Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$

Question Can we say anything about Betti numbers or minimal resolution of $H_{\mathfrak m}^i(R)?$

In particular, can we say anything (whether the Betti numbers are nondecreasing or strictly increasing, direct summand of the syzygies) about $H_{\mathfrak m}^0(R)?$

P.S. Any reference will be extremely helpful.