do, but it has no batteries right now. I don't know. Hold on a second again. Everyone do you want to try the Elvis? What are you waiting for? I just want to see what it looks like. Are we good? Almost? Sure, I'm just out. It's okay, I can get it for you. These are new. Oh, those are new? We just wanted to make sure that we can make sure it wasn't for Quiltman. Awesome, are we good? Hey, I'm back. All right, so some of the things that are down the line that we hope for Gutenberg are full front-end editing, like I had mentioned, and also true post templates, meaning that you're developing a program something to where when you open a post, all the information that you typically put in it is right there. So if it's something like a product review that you do and you always want the certain images and you want to keep the same headings or headings in the same place or whatever, it can already be loaded there for you when you open the post. You don't have to go and pull in all the blocks again all over again and do it from scratch. And that's kind of where we're going with a lot of this. So, preparing for Gutenberg. What should you do? You should get excited for one. Everything's kind of changing and in a good way, I think. There's a lot of critics out there for Gutenberg and a lot of critics for the way it's being handled, but at the end of the day, it's still coming and it's still good as a piece of software. So I think we need to be excited about it and ready for it. So how can you get ready with your personal site? Well, number one, you need to be up-to-date on everything. You make sure your WordPress installation is up-to-date which is a given. You should be doing that anyways for security reasons. Make sure that all your plugins are up-to-date. If you have any legacy plugins, things that aren't being maintained anymore or haven't been updated in a very long time, you might want to consider replacing them because one, it can be a security risk for you and two, no one wants to use software that nobody's looking at anymore. There's always bugs in software. They will always be there. And if nobody's maintaining your software, that means you're using software that has bugs that nobody's fixing. And generally, there's always an alternative. It's very rare that you'll find a plugin where there is no alternative for it and if you do, maybe you should write one. I don't know. Number four, play with Gutenberg, check it out, look at it, just get familiar with it. You don't have to commit to it yet. There's a thing called Frontenberg out there, kind of sounds weird, but it just lets you play with Gutenberg on the front end. There's a couple of sites out there. The official Gutenberg informational site for oneonwordpress.org is a Gutenberg front-end instance. You can go and you can read about Gutenberg while playing with it. And then there's another one on the WordPress VIP site and I'll link to both of those. But get out there and play with it, look at it. It's not that scary, especially once you mess around with it. You might even like it, who knows? If you're looking to upgrade to it or to add it to your site, which I think you should consider, you wanna set up a staging site or a local development site because you do not want to put it on your production site without having tested it. The issue that a lot of people have with Gutenberg and the issue that is real with Gutenberg is that there are a lot of themes and plugins that have not updated or added compatibility with it. You may use some of those, I don't know. It's, and then there's some argument for how short of a timeline Gutenberg has been stable versus how long plugin authors have had to update to it. So you may wanna wait a little while. If it's not compatible now, it will be in the future hopefully. But you wanna set up something local or something staging, which just means not your main site and test it. You wanna make sure it's compatible with all your plugins. Test all your functionality. I am not telling you to go download it on your production site right now and have fun because I don't wanna be responsible for anything that happens. But once you have tested, once you have checked it out, made sure it's compatible, maybe updated your stuff or added functionality that makes it compatible. You might need a developer for some of it if you have very customized platforms. But once you've done all that, download it, use it. It's awesome. I use it on my site. I really enjoy it. In my job, I do a lot more development than I do content writing. And by a lot more, I mean 100% more. I don't do any content stuff. And I really enjoy playing with Gutenberg. I think it's really fun. There's a lot of possibilities that didn't exist before. But yeah, so that's what you gotta do. There's a lot of stuff. I know the technical part of this is the staging or local development. A lot of hosts will set up staging sites for you. There's a lot of articles on how to do local development stuff. It's really not that scary. So just go do it. Here are some resources. All the plugins I mentioned, all the sites I mentioned are here. There's a link to slides down at the bottom. Yeah, that's it. Any questions? Yeah. So, well thank you. Sure, so the question is, if you currently, to paraphrase, if you currently use a different kind of editing platform, like a page builder, I assume is what you're talking about, should you switch to Gutenberg or stick with that? What are you using, do you mind me asking? Divi, I don't know about Divi specifically, whether they have added compatibility for it or not. I know Beaver Builder has added compatibility and I believe Visual Composer is working on it if they haven't already added it. And I assume Divi probably is too. If you're already familiar with it, there's really no reason to switch unless it just gets to a point where they just stopped supporting it, which I doubt. Initial support for Gutenberg and Divi is coming. So you shouldn't have to worry about it. All the page builders