---
tag: 3B1743XS7S
kind: concept
name: Betti numbers of a module
msc: [13D02, 13D07]
sources:
  - url: https://en.wikipedia.org/wiki/Betti_number
---

## Summary

The Betti numbers of a finitely generated module over a local ring are the ranks of the free modules in its minimal free resolution.
The same holds for graded modules over a polynomial ring.
They are computed as the dimensions of the [[Tor|DDQRERAQCV]] modules against the residue field.
They are the basic numerical invariants of syzygies, and they refine the Hilbert function and the [[projective dimension|MR9QCH1PNA]].

## Explanations

### L1 human

To describe a module, one picks generators, then relations among them, then relations among the relations, and so on.
If one picks as few as possible at each step, the counts do not depend on the choices.
These counts are the Betti numbers.
For the residue field $k$ of $k[x, y]$, one needs $1$ generator, $2$ relations $x$ and $y$, and $1$ relation among them, so the Betti numbers are $1, 2, 1$.
The name comes from the Betti numbers of topology, which count holes in a space.