Abstract Nonsense

Crushing one theorem at a time

Chain Homotopy


Point of Post: In this post we discuss the notion of chain homotopic chain complexes and prove that they give isomorphic homology objects.

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Motivation

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We know that if we are given two chain complexes \mathbf{C} and \mathbf{D} and a chain map f=\{f_n\} this induces an arrow H_n(f):H_n(\mathbf{C})\to H_n(\mathbf{D}). A fairly obvious question then is when two different chain maps f,g:\mathbf{C}\to\mathbf{D} induce the same map on homology objects. This leads us naturally to the notion of a chain homotopy between which roughly states that f and g differ by something that trivially gets sent to zero.

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April 29, 2012 Posted by | Algebra, Homological Algebra | , , | Leave a comment

The Long Exact Sequence


Point of Post: In this post we construct the, ever-powerful, long-exact sequence of homology objects induced by a short exact sequence of chain complexes.

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Motivation

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We shall discuss one of the most useful aspects of homology–it’s relation with long-exact sequences of chain complexes. Throughout all of mathematics being able to put an object X in the middle of a short exact sequence 0\to Y\to X\to Z\to 0 where Y,Z are known allows one to glean information about X from information about Y,Z. Why should chain complexes be any different? Namely, if it’s computationally advantageous to try to put a module or a group in the middle of a short exact sequence, it stands to reason that putting a chain complex in such a sequence would be equally fruitful. While this is all just R\text{-}\mathbf{Mod} intuition it actually turns out to be quite correct in general. Namely, we shall see that the ability to form a short exact sequence 0\to \mathbf{C}'\to\mathbf{C}\to\mathbf{C}''\to 0 shall tell us that there is a fairly substantial connection between the homology objects of \mathbf{C},\mathbf{C}', and \mathbf{C}''. This shall be very useful computationally but it shall also enable us to prove some pretty theoretical things as well–in particular it shall prove to be a very powerful tool in relating certain levels of the derived functors we are going to create. It also has huge uses in topology in as basic ideas as the long exact sequence of relative homology groups.

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Hopefully the above motivates you to care about what I am about to talk about. Unfortunately, this is perhaps famous for being one of the most annoying, messy, and just nonsensey part of homological algebra. It is just infeasible to prove this result directly in general categories and so we are going to be forced to prove it for R\text{-}\mathbf{Mod} and appeal to the metatheorem to give us the general result.

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April 24, 2012 Posted by | Algebra, Homological Algebra | , , , | Leave a comment

Exact Sequences and Homology (Pt. IV)


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

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , , | Leave a comment

Exact Sequences and Homology (Pt. III)


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

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , , , | 2 Comments

Exact Sequences and Homology (Pt. II)


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

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , , | 1 Comment

Exact Sequences and Homology (Pt. I)


Point of Post: In this post we discuss how to define exactness for chains in a general abelian category and then discuss the homology objects associated to a chain.

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Motivation

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Last time we discussed the notions of chain complexes in abelian categories (amongst other things). This time we are going to discuss the notion of exact sequences which, in essence, are the “best kind” of exact sequences one can reasonably expect from a general chain complex. We should have a pretty good idea about what exactness means in our favorite categories like \mathbf{Ab}–it’s just the old image equals kernel routine. Of course, going from our favorite abelian category to general ones is a task which, by now, should be obvious isn’t always quite easy or obvious. Indeed, how exactly do we define “image equals kernel” when a) our objects aren’t necessarily sets, b) kernels are objects defined only up to isomorphism, and so even if they were sets there is no reason that kernel has to be literally contained inside image, c) . As has been a theme in our development of abelian categories we can replace the notion of “literal equality” in our more standard, tame categories with the notion of “canonical isomorphism”. Though, we shall see that while \text{im }f=\ker g shall be meaningless in a general abelian category, that there will be a canonical maps \text{im }f\to\ker g whose invertibility shall be equivalent to being exact. Here is where it shall be extremely important that we are dealing with abelian categories. Namely, we shall see that in general we shall only get a canonical map \text{coim }f\to\ker g and it’s the fact that there is a natural isomorphism \text{coim }f\xrightarrow{\approx}\text{im }f that allows us to construct our canonical map \text{im }f\to \ker g.

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Once we have defined exact sequences we shall define the homology objects of a given chain. Roughly, these will be measures of how far away a given point in a chain is from being exact. What this shall mean is that for each chain \mathbf{C} in \mathbf{Ch}(\mathscr{A}) we shall associated objects H_n(\mathbf{C}) for n\in\mathbb{Z} such that \mathbf{C} will be exact at C_n if and only if H_n(\mathbf{C})=0.

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Ok, so I think it’s about time that I tried (no doubt, to no avail) to explain the reasons that I have come to understand homology is important and why we care about it.

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , | 2 Comments

Chain Complexes (Pt. II)


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

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April 3, 2012 Posted by | Algebra, Homological Algebra | , , , , | Leave a comment

Chain Complexes


Point of Post: In this post we discuss the notion of chain complexes in additive and abelian categories

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Motivation

Now that we have finally defined abelian categories we can start discussing the objects that interesting us–namely chain complexes. Chain complexes should be very familiar to anyone who has done the slightest bit of algebra or topology should be well acquainted with chain complexes. Indeed, chain complexes show up in the form often times when one is able to describe an object X as being “put together” from objects A and B. For example, the existence of a short exact sequence 0\to A\to X\to B\to 0 in \mathbf{Ab}  indicates that X is some kind of “combination” of A and B in such a way that X/A\cong B–in the best case scenario X\cong A\oplus B.  Thus, a common technique to classify all algebraic objects satisfying property P is to prove that all such objects can be put in a short exact sequence 0\to A_0\to X\to B_0\to 0 where A_0,B_0 are known, and thus we have reduced our questioning to find exactly which X can fit into that sequence. In general, putting X into a long complex \cdots \to A\to X\to B\to\cdots indicates that we have decomposed X, and that how “nice” this decomposition is depends on how the rest of the chain plays out.

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Chain complexes come up in topology in a lot of contexts, probably the one which is closest to the surface being the singular chain complex coming from the singular homology of a space. That said, they also come up when doing slightly more exotic constructions like the DeRham cohomology of a space.

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Regardless, in all of the “basic” mathematics that we have seen chain complexes come up in, they indicate a sort of “breaking up” of a space into smaller, more understandable pieces. In this post we take these basic ideas of chain complexes and abstractify them so that we can discuss them in general additive, and thus in abelian, categories.

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April 3, 2012 Posted by | Algebra, Homological Algebra | , , , | 2 Comments