out there that are popular and being used and being maintained actively that have companies backing them, know about Gutenberg for sure. And they're definitely making sure that they run well on it. Again, I don't think you should switch necessarily unless for some reason you just want to. I don't think there's a need to really. Everything should still work fine. In fact, Gutenberg may enhance the way it works. Does that answer your question? Yeah. Good question, I meant to cover that. The question was whether the Classic Editor will eventually be phased out or if it's always gonna be optional. The Classic Editor has actually been added to the plugin repository as a plugin just like Gutenberg currently is. When Gutenberg eventually gets adapted into Core, it'll be optional for a time. And you'll still be able to use the Classic Editor. Eventually, the idea is to disable the Classic Editor in WordPress altogether. But as long as that plugin is maintained, the Classic Editor should be available. For the most part, I doubt that there will ever be any kind of attempt to make sure it does never work again. So you could feasibly still run it. And I believe that, if I remember correctly, this is, don't take my word for it. But I believe that they will still be maintaining a version of WordPress with the Classic Editor for several security releases after. So if you don't want to update to 5.0, for instance, you could probably stay on 4.9, whatever, and still get security updates for a while. Eventually that will stop, but they'll have it for a while. Yeah. No, actually, that's a good question. The theme support documents will actually tell you how to add theme support for Gutenberg to a theme. Like some of the optional features that I would mention with the default styles and all those, that's where actually you'll find that stuff. That's a good question. Most of them, if you're on the plugin repository, for instance, most of them will advertise that they work with Gutenberg. But other than that, there's really not a good system right now. Like you couldn't go and say at the moment that I know of Gutenberg compatibility, I want to see only those and see those. Sure, yeah. Yeah. You should not, if you haven't tested it, if you have tested it and it works fine. Well, that's what I was talking about earlier. You set up a staging site or a local development site to test outside of your production site, which is a little more of a technical step. And you could have a developer do it if you needed. You could, you could. But something might break. It's possible, I don't know. It depends on what plugins you're using. You could go through all your plugins and check Gutenberg compatibility. If you use a lot of really popular plugins, or like name brand plugins or whatever, whatever that means, like Yoast or Ninja Forms or Gravity Forms or WooCommerce, all of those are Gutenberg compatible for the most part. Does that answer your question? OK, yep. So that will be based on what your theme styling is. As far as the default Gutenberg styling is, as far as I know, it's all mobile responsive. I haven't tested everything out, so I couldn't tell you for sure. But it really depends on your theme styling, because Gutenberg still is functionality. It's not styling necessarily, per se. Good. I would think so. I've used it on 2017, like on a testing site. I've never had issues with it, but I haven't used it extensively, so I couldn't tell you for sure. Any other questions? Yeah. Yeah. Yeah, so the wide alignment, I'm not sure what the actual CSS settings are for that. But by default, Gutenberg kind of restricts the blocks to a default kind of a content width or whatever. So when you align left, right, or center, and then when you do the wide align, it sets it to, I think, 80 or 90%. And then you can do full align, and it's 100%. But that's all still within the content container on your site. So it's not going to go outside the postfield. Yeah. Right. Anybody else? Yep. That's a good question. You shouldn't, but I don't know, honestly. I'm not on the Core Gutenberg team. I don't know how it would work on a legacy site that's been around for a really long time that hasn't been touched in a while. That might and probably would need some special considerations. It won't need to be rebuilt for sure. I don't think there's an instance where it will need to be built from scratch unless there's just something horrendous in it already that's keeping it old. For the most part, you should be able to update. And if you have issues for the foreseeable future, the classic editor plug-in will be available. So you could always revert back that way. Yeah, and a good point that you made me think of. The Gutenberg editor, I don't know if all of you are aware, most of you should be, because your sites are probably automatically updated to 4.9.2 two days ago. And in the installation, if the user, so this is something to consider if you have clients, if the user has the ability to edit plug-ins and Gutenberg is not installed or activated, they will receive a prompt to install and try Gutenberg. If it is installed but not activated, anyone who can edit posts will also receive that prompt. Anyone, so the new 4.9.2 adds a prompt, 4.9.8, sorry, I'm way off of my version numbers. 4.9.8 adds a prompt to try Gutenberg in the dashboard. So when you log in, it'll be like, hey, try Gutenberg, we've got this new thing and it's really cool. And that will be shown to anyone who can edit plug-ins or activate or deactivate plug-ins. And if it's installed on the site already, it will be shown to be like it'll say, hey, activate Gutenberg to anyone who can edit posts. But if your clients don't have the privileges of adding plug-ins, which a lot of people don't give them for, I would think, good reasons, then you don't have to worry about it. But just so you know, that is out there, so a lot of people will be seeing it and asking questions. And I think that's all my time, so. Thanks, guys, it's been fun. Thank you.


