1. CMB 2015 (vol 58 pp. 664)
 Vahidi, Alireza

Betti Numbers and Flat Dimensions of Local Cohomology Modules
Assume that $R$ is a commutative Noetherian ring with nonzero
identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$module.
In this paper, we first study the finiteness of Betti numbers
of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some
inequalities between the Betti numbers of $X$ and those of its
local cohomology modules. Finally, we present many upper bounds
for the flat dimension of $X$ in terms of the flat dimensions
of its local cohomology modules and an upper bound for the flat
dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of
the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.
Keywords:Betti numbers, flat dimensions, local cohomology modules Categories:13D45, 13D05 

2. CMB 2011 (vol 56 pp. 459)
 Athavale, Ameer; Patil, Pramod

On Certain Multivariable Subnormal Weighted Shifts and their Duals
To every subnormal $m$variable weighted shift $S$ (with bounded
positive weights) corresponds a positive Reinhardt measure $\mu$
supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for
$m \geq 2$, the dimensions of the $1$st cohomology vector spaces
associated with the Koszul complexes of $S$ and its dual ${\tilde S}$
are different if a certain radial function happens to be integrable
with respect to $\mu$ (which is indeed the case with many classical
examples). In particular, $S$ cannot in that case be similar to
${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal
$m$variable weighted shift $S$ cannot be similar to its dual.
Keywords:subnormal, Reinhardt, Betti numbers Category:47B20 
