Betti numbers of a module

Remarks

Equivalently, $\beta_i(M)$ is the rank of the free module $F_i$ in a minimal free resolution $\cdots \to F_1 \to F_0 \to M \to 0$, that is, one whose differentials have all entries in $\mathfrak{m}$.

For $R = k[x, y]$ and $M = k = R/(x, y)$, the Koszul complex $$0 \to R(-2) \to R(-1)^2 \to R \to k \to 0$$ is a minimal resolution, so $\beta_0 = 1$, $\beta_1 = 2$, $\beta_2 = 1$, and the nonzero graded Betti numbers are $\beta_{0,0} = 1$, $\beta_{1,1} = 2$, $\beta_{2,2} = 1$.

The ranks in a resolution that is not minimal are not the Betti numbers. For $M = R/(x)$ over $R = k[x]$, the resolution $0 \to R \oplus R \to R \oplus R \to M \to 0$, with the second summand mapped identically, has ranks $2, 2$, while $\beta_0 = \beta_1 = 1$.

$\beta_0(M)$ is the least number of generators of $M$, and the largest $i$ with $\beta_i(M) \ne 0$ is the projective dimension of $M$ when it is finite.