2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.