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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 $r$).

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    $\begingroup$ Locally $s$ is given a collection of functions $f_1,f_2\ldots $. $s$ would be regular if the sequence is regular in the usual sense of commutative algebra. Concerning your 3rd paragraph, it's a bit more complicated. See p 151 of Grothendieck, La theorie des classes de Chern. The last paragraph seems pretty far off. The closest correct statement involves the relation between $K_0(X)$ and $CH^*(X)$. $\endgroup$ – Donu Arapura Nov 22 '20 at 17:11
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    $\begingroup$ Of course the local description globalises to a definition in terms of the Koszul complex: a section $\mathscr E^\vee \to \mathcal O$ is regular if and only if the Koszul complex $0 \to \wedge^r \mathscr E^\vee \to \ldots \to \mathscr E^\vee \to \mathcal O$ is exact. $\endgroup$ – R. van Dobben de Bruyn Nov 22 '20 at 21:13
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The question of what regularity means was addressed in the comments, and I don't have any more to add. In particular, if $s$ is a regular section of a rank $r$ bundle $\mathcal{E}$, the class of $Z(s)$ in the Grothendieck group $K_0(X)$ is $\sum (-1)^i [\wedge^i \mathcal{E}^\vee]$. When $X$ is a nonsingular variety, $r$th Chern class of this is $(-1)^r(r-1)![Z(s)]$, where $[Z(s)]$ is the class in $CH^r(X)$. See, for example, page 151 of Grothendieck, La théorie des classes de Chern.

Finally, let me point out that the (normalized) $r$th Chern class generally won't give a bijection between the pointed set of isomorphism classes of rank $r$ bundles and $CH^r(X)$. To see this, let $X=\mathbb{P}^2$ and $r=2$. Then I'll leave you to check that $$c_2(T_X)= c_2(O(1)\oplus O(3))\in CH^2(X)\cong \mathbb{Z}$$ but the bundles are not isomorphic. Probably the best you can hope for along the lines you are suggesting is the isomorphism $K_0(X)\otimes \mathbb{Q}\cong CH^*(X)\otimes\mathbb{Q}$, when $X$ is a nonsingular variety.

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