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  "body": "\n## Summary\n\nThe Betti numbers of a finitely generated module over a local ring are the ranks of the free modules in its minimal free resolution.\nThe same holds for graded modules over a polynomial ring.\nThey are computed as the dimensions of the [[Tor|DDQRERAQCV]] modules against the residue field.\nThey are the basic numerical invariants of syzygies, and they refine the Hilbert function and the [[projective dimension|MR9QCH1PNA]].\n\n## Explanations\n\n### L1 human\n\nTo describe a module, one picks generators, then relations among them, then relations among the relations, and so on.\nIf one picks as few as possible at each step, the counts do not depend on the choices.\nThese counts are the Betti numbers.\nFor 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 $1, 2, 1$.\nThe name comes from the Betti numbers of topology, which count holes in a space.\n",
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