 This lecture is part of an online course on commutative algebra and will be a fairly short survey of different properties of modules. So first of all, I'll talk about finiteness conditions for modules. So here are the most common finiteness conditions. The most obvious one is finitely generated, and if a module is notarian, and it's obviously finitely generated, there's also a condition that a module can be finitely presented, which again has a fairly obvious meaning. This just means there's a map from rm to rn to m is to zero. And there's a slightly more technical condition that says a module is coherent. Coherent means a module is finitely generated, and it means that a module has the extra condition that if you've got a finitely generated free module mapping to m, then the kernel of this map is also finitely generated. So this map here doesn't have to be onto. Coherent modules don't actually appear all that much in commutative algebra, but they appear quite a lot if you do sheath theory over complex analytic manifolds. There it turns out that the rings you get are generally not notarian, and coherence turns out to be a useful property. We also mentioned the property of having finite length, which is equivalent to a module being notarian and artinian. So this is a summary of the more common properties of modules, sorry, the more common finiteness conditions of modules. And you can also remark that these are all equivalent over a notarian ring. So you'll notice we've had quite a lot of theorems that apply to finitely generated modules over notarian rings, and that which is probably the most common and useful sort of finiteness condition. So the next sheet is going to be about various properties of modules that are weakening of the condition for being a free module. So free modules are really useful, but they're not actually all that common. So we need various variations of them. So this is this is the sort of module cheat sheet. And let's start with free modules at the top. So a free module is, of course, one just where m is the sum of copies of the ring r. Then we can get stably free. So stably free modules just mean that m plus r to the n is free for some n with n finite. What we're going to do later in the next few lectures is discuss each of these properties in more detail. So what I'm doing at the moment is just giving a sort of quick overview of the things we're going to be doing in the next lectures. Next, we can have modules that correspond to vector bundles, which are locally free modules. And it turns out there are two slightly different concepts of locally free modules. So these are locally free in the Zariski topology. What this means is that m fi to the minus one is free over the localization of r at some element of fi, where the ideal generated by the elements f1 up to fn is just r. So informally, the spectrum of r is covered by a finite number of open sets. And over each of these open sets, the module looks free. And locally free modules are also projective. So we recall projective means if you've got a map from a to b to zero of modules and a maps on to b, and your module maps on to b, then this can be lifted to a. And projective modules satisfy a slightly different version of the locally free condition. This says that the stalks are locally free. So this means mp is free for all p in the spectrum of m. And if the stalks are locally free, this turns out to mean that the module is flat. So flatness means if a is a submodule of b, this implies that if you tense with m, this is still exact. And finally, flatness implies that the module is torsion free. If you're working over a ring that isn't an integral domain, there are several slightly different notions of torsion free. So one might say that if a m is equal to naught for a regular, in other words, not a zero divisor, this implies that m is equal to zero. So these are probably the more common properties. And now I just want to say a little bit about relations between them. So first of all, these four properties are all the same for finitely presented modules. And in fact, finitely presented modules with these properties turn out to be the analog of finite dimensional vector bundles. We can be slightly more precise. Flat implies stalks locally free for finitely generated modules. That implies projective finitely presented modules. And that implies locally free for finitely generated modules. Notice that here we have finitely generated modules. And here we have the slightly stronger condition that the module has to be finitely presented. And this actually fails with finitely generated modules. And then these conditions here are all the same over local rings, for principal ideal domains. And well, you might say which of these properties is the most important. I mean, you know, you sometimes get people on shows told they're being sent to a desert island for a year and only allowed to take one book in which they take. Well, if you're sent to a desert island for a year and only allowed to take one of these properties or modules with you, which would you take? And the answer I think is fairly key. You should take flat modules. So this is this is the really key property. It's a bit surprising that flat is so important because it's in fact the most technical and the least intuitive property. In fact, people have been doing commutative algebra for several decades before flatness was even defined. The reason flatness turns out to be so important is that well, in order for a property to be useful, you must have lots of modules with that property. I mean, a property that no modules have is just not very useful. So so flat modules and torsion free modules are actually rather common. On the other hand, a property must make modules quite nice. And it turns out that all the properties from flatwoods upwards are sort of fairly nice properties and torsion free modules don't really behave all that nicely. So so flat modules are in the sort of sweet spot where they're really well behaved. And on the other hand, there are lots and lots of them. So that makes flatness a really useful property. Okay, I think that's the end of the cheat sheet. So what I'll be doing in the next few lectures is going through these properties one by one and saying a bit more about each of them.