Chronicles of Chain Complexes

[quasi-coherent]

Hey, Welcome to my maths blog!

I am a masters student in Mathematics. Currently, I am interested in anything to do with sheaves!

The background picture is from the book 'Homotopical Topology' by Anatoly Fomenko and Dmitry Fuchs.

Icon credits: @anonymous-leemur

[kata4a]

algebra for age regressors

[kata4a]

do you see my vision?

[synoddiane]

[janmusija]

here's my shitty low effort contribution that I sketched up while bored. any reader should feel free to improve this

[sevenfromelevenisfour]

I crocheted a series of hyperbolic planes and strung them up into a garland. Each module is one more round than the one before, and each round has twice the stitches as the round before.

I made eight iterations; the largest one took an entire skein of cotton yarn. I really liked how this shows the change in curvature as cross section of the plane expands.

[todays-xkcd]

I can't believe Candles of Vienna caved to commercial pressure and added the Goku expansion.

Types of Board Game [Explained]

Transcript Under the Cut

Types of Board Game

[Under this header text, the comic contains 8 panels. Each of them is labeled at the top with a short description of the board game being played and features (from left to right) Cueball, Ponytail, Megan, and White Hat sitting on chairs around a table trying to play it.]

Boring Megan: Each turn, roll a die and move your token. Turns proceed clockwise around the table until we get bored and go home.

Abstract Cueball: Each turn, you can place any number of red triangles or blue squares on a hexagon, or move any hexagon to a...

Hyperspecific Theme Ponytail: It's October 2, 1814. The Congress of Vienna convenes. You are each in charge of distributing and lighting candles for the opening ball, which was held at these three locations...

Overcomplicated White Hat: It's a cross between *Twilight Imperium* and *Cones of Dunshire,* but implemented entirely in category theory. Every cone is a monad, and...

Cooperative Megan: We're working together to sort these decks of cards using only hand gestures. After that, we'll silently organize my junk drawer.

Branded Cueball: You can play as Phoebe, Chandler, Monica, Rachel, Ross, Joey, or, due to an ill-advised tie-in, Goku.

Party Ponytail: Each of the cards in your hand has a bad word on it. On the count of three, yell the...

Social Deduction Megan: Remember, per our *Find the Secret Murderer* house rules from last week, discovering that a player had committed a real-life murder does *not* count.

[ashmindpalace]

when i was first learning comm alg this was unironically how i remembered details of exact sequences and now that i've been studying it for some time i still think of this whenever i see a short exact sequence

[fireemoji1f525]

Are you an aspiring math major, but don't know what your field of study should be?

@algebraic-dumbass had the idea to make a flow chart of math sub-fields, so perhaps this will help any future mathematicians decide what to study!

[algebraic-dumbass]

beautiful, but where's my algebra (besides galois and category theory)?

[pv1isalsoimportant]

It's pretty depressing that I can't remember the contents of most of the courses I took over the past 5 years.

[m---a---x]

My dark ages was during my masters. I forgot almost everything i learnt during my bachelors. But now i'm in contact with a bit more diverse field of math and a lot of it is coming back to me

[acid-lich]

A daring escape past y^2=x^3+x+3 over F14641

I was sent a lovely little paper last June about visualizing elliptic curves over finite fields and you just know I had to draw about it. Anyway now it’s space and piracy is involved!

[jeremylovescandace]

irl oomfs just ignore the new fake name im turning this acct into a studyblr for my quarter life crisis

[spectracled-eider]

Introduction to Homological Algebra 1: Chain Complexes and Exact Sequences

At its heart, homological algebra is the study of chain complexes, exact sequences and measuring how chain complexes fail to be exact. In this post we shall introduce the first two concepts and talk about how to measure the failure of exactness (homology) in the next post. Later, we will also want to study how well functors preserve exactness and this will amount to measuring how certain complexes fail to be exact.

I will be assuming some familiarity with category theory and module theory. I have a post about category theory here and I may write a primer post on module theory in the future. I will recall the definition of a module over a ring before we proceed!

R-modules:

Definition 1.1:

Let R be a ring. A (left) R-module is an abelian group (M,+) with a scalar mulitplication R×M->M such that

If R=ℤ, ℤ-modules are precisely abelian groups. If R=k is a field, k-modules are vector spaces over k. If R is not commutative, we also get a distinct notion of right R-modules where scalar multiplication is on the right. The only axiom which changes is (4) which becomes (ms)r=m(sr). When R is commutative, sr=rs so the objects are the same.

Definition 1.2: A function f:M->N between left R-modules is said to be an R-module homomorphism (or an R-linear map) if it is a group homomorphism, i.e. f(m+m')=f(m)+f(m'), such that f(rm)=rf(m) for all m∈M and r∈R. If instead M and N are right R-modules, we require f(mr)=f(m)r.