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Firefighters battled blazes in Italy's Sicily on September 23, which hit the Palermo and Messina areas, in the northern part of the island.Two people are reported to have died during the wildfires, lashed by strong Sirocco winds and high temperatures, and hundreds of tourists were evacuated from a hotel in the well-known tourist resort of Sifalu. Italian firefighters said the rescue force has been strengthened with the doubling of shifts in all the commands in Sicily, with 300 firefighters on, duty to extinguish the fires in the area.


Welcome to All Hands Update. I'm Petty Officer Tony Rosa. These are your headlines from around the fleet. Based on feedback from the fleet, updates have been made to the Command Advancement Program or CAP. As of October 1st, Command Triads now have more control to reward sailors who display sustained superior performance. CAP continues to advance eligible rated sailors in recognition of their superior performance to E-4, E-5, and E-6. For more details on updates made to the CAP program, visit the Chief of Naval Personnel's website. The Department of Defense recently celebrated National Hispanic Heritage Month. Hispanic Heritage Month recognizes the contributions of Hispanic and Latino Americans to the U.S. by celebrating their heritage and culture. Today, our active and reserve components are made up of more than 58,000 sailors with Latin heritage.


This lecture is part of an online course on algebraic geometry and will be about the Zariski tangent space. So, in previous lecture, we defined the tangent space of a variety embedded in, say, affine space a to the m. And we use this to define singular points, where a singular point was one where the tangent space of the variety was in some sense of the wrong dimension. The problem with this is that we can embed the same variety into affine space in many different ways. And it's not at all obvious that the tangent space of a point p in the variety will have the same dimension independent of which space we embed the variety into. So what we want to do is to find a definition of the tangent space that only depends on the variety v and the point p and doesn't depend on how we embed v into affine space. So we have the following definition. The tangent space at point p is the dual of the cotangent space at p.  So this definition appears to be circular. However, in differential geometry, the tangent space is basic and you define the cotangent space to be its dual. But in algebraic geometry, it turns out the cotangent space is basic and you define the tangent space to be its dual. So how do we define the cotangent space? Well, the cotangent space is defined to be m over m squared. Okay, well, what's m? Well, m is the maximal ideal of the local ring at the point p. So you remember the local ring at a point p can be thought of as the set of, as the ring of functions of the form f over g where g is not equal to zero at the point p. So informally, we can think of the local ring as being rational functions defined at the point p. And the maximal ideal m can then be thought of as functions vanishing at the point p. And the key point of this is that this depends only on the local ring and the local ring of a variety at a point doesn't depend on the embedding of the variety. It only depends on the variety itself. So this gives us a definition of the tangent space that doesn't depend on the embedding, but we still have the problem. Is it the same as the earlier definition we gave? So we have to check that this is equivalent to the earlier definition. So let's look at the earlier definition. If we take a point p, we may as well take it to be the origin for simplicity. And we have a variety contained in a to the n with coordinates x1 up to xn. The maximal ideal of the local ring at p is just generated by the functions x1 up to xn, obviously. So v is going to be the set of zeros of some functions f1, f2 and so on. So these will be the polynomials defining the variety v. And now let's calculate m over m squared. Well, m will be the ideal generated by x1 over up to xn. I'm not quite sure whether we're working in the local ring or the full polynomial ring, but it doesn't really matter very much at this point. And we quotient out by the ideal generated by f1 up to fn in order to get the variety v. And we also have to quotient out by all monomials of degree greater than or equal to 2. So the monomials of degree greater than or equal to 2 are sort of giving you m squared. And this is giving you v. And of course, this is giving you m. And now we can see that this is the same as the vector space spanned by x1 up to xn, because that's just this ideal divided by all monomials of degree at least 2 quotient out by the linear