Let $X$ be a scheme. Let $\mathcal E$ be a locally free sheaf of rank $r$ on $X$ and let $s$ be a section of $\mathcal E$. Then the zero scheme of $s$ is defined as follows: Consider the homomorphism $\mathcal O_X \to \mathcal E$ induced by $s$, taking duals, we obtain $\mathcal E^\vee \to \mathcal O_X$. Then $Z(s)$ is defined to be the scheme associated to the sheaf of ideals $\mathop{\mathrm{im}}(\mathcal E^\vee \to \mathcal O_X)$.

First question: is there some natural conditions of regularity defined on $s$. For example, when $\mathcal E = \mathcal L$ is an invertible $\mathcal O_X$-module, then $s$ is said to be **regular** if and only if the induced homomorphism $\mathcal O_X \to \mathcal L$ is injective, and in this case $Z(s)$ is an effective Cartier divisor on $X$. So I think the condition of regularity should satisfy that if $s$ is regular, then any generic point of irreducible components of $Z(s)$ has codimension $r$ in $X$.

Now suppose we have already defined some conditions of regularity. Assume now that $X$ is a smooth projective variety and let $s$ be a "regular" section. Consider the $r$-cycle associated to $Z(s)$. Prove the following statement: The class of the $r$-cycle associated to $Z(s)$ in the $r$-th Chow group $\mathop{\mathrm{CH}}^r(X)$ equals the $r$-th Chern class $c_r(\mathcal E)$ of $\mathcal E$.

Hence the linear equivalence class of $Z(s)$ is independent of the choice of $s$ and we get a well-defined map $$\{\text{locally free sheaves of rank } r \text{ on } X\} / \{\text{isomorphisms}\} \to \mathop{\mathrm{CH}}\nolimits^r(X),$$ is this map bijective? (When $r = 1$, we obtain an isomorphism $\mathop{\mathrm{Pic}}(X) \to \mathop{\mathrm{Cl}}(X)$.)

If the map above is bijective, then the group structure on $\mathop{\mathrm{CH}}\nolimits^r(X)$ should induces a group structure on $\{\text{locally free sheaves of rank } r \text{ on } X\} / \{\text{isomorphisms}\}$, and what is it? (When $r = 1$, it is tensor products of invertible $\mathcal O_X$-modules. However, we cannot simply take the tensor product of two locally free sheaves of rank $r$).