With this definition, one can show that the identity map and the composition of R-module homs are both R-module homs. Hence we have a category of (left) R-modules and R-module homs which we denote R-Mod. We denote the category of right R-modules and R-module homes Mod-R.

In what follows, R-module will always mean left R-module unless stated otherwise.

Chain Complexes:

Definition 1.3:

Examples 1.4:

We may also consider complexes where the operators increase in degree instead and we call these cochain complexes. Mathematically, these are the same up to relabelling but it is often useful to treat them as distinct as we shall see later on.

We get the following immediate result.

Lemma 1.5:

One might care about complexes where this is actually an equality and indeed this is our definition of an exact sequence!

Definition 1.6:

From this definition, we see that any exact sequence is also a chain complex. We distiguish short exact sequences for two reasons. One is that in a sense (see the next example) short exact sequences are the shortest "interesting" exact sequences. The second is that they turn out to be very important in homological algebra. Also note that to show a sequence is short exact, we need only check that φ is injective, imφ=kerψ and that ψ is surjective.

Examples 1.7:

Comparing examples 1 and 7, we see that short exact given R-modules A and C, we don't necessarily have that all short exact sequences 0->A->B->C->0 must have B≅A⊕C. Now comparing with examples 3 and 4, we see that short exact sequences are indeed the shortest exact sequences where something more interesting can happen. However do not discount the utility of examples 3 and 4!

Definition 1.8:

One may now wonder whether we can tell when a short exact sequence is split and indeed the next result gives a full classification!

Proposition 1.9 (Splitting Lemma):

Chain Maps:

As is common in maths, now that we have seen some objects we should should talk about maps between these objects. This leads us to talking about chain maps.

Definition 1.10:

As one might expect, chain complexes over R with chain maps forms a category, which we denote by Ch. This is the result on the next lemma.

Lemma 1.11:

Note that the composition of chain maps is also associative. This follows from the associativity of the composition of set functions.

This allows us to talk of isomorphisms of chain complexes. A chain isomorphism is an isomorphism in Ch. Equivalently, a chain isomorphism is a chain map such that each map in the sequence is an isomorphism.

We conlude with a useful result involving chain maps between exact sequences.

Theorem 1.12 (The Five Lemma):

The first of these results is sometimes referred to as the Short Five Lemma.

Note that in the proof of the splitting lemma, in particular in (ii)⇒(i), we need not have checked that f is an isomorphism or that B=φ(A)⊕η(C). If, after constructing f as we did, we check that the following diagram commutes, then the short five lemma implies that f is an isomorphism:

In the next post, we shall discussion homology (and cohomology) and how it measures the failure of exactness as well as a few very important results!

[locally-normal]

Great post!! I don't know how you want people to interact with these so forgive me if my reaction is a faux pas and I'm doing the equivalent of asking a question that's actually a statement. It's a risk but hey maybe my comment is interesting.

I was also presented with homology as measuring the failure of exactness. I think it's a little incomplete though, it begs the question, why care about exact functora? More fully, exact functors are the functors in the derived category, which is the homotopy category of Ch, ie, the category of chain complexes with chain maps but where we formally consider quasi isos (homology isos) as isos in the category (formally add inverses). There's other functors which aren't exact but which we want to pass to the derived category anyway. The homology provides this functor!

Alright admittedly this may seem less motivating. But we want to understand the homotopy category because we like topology... it turns out in algebraic geometry the derived category fills a similar role, it captures varieties over R up to wiggling.

[mathematikerin-freyja]

I'm wondering if you could provide an explicit example of this derived category accounting allowing for wiggling? I find, in my attempts to learn this language of derived categories, that I am really struggling to visualise or even just algebraically construct examples of anything from which to begin to get a sense of concreteness from.

[smbc-comics]

Tragically, once you introduce costly signaling, all mathematics is impure.

Check out the bonus panel on the site!

SMBC ◆ PATREON ◆ INSTAGRAM ◆ BLUESKY ◆ STORE

Buy this comic as a print!

[no-sharks-on-the-sun]

[adventures-in-mathematics]

[vacuously-true]

Cats are experimental topologists

[marigold-22]

All algebraic topology courses should just be about simplical sets and then show at the end that simplical sets and topological spaces are quillen equivalent

[mathematikerin-freyja]

The golden guild of infinity-category theorists endorses this vision.

[mathematikerin-freyja]

I intensly miss the mathematical infodumping of the student common room in my old mathematics undergraduate program. I want a thing where mathematicians give talks, less on their research, and more on just generally enjoyable mathematical things. Like Numberphile but pitched at the level of graduate students or something.

[mentalknot]

We need a “performative math competition”

I wanna see people carrying category theory textbooks as “light morning reading,” equipped with their black coffee and trench coats

Hartshorne is a “chill bedtime story.”

They carry Hagoromo chalk like it’s an expensive cigar

And of course, read the daily math subjectClass mailing list, as if it’s the Sunday newspaper

Oh and…

they’re not like other mathematicians

they do rock climbing.