Let $N$ be a module of finite length over a regular local ring $R$ of characteristic $0$. Let $M$ be an $R$-module which fits into a short exact sequence $0\to N^{\oplus a}\to M \to N^{\oplus b}\to 0$ for some positive integers $a, b$. Then, is it true that $\sum_{i=0}^\infty \beta^R_i(M) \ge \sum_{i=0}^\infty \beta^R_i(N) $ ?
Here, $\beta_i^R(-)$ denotes the Betti numbers, hence each of the infinite sums are actually finite since $R$ is regular.
