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.
It is hard to give a useful answer. I suspect whatever you want/need would be more specific. In particular, details on how such $R$ arises in your research would make it easier to say something more concrete. But anyhow, there are a couple of remarks and references one can point to.
First, your ring is not Cohen-Macaulay. So any nonzero finite length module would have infinite minimal resolution. Typically, the Betti numbers are expected to have eventual exponential growth (see Avramov's surveys from the reference below).
One of the outstanding open problems in this area is whether the Betti numbers are always eventually non-decreasing for any module. There is a related question of whether the syzygies will eventually have full dimension (equals $\dim R$). I don't expect the fact that your modules are local cohomology to make a big difference. Here is a nice recent paper on both questions mentioned above. The bibliography also contains some useful references, particularly the surveys of Avramov.
