Let $X$ be a smooth projective variety (over $\mathbb C$ to fix ideas) and $Z\subset X$ a locally complete intersection of codimension $k$. We fix the notation $U:=X-Z$ for the complement and $j: U\to X$ the open immersion.
In Bloch's article "Semi-Regularity and de Rham Cohomology", he constructed the de Rham class of $[Z]\in H^{2k}_{dR}(X/\mathbb C)$ explicitly. There is however one construction which seems obscure to me:
Let $V\subset X$ be an open subset such that $V\cap Z\subset V$ is defined by $k$ functions $f_1,\ldots, f_k$ on $V$. If we write $V_i:=\{x\in V: f_i(x)\neq 0\}$, then $\{V_1,\ldots, V_k\}$ gives an open cover of $U\cap V$. Furthermore, we can define a closed $k$-form on $V_1\cap \ldots \cap V_k$ by $$\frac{df_1}{f_1}\wedge\ldots\wedge \frac{df_k}{f_k}. $$ This $k$-form, which is clearly a closed $k-1$-cochain in the Čech cohomology, gives us a class in $H^{k-1}(U\cap V,\Omega_X^k)$, which gives us further a class in $H^0(V,R^{k-1}j_*\Omega_U^k)$, then further a class in $H^0(V,\mathcal H^k_Z(X,\Omega_X^k)$ where $\mathcal H^k_Z(X,\Omega_X^k)$ is the sheaf version local cohomology defined by $$\mathcal H^k_Z(X,\Omega_X^k):=\lim_{i\in\mathbb N}\mathcal Ext_X^k(\mathcal O_X/\mathcal I_Z^i, \Omega_X^k). $$
It is said that this class does not depend on the choice of functions $f_1,\ldots, f_k$. In other words, if we have $g_1,\ldots, g_k$ is another system of functions defining $Z\cap V\subset V$, then $\frac{df_1}{f_1}\wedge\ldots\wedge \frac{df_k}{f_k}$ and $\frac{dg_1}{g_1}\wedge\ldots\wedge \frac{dg_k}{g_k}$ will define the same class in $H^0(V,\mathcal H^k_Z(X,\Omega_X^k)$.
This is the part that I do not understand. In the article, references are given: they are FGA exposé 149 and Hartshorne's Residues and Duality page 176. But they do not seem to me to be helpful.
p.s. I decide to add this comment which may be trivial. I wanted to say that $k=1$ case is relatively easy. In fact, if we have a divisor locally defined by a function $f$, then another defining function $g$ has $g=tf$ with $t$ an invertible function. Then $$\frac{dg}{g}=\frac{df}{f}+d\log t. $$ $t$ being invertible, $d\log t$ is locally well-defined one form and thus is an element $H^0(V,\Omega_X^1)$. By the exact sequence $$\Omega_X^1\to j_*\Omega_U^1\to \mathcal H_Z^1(X,\Omega_X^1) $$, the image of $d\log t$ in $H^0(V,\mathcal H_Z^1(X,\Omega_X^1))$ is zero. Hence, $\frac{dg}{g}$ and $\frac{df}{f}$ have the same class in $H^0(V, \mathcal H_Z^1(X,\Omega_X^1))$.