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Stream: learning: reading & references

Topic: teaching modules and homological algebra

Mike Shulman (Dec 04 2023 at 18:10):

Next fall I'm going to be teaching a second-semester course in abstract algebra, for students who've seen some basic group theory and ring theory. I have a lot of freedom in choosing topics. I'd like to give the students some exposure to category theory, and so far the most promising direction I've thought of is to do some theory of modules and exact sequences, moving in the direction of homological algebra. I think this would make a good class, and I even found a good-looking textbook that's free online: Module Theory: an approach to linear algebra by T. S. Blyth.

What it's missing is motivation: what is all this good for? As a topologist by training, the applications I think of for homological algebra are all topological, involving homology and cohomology of spaces, but my students won't necessarily have had any topology. There ought to be more accessible examples related to things in algebraic geometry, but not being an algebraic geometer I don't have them at my fingertips, and I'm having trouble finding a lot of them that wouldn't require so much background that the class would essentially become a class in commutative algebra and algebraic geometry. E.g. I'd like to avoid talking about regular rings or chain conditions. Can anyone suggest some accessible applications of modules and homological algebra? Or a good book where I can find some?

Tobias Fritz (Dec 04 2023 at 18:46):

This is arguably topology again, but I find the connection between (simplicial) homology and cohomology and electrical networks to be very instructive. Of course this will require some understanding of electrical circuits, but then again every area of application comes with its own required background.

Tobias Fritz (Dec 04 2023 at 18:48):

(Minor correction: it's homology and cohomology of polyhedral 2-complexes rather than simplicial.)

Todd Trimble (Dec 04 2023 at 18:55):

Not an answer to the question, but bravo to teaching a second semester algebra course that isn't all about Galois theory. Nothing against Galois theory, except that I think it's traditionally overemphasized for second semester algebra courses, at least in the US. There is just as much if not greater need for module theory.

Of course, one topic is the abstract algebra that underlies the Jordan canonical form, Cayley-Hamilton, structure of finitely generated abelian groups, etc. in terms of modules over a PID.

David Egolf (Dec 04 2023 at 19:24):

Thanks for pointing out that free online textbook! This is stuff I want to learn!

For what it's worth, the reason I want to learn this stuff is because there are a lot of intriguing books I want to be able to understand. These are on topics including: algebraic topology, algebraic geometry, homotopy theory, categorical homotopy theory, and abelian categories.

For me, unlocking the ability to read books (that were previously incomprehensible to me) is a pretty big source of motivation, even when I don't really know what's in those books just yet!

Graham Manuell (Dec 04 2023 at 22:31):

I struggled for a long time to find motivation for homological algebra and I am still not completely happy with it. One thing I do think is easy to motivate is extension problems, so this could be used to at least motivate Ext^1.

Fwiw, for modules themselves I think being a common generalisation of vector spaces and abelian groups is probably sufficient.

David Egolf (Dec 04 2023 at 23:37):

Regarding motivation for modules, I recently learned that modules can be thought of as a kind of monoid action. That's helped me remember the definition of a module. For example, this helps me remember that we need $(rr')m = r(r'm)$ and $1 m = m$ for all $r,r' \in R$ and $m \in M$ if we are going to have an $R$-module over $M$.

Mike Shulman (Dec 04 2023 at 23:40):

Abstract/intrinsic motivations like "modules are a generalization of vector spaces and abelian groups" and "a module is a monoid action that's compatible with addition" are all very well, but I'd also like some more concrete frog-type applications. What problems can you solve with modules and homological algebra that you could already have been interested in before you knew what modules and homological algebra were?

Kevin Arlin (Dec 05 2023 at 01:22):

Wait, what's a frog-type application?

Spencer Breiner (Dec 05 2023 at 01:33):

birds and frogs

John Baez (Dec 05 2023 at 12:14):

What problems can you solve with modules and homological algebra that you could already have been interested in before you knew what modules and homological algebra were?

I would not attempt to teach homological algebra without doing its application to the (co)homology of spaces, since with it there are stunning applications, while without it the applications all seem more technical.

John Baez (Dec 05 2023 at 12:17):

The old first edition of Rotman's An Introduction to Homological Algebra (click to download it) has chapters called "Specific rings", "Extensions of groups", "Son of specific rings" and "The return of cohomology of groups", which apply homological algebra to these topics. They're nice. But I must say the results feel rather pale and wan compared to, say, showing there's no retraction from the disk to the circle.

John Baez (Dec 05 2023 at 12:19):

(The most recent edition of this same book is much larger, and it feels bloated to me. The additional contents seems to be mainly category theory - so, laudable in spirit but not much better at explaining the applications of homological algebra.)

Todd Trimble (Dec 05 2023 at 13:14):

This feels kind of limited, but there's a homological way to prove the characterization of Pythagorean triples by something called Hilbert's theorem 90, having to do with the cohomology of Galois groups.

If $G$ is the Galois group of a Galois extension $K/k$ and $G$ is cyclic, then one version of the theorem says that $H^1(G, K^\ast) \cong 0$.