parts of f1 up to fn. Because of course, if you're if you're quotient out by all monomials of degree at least 2, then you can ignore all the nonlinear parts of these polynomials. And we can see that this is just the dual of the tangent space defined earlier. So this confirms that our new definition of the tangent space really is compatible with the old definition of the tangent space. And I should say this vector space can be defined to be r over m and which is equal to k. And why do we bother doing that? Well, it turns out this definition actually makes sense for r is any local ring. We can take the maximal ideal form the quotient m over m squared. And this will be a vector space over the field of quotients of the local ring by its maximal ideal. In fact, this space here is called the Zariski cotangent space for any local ring. And of course, you define the Zariski tangent space of a local ring just be the dual of the cotangent space. So you see that the cotangent space is the more basic object because it's just defined like this. You have to define the tangent space in terms of the cotangent space, which is the opposite way round. There's another way of looking at the local ring or rather the cotangent space of a local ring. So here we take a local ring r and we can form m over m squared for m a maximal ideal, which is a vector space over the field r over m. And the dual of m over m squared can be identified as ring homomorphisms from r to this funny ring k epsilon over epsilon squared. So what's this ring here? Well, it's a two dimensional vector space over k, of course, with a basis consisting of elements one and epsilon. With epsilon squared equals zero. So the ring homomorphism will take m squared to zero and it just takes m each element of m to a multiple of epsilon and as multiples of epsilon are from a one dimensional vector space, we can see this really is an element of the dual of m over m squared. We can also try and picture what does this ring look like. So if we've got any polynomial ring in various generators and we quotient out by an ideal, then this quite often corresponds to a variety or an algebraic set consisting of the points where i is zero. Well, this doesn't really work in this case. The trouble is this only works nicely when the ideal i contains no nil potents and this ideal does actually contain nil potents so it doesn't quite correspond to an algebraic set. Instead it actually corresponds to a scheme called the spectrum of k epsilon over epsilon squared that we'll be looking at a bit later. Well, we can try and imagine what this looks like. Well, first of all, we should take all maximal or prime ideals of this and there is only one prime ideal, which is the ideal generated by epsilon squared. So it's spectrum strictly speaking looks like a point, but it's not really quite a point.  Somehow it's really got you can think of it as being something like a point for a little bit sticking out of it where the little bit sticking out is something to do with this this element epsilon. So we should think of the spectrum of k of epsilon over epsilon squared is not quite a point, but a point with a with a tangent vector sticking out of it. And then maps from the spectrum of this to a variety consists of picking a point of the variety and then choosing a sort of tangent vector at this point. So there's yet another way of looking at tangent vectors or there's a risky tangent space. Suppose you look at vector fields on a smooth manifold be over the reals. Then let's look at the ring. C of v, which is just smooth functions. Then if we look at the space of all vector fields, smooth vector fields, of course, then the space of vector fields is just a module. Over the ring of smooth functions on the manifold because of course, if you've got a vector at a point we can just multiply it by the value of that function at that point. And we can define a module w of cotangent vector fields just to be the dual of the module of vector fields. And now you remember from differential geometry that there's actually map D, which goes from functions to cotangent vector fields. So it's going from the ring C of v to its module. This takes F to the one form DF. So here this is a ring. This is a module. You have to be a little bit careful because D is not actually linear over C of v. Instead, it satisfies Leibniz's rule DFG equals FDG plus DF times G. So if it were linear, it would satisfy this rule here, but it's now satisfying Leibniz's rule so it's not linear over C of v. It's linear over the field of real numbers, of course. So more generally in algebraic geometry, instead of the ring of smooth functions, we might look at the ring of polynomials over a variety, which should be some ring like this. And now what we want to do is take