Abstract :
Let $I$ be an ideal in a regular local ring $(R,\n)$, we will find
bounds on the first and the last Betti numbers of
$(A,\m)=(R/I,\n/I)$. if $A$ is an Artinian ring of the embedding
codimension $h$, $I$ has the initial degree $t$ and $\mu(\m^t)=1$,
we call $A$ a {\it $t-$extended stretched local ring}. This class of
local rings is a natural generalization of the class of stretched local rings studied by Sally, Elias and Valla. For a $t-$extended stretched local ring,
we show that ${h+t-2\choose t-1}-h+1\leq \tau(A)\leq {h+t-2\choose
t-1}$ and $ {h+t-1\choose t}-1 \leq \mu(I) \leq {h+t-1\choose t}$.
Moreover $\tau(A)$ reaches the upper bound if and only if $\mu(I)$
is the maximum value. Using these results, we show when
$\beta_i(A)=\beta_i(\gr_\m(A))$ for each $i\geq 0$. Beside, we will
investigate the rigid behavior of the Betti numbers of $A$ in the
case that $I$ has initial degree $t$ and $\mu(\m^t)=2$. This class
is a natural generalization of {\it almost stretched local rings}
again studied by Elias and Valla. Our research extends several
results of two papers by Rossi, Elias and Valla.