Todd Trimble (Dec 05 2023 at 13:15):

For cohomology of cyclic groups $G$ of with generator $g$ of order $n$, there's a projective resolution

$\ldots \to \mathbb{Z}G \overset{N}{\to} \mathbb{Z}G \overset{g-1}{\to} \mathbb{Z}G \overset{N}{\to} \mathbb{Z}G \overset{g-1}{\to} \mathbb{Z}G \to \mathbb{Z}$

where $N$ is the norm map given by multiplying by $1 + g + \ldots + g^{n-1}$. The statement $H^1(G, A) \cong 0$ would say that if $N(a) = 0$ (or multiplicatively, $N(a) = 1$, then there exists $b \in A$ such that $gb - b = a$ (or multiplicatively, $\frac{gb}{b} = a$.

Todd Trimble (Dec 05 2023 at 13:15):

In the very special case of Hilbert theorem 90 where we consider the $\mathbb{Z}_2$ extension $\mathbb{Q}(i)/\mathbb{Q}$, it says that if $a + bi$ has norm $1$, i.e., $a^2 + b^2 = 1$, then there exists $c + di$ such that

$\frac{c-di}{c+di} = a + bi$

and out of this pops the characterization of Pythagorean triples.

Todd Trimble (Dec 05 2023 at 13:15):

While on the topic of cohomology of cyclic groups, there's also the description of carrying in basic arithmetic in terms of second cohomology.

Patrick Nicodemus (Dec 05 2023 at 14:14):

Hilbert's theorem that a polynomial ring is of finite cohomological dimension is one of the earliest if not the earliest results. It would be good to find some geometric intuition to motivate it

Patrick Nicodemus (Dec 05 2023 at 14:15):

Prequisite knowledge of topology isn't important if you just treat simplicial complexes in a purely combinatorial way and ignore homotopy invariance

Patrick Nicodemus (Dec 05 2023 at 14:16):

I think that Irving kaplanskys book on ring theory uses homological methods. I will check later

Patrick Nicodemus (Dec 05 2023 at 14:17):

Rotmans book on homological algebra is your best bet as a general reference for student examples omo

Patrick Nicodemus (Dec 05 2023 at 14:20):

This is not a concrete answer but a further specification of your question is- what are some important cases where the tensor and Hom functors fail to be exact and what's the practical impact of that?
Examples should probably be computational in nature. We want to describe X by generators and relations so we can compute with it but "characterizing X by generators and relations doesn't play nice with functors that aren't right exact".

Patrick Nicodemus (Dec 05 2023 at 14:21):

That is the generators and relations presentation of X doesn't help much in describing FX if F doesn't play nice w quotients

Patrick Nicodemus (Dec 05 2023 at 14:25):

Again not topology proper but if they have a good background in calculus then you can talk about de Rham cohomology of the punctured plane, using standard undergrad language of vector fields curl divergence flux and gradient. I would imagine open submanifolds of R^3 and R^2 would give interesting examples without having to explain what is meant by a general manifold.

Patrick Nicodemus (Dec 05 2023 at 14:26):

Compactly supported de rham cohomology can also be discussed without too much background knowledge if the students have background in analysis in R^n

Patrick Nicodemus (Dec 05 2023 at 14:27):

Again my point is to avoid discussing general manifolds and the concept of a differential form and try and reuse as much terminology as possible from calculus and analysis

Mike Shulman (Dec 05 2023 at 14:40):

Yes, one can develop some topology in a purely combinatorial way, but this is supposed to be a class on algebra, not on combinatorial topology.

Mike Shulman (Dec 05 2023 at 14:42):

But maybe; I'll think about it, thanks.

Mike Shulman (Dec 05 2023 at 14:42):

de Rham cohomology using undergrad multivariable calculus is an interesting idea!

JR (Dec 05 2023 at 15:25):

Mike Shulman said:

de Rham cohomology using undergrad multivariable calculus is an interesting idea!

From Calculus to Cohomology

Evan Patterson (Dec 05 2023 at 18:22):

Recently some people have been studying systems theory using homological algebra. They had a workshop about it last year: https://web.northeastern.edu/martsinkovsky/p/Conferences/Ghent2023/AA-23.html

Here's an abstract from Sebastian Postur that caught my eye:

Systems theory is the study of systems. Examples of systems are a pendulum or a solar system. There is no all-encompassing mathematical notion of what counts as a system. There are rather different mathematical approaches to model, reason about, and compute with systems. In these lectures, we deal with Willems approach to systems theory combined with the methods of algebraic analysis. This combination is called algebraic systems theory. It means that systems are modeled as solution sets of finitely many linear equations over a ring, usually a ring of differential operators, and that the solutions are taken within some fixed module, usually a module whose elements may be interpreted as trajectories. Willems coined the term "behavior" for such solution sets. Whenever our ring is noetherian and our fixed module is an injective cogenerator, a good notion of an abelian ambient category for behaviors is known in algebraic systems theory: namely the opposite category of finitely presented modules over our ring. In that good case, intrinsic features of behaviors, like being controllable or autonomous, translate into intrinsic features of finitely presented modules, like being torsion-free or torsion. In these lectures, we propose a setup for algebraic systems theory that also works for an arbitrary fixed module over an arbitrary ring. This setup is based on functor categories instead of module categories. This functorial setup overcomes some deficiencies that arise within the module theoretic approach, like behaviors not being isomorphic in situations where they clearly should be isomorphic. Moreover, we explain what being controllable, autonomous, or observable could mean in this functorial setup. [Note for experts: our proposed abelian ambient category for behaviors is the functor category of the definable subcategory generated by our fixed module.]

I'm not familiar with this work myself, so all I can say is that it seems interesting. I like the idea of doing something with homological algebra that feels really applied, not just within mathematics itself.