Let $I$ be a non-zero ideal in a regular local ring $(R, \mathfrak m,k) $ (where $k:=R/\mathfrak m$) . The socle of $R/I$ is Hom$_R(k, R/I) \cong (I:\mathfrak m)/I$ , which as evident has a natural $k$-module structure. Let $s$ denote the $k$-vector space dimension of the socle of $R/I$ i.e. $s:=\dim_k (I:\mathfrak m)/I$ .
My question is: Can we express $s$ in terms of the Betti numbers of $I$ or $R/I$ ? Even if there's no general formula, I would like to know if any thing is known at least when $R$ has dimension $2$ or $3$.
NOTE: The Betti numbers of a finitely generated module $M$ over a Noetherian local ring $(R, \mathfrak m,k) $ are $b_i (M) = \dim_k$Tor$_i^R (M,k)=\dim_k$ Ext$^i_R (M,k)=$rank $F_i$ , where $F_i$ is the $i$-th free module in a minimal free resolution of $M$. So for example, $b_0(M)=\mu (M)$ is the minimal no. of generators for $M$. Also, the definition of colon ideal is $(I:J):=\{r \in R | rJ \subseteq I\}$.
Best Answer
Since $R$ is regular, there is the Koszul complex $K^\bullet$ resolving $k$. Assume $\mathfrak{m} = (x_1, \ldots, x_d)$ where $d = \dim R$.
The homology of $K^\bullet \otimes M$ is $\mathrm{Tor}^R_i(k, M)$. On the other hand, the last homology is the kernel of the map
$M \to M^d$ according to $m \mapsto (x_1m, \ldots, x_d m)$
which is clearly isomorphic to $\mathrm{Hom}(k, M)$. With $M = R/I$, this shows that the dimension of the socle is the top Betti number $\beta_d(R/I)$.