an analog of this. We want to find a map from R to a module W, which will be a module over R. And we want to think of W as being the module of cotangent vectors. So we're just going to, the idea is we are going to take this construction of cotangent vector fields over a smooth manifold and try and find an analog of it for algebraic varieties. At least I find one. So how do we construct the module W? Well, W is generated over as a module over R by elements of the form DR for R in R. And it satisfies the relations. First of all, DR is equal to zero for R in the field K we're working over. And secondly, DFG is equal to DF times G plus F times DG. And now we can define W to be the module with these generators and relations. So let's work out an example of this. Suppose we take R to be affine space KX1 up to XN. Then it's fairly easy to check that W is just the free module on elements DX1 up to DXN. And having defined W as the module of cotangent vectors, we can also define the module of tangent vector fields. Just to be the module of all homomorphisms over R from W to R, which is the sort of dual of W. If we want the tangent space of this at a point, we can just localize this module which gives us a module over the local ring and then take the Zariski tangent space of that. Well, it's a bit difficult. I mean, in general, this module of cotangent vector fields is kind of a bit complicated. It's usually not a free module over the ring. In fact, it's quite rare for it to be a free module. It's only a free module if the cotangent bundle happens to be a sum of trivial bundles, which is actually quite rare. So it's useful to have an alternative description of the module of cotangent vector fields. And we can define the module W like this. Suppose we take the homomorphism from R tensor R to R, so all tensor products will be over K. So this just takes A tensor B to AB. And we're going to define I to be the kernel of this map. And then I'm going to claim that we can define W to be I over I squared. So here I is an ideal of R tensor over R. And we need to define a map D from R to W, of course.  And I also need to say how W is an R module. So I'm going to say R acts by multiplication on the left R in R tensor with R. So we could make this into an R module by having R acting on the right or on the left. And I'm just going to fix acting on the left. So we can easily check Leibniz rule DRS equals RDS plus DR times S. The big problem is to check the universal property. So what we mean is that by this is that we want to show that this really is the module that generates this relation. In other words, suppose we've got any module M with a map from D from R to M obey Leibniz's rule. We've also got a map from a module W from R to W. And we want to show there's a unique map from W to M making this diagram commute. So we can do this as follows. We've got our ideal I containing R tensor with R which maps to R. And we've got a map from R to M given by D. So we can define a map F from R tensor R to M by just mapping A tensor B to A D B. So this gives a map from I to M. And what we want is not a map from I to M but a map from I over I squared to M. So we need to check that this map F is not on I squared. And to do this, we just do an explicit calculation. So pick some element sum of Si tensor Ti in I and pick another element sum of Ui tensor Uj tensor Vj in I. So this means that the sum of Si Ti is equal to naught and the sum of Uj Vj is equal to naught because that's the definition of being in I more or less. And now we calculate F of sum of Si tensor Ti times sum of Uj tensor Vj. So this is an element of M squared and we want to show that F of this is equal to zero. So is this equal to zero? Well, let me just calculate it. So this is the sum of Si Uj d Ti Vj. And now we apply Leibniz as normally find this is the sum of Si d Ti times sum of Uj Vj plus sum of Si Ti times sum of Uj d Vj. So now we notice that this sum here is zero because we said so up here. And we notice that this sum here is zero because we said it was zero up there. So this expression here is equal to zero, which is what we wanted to prove. And now it's easier to check from this that this does indeed make W into the universal module so that there's a map from D from R to W satisfying Leibniz's rule. So in general, we say that a local ring is regular if the dimension of the Zariski tangent space is equal to the dimension of the local ring that we may discuss this in more detail later. And we say that a variety is non-singular at a point of its local ring is regular. Okay, well that's all for the Zariski tangent space for this lecture. Next lecture will be about some particular examples of singularities called Duval singularities.


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