# Dimension of the socle of the first local cohomology module

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.

• I would check the Brodmann-Sharp book on Local Cohomology. Apr 24 '20 at 16:16