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.



A question about Complete Intersections
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](https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455121) 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](http://www.sciencedirect.com/science/article/pii/S0022404997001035)).


