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.



Minimal resolution of local cohomology module
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](http://www.ams.org/journals/proc/2013-141-07/S0002-9939-2013-11510-8/home.html) on both questions mentioned above. The bibliography also contains some useful references, particularly the surveys of Avramov. 


