Abstract Nonsense

Crushing one theorem at a time

Flat Modules (Pt. I)


Point of Post: In this post we discuss, in some detail, flat modules including equational characterizations and Lazard’s theorem.

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Motivation

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We have so far discussed two kinds of modules that make certain half-exact (i.e. right exact or left exact) functors exact. Namely, we have discussed projective modules as the modules that make the Hom functor \text{Hom}_R(P,\bullet) exact, and we have discussed injective modules as the modules that make the contravariant Hom functor \text{Hom}_R(\bullet,I) exact. Both of these are very important since projective and injective modules are the “approximating” modules used to define powerful notions like derived functors. That said, they aren’t the most “functional” modules that make certain functors exact. In particular, in basic module theory there are three main functors, two of which are the covariant and contravariant Hom functor. The last, of course, is the tensor functor M\otimes_R\bullet. The tensor functor shows up everywhere in mathematics and is, dare I say it, “more important” (whatever that really means) than the two Hom functors [note: this is just my limited opinion on the matter]. Thus, it would seem to behoove us to understand what kind of modules make the functor M\otimes_R\bullet exact (it is already right exact).

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This naturally then leads us to considering flat modules, which are precisely those modules that make the relevant tensor functor exact. Flat modules show up in geometry in some pretty interesting ways (none of which I am really able to speak to at this point).

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May 4, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , | 2 Comments

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. IV)


Point of Post: This is a continuation of this post.

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December 28, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , | 1 Comment

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. III)


Point of Post: This is a continuation of this post.

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December 28, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , | 2 Comments

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. II)


Point of Post: This is a continuation of this post.

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December 28, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , , | 3 Comments

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. I)


Point of Post: In this post we define the category of directed/inverse system of modules over a given ring and then discuss how the construction of the direct/inverse limit is a functor from this category to the category of R-modules. We will then show how this functor is “exact” in the sense that it preserves exact sequences.

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Motivation

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We’d now like to take a more theoretical approach to looking at direct/inverse systems, and their limits. Roughly what we we shall see is that direct systems over a fixed preordered set form a nice little category and limits shall be functors on this category. Moreover, what we shall see is that there is a natural notion of exactness for chains of direct/inverse systems, and we shall see that the limit functor preserves exactness when passed into the target category (which will be just R\text{-}\mathbf{Mod}). The reason for this abstraction is nothing more than a desire to put phrase a common construction in the convenient language of category theory. In fact,  not only will we see that this phrasing will be useful in and of itself, but it shall serve as a prime example of a more general categorical construct in the future, and this functor shall even occupy much of our time once we start talking about homological algebra (or the left derivation of this functor).

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December 27, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , | 4 Comments

Relationship Between Hom and Limits (Modules)(Pt. II)


Point of Post: This is a continuation of this post.

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December 26, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , | Leave a comment

Relationship Between Hom and Limits (Modules)(Pt. I)


Point of Post: In this post we discuss how the Hom functor relates to limits. This shall serve, later on, to be a prime example of adjoint functors.

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Motivation

Now that we have properly defined notions of direct  and inverse limits of modules there are several natural questions we can ask, relating to previous topics. Perhaps one of the natural questions one might ask is how direct and inverse limits react with the Hom module (group). We should already have a good idea what’s going to happen based on the case when we are taking the direct and inverse limit over trivial directed and inverse systems. In other words, we already know what happens in the case of products and coproducts. This suggests that perhaps something of the form \text{Hom}(\varinjlim \bullet,\bullet)\cong \varprojlim\text{Hom}(\bullet,\bullet) and \text{Hom}(\bullet,\varprojlim\bullet)\cong\varprojlim\text{Hom}(\bullet,\bullet). Of course, we have to define what precisely what we mean here for, obviously as it stands, this makes absolutely no sense–what are the systems we are taking the limits over on the right hand side of both these isomorphisms? Regardless, the idea of why this “should” be true is clear enough. Namely direct limits are constructed to be such that mappings out them are completely determined by a set of mappings out of each of the individual terms of the limit, and similarly inverse limits are such that mappings into them are dtermined by a set of mappings into each of the individual factors. This roughly tells us that \text{Hom}(\varinjlim\bullet,\bullet)\sim\text{Hom}(\bullet,\bullet) where \sim denotes “related by some operation”, and similarly for the other entry with inverse limit. What suggests that in both cases we would get inverse limits is that there are natural projections maps \text{Hom}(\varinjlim\bullet,\bullet)\to\text{Hom}(\bullet,\bullet) and \text{Hom}(\bullet,\varprojlim\bullet)\to\text{Hom}(\bullet,\bullet) which heavily suggests we should be doing some kind of inverse limit (whenever one see mappings into things, inverse limits should be an immediate thought). Perhaps a bigger hint, or those who are in the know (or have done things in a slightly different order than I am posting), is that we know the maps M_\alpha\to M_\beta and M_\beta\to M_\alpha (where \alpha\leqslant\beta) in the \varinjlim and \varprojlim case respectively naturally induce maps \text{Hom}_R(M_\beta,N)\to\text{Hom}_R(M_\alpha,N) and \text{Hom}_R(M_\beta,N)\to\text{Hom}_R(M_\alpha,N) respectively since, as we’ve already discussed, \text{Hom}_R(\bullet,N) and \text{Hom}_R(N,\bullet) are contravariant and covariant functors respectively. This heavily suggests that we are going to be doing some kind inverse limit. Making all of this rigorous is slightly annoying, but this the basic idea. Either way, the proof shouldn’t seem to foreign since we have already tackled the special case of products and coproducts.

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December 26, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , | 2 Comments