\[ \newcommand{\blank}{\underline{\hspace{2em}}} \newcommand{\bk}{\underline{\hspace{.5em}}} \newcommand{\T}{\mathbf T} \]

This is a response to Daniel Litt’s challenge to explain the concept of short exact sequences at an elementary level. In this post, I’ll try to explain the general idea behind short exact sequences and what they’re used for. I’ll get into more precise definitions in subsequent posts.

This concept comes from an area of math called abstract algebra. At least at the outset, this kind of math is more about describing commonalities between different situations, rather than “calculations”. So there’s a lot of new terminology all at once, but once you ingest it, we’re not doing very much with it yet.

Overview

A short exact sequence describes a relationship between three “data types” $A,$ $B,$ and $C.$ It’s written as \[ 0 \to A \to B \to C \to 0, \] or sometimes \[ 0 \to A \xra f B \xra g C \to 0 \] if we need explicitly say how to compare $A$ to $B$ and $B$ to $C$.

In many cases,

  • $C$ describes the possible results of some process;
  • $B$ describes some change accumulated during the process;
  • $A$ records when two different processes give the same result.

In all cases, $B$ is “bigger” than $A$ and $C$.

Here are some examples:

  • If you’re flipping a switch, the number of times you flip the switch could be $0,1,2,3,\dotsc$ But the end result only depends on whether you flipped it an even or odd number of times (and the initial state). We would describe this using the short exact sequence

    \[ 0 \to \{\text{even numbers}\} \to \{\text{whole numbers}\} \to \{\text{even},\text{odd}\} \to 0 \]

    or in mathy notation

    \[ 0 \to 2\N \to \N \to \{\text{even},\text{odd}\} \to 0. \]

  • Rotating an object $360^\circ$ brings it back to its starting position. More generally, two rotations that differ by a multiple of $360^\circ$ (e.g. $20^\circ$ and $740^\circ$) will bring the object to the same position. We would describe this using the short exact sequence

    \[ 0 \to \{\text{whole multiples of $360^\circ$}\} \to \{\text{decimal numbers}\} \to \{\text{angles}\} \to 0. \]

    or in mathy notation

    \[ 0 \to 360\Z \to \R \to \{\text{angles}\} \to 0. \]

    The difference between $20^\circ$ and $720^\circ$ might be important e.g. for calculating the energy expended in turning the object; for other purposes, you might only care where the object ends up.

Asking when two different inputs give the same output is a very common question in math. One of the special things about short exact sequences is that we get a uniform description of the answer: two inputs give the same output if and only if they differ by an input which “does nothing”.

Usage

There are a few different ways we make use of short exact sequences:

  • Even though $B$ is “bigger” than $C$, it might be easier to understand—in the same way that a shoelace laid out is “simpler” than one tied into a ball, despite taking up more space. The second example above (using numbers to represent angles) would fall into this category.

  • In other cases, $B$ is something complicated, and we use a short exact sequence to “break it up” into the easier pieces $A$ and $C$. We can then understand $B$ by understanding $A$ and $C$ individually, as well as how they fit together to make $B$.

    An example of this situation: When adding two (whole) numbers, we only need to know the last digit of the summands to find the last digit of the sum, e.g.

    \begin{gather*} \blank7 + \blank2 = \blank9,\\ \blank8 + \blank3 = \blank1. \end{gather*}

    We can almost do the same with the next digits, but not quite:

    \[ \blank2\bk + \blank3\bk \]

    could be either $\blank5\bk$ (e.g. 27 + 31 = 58) or $\blank6\bk$ (27 + 34 = 61), due to “carries” from the ones digit.

    This situation is described by the short exact sequence

    \[ 0 \to \{\text{multiples of 10}\} \to \{\text{whole numbers}\} \to \{0,1,2,3,4,5,6,7,8,9\} \to 0. \]

  • We can also use short exact sequences to describe when a system of equations can be solved, and if so, how many solutions there are. (Actually, to do both of these at the same time we need “long” exact sequences, but let me postpone that discussion.)

    For example, the solutions to the equation \[ 2x - y = 0 \] form a line through the origin, given by multiples of the vector $(1,2)$. If we replace $0$ on the right-hand side by another number, e.g. \[ 2x -y = 2, \] then we can still solve the equation, and in fact the solutions form a line parallel to the previous one.

    We would describe this by the short exact sequence

    \[\xymatrix{ 0 \ar[r] & \R \ar[r]^-{\begin{bmatrix}1\\2\end{bmatrix}} & \R^2 \ar[r]^-{\begin{bmatrix}2 & -1 \end{bmatrix}} & \R \ar[r] & 0. }\]

  • Sometimes we’re trying to “calculate” $B$, and knowing $A$ and $C$ gives us constraints on what it could be. That is, we end up looking at

    \[ 0 \to A \to {?} \to C \to 0 \]

    and are trying to find out what $?$ could be. This is often very hard: there may be many different ways to “mix” $A$ and $C$ together.

Up next

To be more precise about what short exact sequences actually are, we need to explain some other concepts first:

  • the “data types” appearing are called abelian groups
  • the arrows between them are called maps