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][1] 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][2]).


  [1]: https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455121
  [2]: http://www.sciencedirect.com/science/article/pii/S0022404997001035