Suppose that $(A,\mathfrak{m})$ is a complete intersection and $\mathbf{x}$ is a minimal basis for $\mathfrak{m}$. Consider the Koszul homologies $H_\bullet(\mathbf{x},A)$. It is well-known that $\text{Vdim}_{K}(H_1(\mathbf{x},A))=\text{embdim}(A)-\text{dim}(A)$. My question is that do we have some information about the vector space dimension of the second Koszul homology $H_2(\mathbf{x},A)$? Something different from the Euler-Poincare characteristic which says that the sum of length of Koszul homologies (multiplied by a power of -1) vanishes.
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If $n=\text{dim}_{K}(H_1(\mathbf{x},A))$ then $\text{dim}_{K}(H_2(\mathbf{x},A))=\frac{n(n-1)}{2}$. This was proved by Assmus in 1958. In general, $H_*(\mathbf{x},A)$ is the exterior algebra over $H_1(\mathbf{x},A)$, even under weaker hypothesis (Corollary 3 in Blanco-Majadas-Rodicio).
