# Koszul homology of Jacobian ideals in a (graded) isolated singularity

Hi Let $f$ be a homogenous polynomial in $R = K[X_1,\ldots,X_n]$. Set $\mathfrak{m} = (X_1,\ldots,X_n)$. Assume $A = R/(f)$ is an isolated singularity. Let $J = (\partial_1(f),\ldots,\partial_n(f))A$ be the ideal in $A$ generated by the Jacobian ideal of $f$. Let

$$K_\bullet = K(\partial_1(f),\ldots,\partial_n(f) ; A)$$

be the Koszul complex. As $A = R/(f)$ is an isolated singularity we have that $J$ is $\mathfrak{m} A$-primary. So we have that $H_i(K) = 0$ for $i \geq 2$. Also $H_0(K) = A/J$.

My question is what is known about $H_1(K)$ ?

Any reference will be greatly appreciated.

-