Torsion and Flat Modules - July 26, 2025
In anticipation for noncommutative geometry, I thought I should probably brush up on some algebra. So this lead me to rings which lead me to modules. Apparently projective modules are a big deal in noncommutative geometry and in algebraic geometry so I thought it best to figure out what they are. Well projective modules were a little nasty for me so I decided to take a step back and look at torsion in modules and flat modules.
Lets start with torsion, its a pretty simple idea and has a nice extension to cohomology (don't be fooled I just learned this yesterday). Lets start with a commutative ring $R$ (usually an integral domain, i.e. no zero divisors). Now lets take a module over $R$ and call it $M$. That is, $M$ is an $R$-module. We say an element $m \in M$ has torsion if there is some nonzero element $r \in R$ such that $r \cdot m = 0$. So intuitively, there are no elements in $R$ that will kill elements in $M$. If $M$ has no elements with torsion, then we say $M$ is torsion free. Lets look at some simple examples. Lets take $\mathbb{Z}$ as a module over $\mathbb{Z}$. Clearly, $\mathbb{Z}$ over $\mathbb{Z}$ as a module is torsion free. If we choose some integer $a \in \mathbb{Z}$, we see there are no non-trivial ways to send $a$ to $0$ using multplicaiton. Specifically, $0$ is the only element such that $a \cdot 0 = 0$. Now lets look at $\mathbb{Q}$ as a $\mathbb{Z}$-module. So we see that there are no rational numbers other than $0$ such that for some integer $a \in \mathbb{Z}$, we get $a \cdot 0 = 0$. Torsion has some kind of geometric interpretation related to the "twisting" of spaces. The theory of torsion goes much deeper I'm sure but the basic idea is simple. I do eventaully want to talk about how it relates to cohomology but that will need an entire entry itself.
I had heard of flat modules before but didn't really know anything about them. So in my infinite wisdom, I thought that a torsion-free module must be flat because it doesn't twist or anything right? Well of course that would be way too simple a definition so I was unpleasantly surprised after learning about flat modules. There does exist a relationship between flat modules and torsion but first lets introduce flat modules. So lets take again our $R$-module, $M$. We say $M$ is flat if for every short exact sequence $$ 0 \to A \to B \to C \to 0 $$ the "tensored" sequence $$ 0 \to A \otimes_R M \to B \otimes_R M \to C \otimes_R M \to 0 $$ is also exact. This seems a little abstract, maybe even useless, but if mathematics has taught me anything, its that invariants are super important. I didn't dig too much deeper into this than beyond the basic definition, but I did want to make a connection to torsion in modules. As it turns out, over an integral domain $R$, every flat module is torsion-free (I mentioned earlier we almost exclusively work with integral domains when discussing torsion, otherwise it wouldn't be interesting). The converse however, is not true. There are many torsion-free modules that are certainly not flat.
I know the importantce of modules cannot be overstated. Unfortunately, I don't yet know enough to give many explicit examples. But we can think of them as the ring theory analogue to vector spaces, so hopefully that gives a glimpse into their significance. All I was trying to do today was learn how to understand the difference between torsion and flat modules and it was a success. Of course my study of algebra, further, modules, will not end here and neither should it for you. I haven't always loved algebra and have often avoided it, but its importance is becoming more apparent by the day. So I urge you to dive deeper into modules, and algebra as a whole.