Let $M$ be a graded $\mathbb{C}[z_0,\dots,z_n]$-module. Using local duality one can show that $$ \dim_\mathbb{C} (\text{soc} H_\mathfrak{m}^1(M))_k = \beta_{n,k+n+1}(M). $$ Here $H_\mathfrak{m}^1(M)$ denotes the first local cohomology module of $M$ with respect to the ideal $\mathfrak{m} = (z_0,\dots,z_n) \subset \mathbb{C}[z_0,\dots,z_n]$, and $\beta_{i,j}(M)$ denotes the graded Betti numbers of $M$.
Has this formula appeared in print somwhere? The closest thing I could find was in https://arxiv.org/pdf/0809.1458.pdf, where is says on page 17 that "...the socle of the first nonzero local cohomology precisely reflects the top Betti numbers."
Edit: I also found an ungraded version of the formula in Theorem 12.4 in Combinatorics and Commutative Algebra by Stanley.
