betti numbers of a finite local module

Assume the Axiom of Choice. Let $M$ be a finite module over a nonzero Noetherian local ring $(R,\mathfrak{m},k)$, and choose a minimal degreewise finite free resolution $F_{\bullet }\to M$, which exists under this assumption. For $i\ge 0$, its Betti number is ${\beta }_i^R(M):={\dim }_k{H}_i(k{\otimes }_R{F}_{\bullet })={\dim }_k{\mathrm{Tor}}_i^R(k,M)\mathrm{.}$ The first dimension is finite because each $k{\otimes }_R{F}_i$ is a finite-dimensional $k$-vector space. The value does not depend on the chosen resolution: choice supplies dependent choice by selecting a successor from each nonempty successor set and recursively iterating that selection, and the Tor comparison for supplied resolutions then identifies their homology. Whenever a minimal Koszul resolution exists, the rank formula identifies these numbers with its Koszul Betti numbers. For $M=0$ all Betti numbers are zero.