Intersection with an invertible sheaf and the first Chern class

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field and let $X$ be a scheme locally of finite type over $k$. Let $L$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves).

1. Integral case. Let $W\subseteq X$ be integral of dimension $n$, with function field $k(W)$ (Sheaf total quotient rings, The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain), and let $s$ be a nonzero rational section of $L{\mathrm{\mid }}_W$ (Rational section line bundle). The Weil divisor of $s$ is ${\mathrm{div}}_L(s)={\sum }_{Z\subseteq W}{\mathrm{ord}}_{{\mathcal{O}}_{W,Z}}(s)[Z]\in Z_{n-1}(W),$ the sum over the codimension-one integral closed subschemes $Z\subseteq W$, with the order function of The order function of a one-dimensional Noetherian local domain read on a local generator of $L$ at the generic point of $Z$; the sum is locally finite, and finite if $W$ is of finite type. The first Chern class $c_1(L)\cap [W]:=[{\mathrm{div}}_L(s)]\in A_{n-1}(W)$ is independent of the chosen nonzero rational section (two such differ by a rational function, whose divisor is rationally equivalent to zero), and therefore defines a codimension-one Chow class; on a smooth $W$ this is the first Chern class in its Chow ring.
2. General case. For an integral closed subscheme $i:W\hookrightarrow X$ of dimension $d+1$ set $c_1(L)\cap [W]:=i_{\ast }(c_1(i^{\ast }L)\cap [W])\in A_d(X)$, where $i_{\ast }$ is the proper pushforward (Proper pushforward of cycles and the norm formula); extend $\mathbb{Z}$-linearly. The resulting operation $c_1(L)\cap -:A_{d+1}(X)\to A_d(X)$ is well defined on Chow groups (this is the content of the next two references) and graded.

Basic properties. (i) $c_1(L{\otimes }_{{\mathcal{O}}_X}N)\cap \alpha =(c_1(L)+c_1(N))\cap \alpha$ and $c_1({\mathcal{O}}_X)\cap \alpha =0$ for all $\alpha$. (ii) If $X$ is pure of dimension $d+1$ and a global section $s$ of $L$ is a nonzerodivisor on ${\mathcal{O}}_X$, its zero scheme $D$ is an effective Cartier divisor and $c_1(L)\cap [X]=[D]$ in $A_d(X)$. More generally the same formula on a pure-dimensional closed subscheme $Y$ requires $s{\mathrm{\mid }}_Y$ to be a nonzerodivisor on ${\mathcal{O}}_Y$. (iii) $c_1(L)\cap -$ commutes with proper pushforward and flat pullback: for $f:X\to Y$ proper, $f_{\ast }(c_1(f^{\ast }L)\cap \alpha )=c_1(L)\cap f_{\ast }\alpha$, and for $f$ flat of relative dimension $n$, $f^{\ast }(c_1(L)\cap \alpha )=c_1(f^{\ast }L)\cap f^{\ast }\alpha$. (iv) $c_1(L)$ depends only on the isomorphism class of $L$.

Cartier Gysin. For a section $s$ of $L$ with zero scheme $j:D\hookrightarrow X$, after any base change define $j^{!}:A_m(X^{\mathrm{\prime }})\to A_{m-1}(D^{\mathrm{\prime }})$ on integral $V\subset X^{\mathrm{\prime }}$ by the Cartier divisor of $s{\mathrm{\mid }}_V$ if $V$ is not contained in $D^{\mathrm{\prime }}$, and by $c_1(L{\mathrm{\mid }}_V)\cap [V]$ if it is; push this class from $D^{\mathrm{\prime }}\cap V$ to $D^{\mathrm{\prime }}$. This works even when the pulled-back zero divisor is not Cartier. It commutes with proper pushforward and flat pullback, two Cartier Gysins commute, and $j^{!}{j}_{\ast }\beta =c_1(L{\mathrm{\mid }}_D)\cap \beta$.

Well-definedness. All the constructions above are local in the integral cycle, so they are defined for locally finite cycles as well. In the integral case, a nonzero rational section $s$ of $L{\mathrm{\mid }}_W$ is a rational multiple of a local generator, so the order function of The order function of a one-dimensional Noetherian local domain is defined at the generic point of every codimension-one integral closed subscheme $Z\subseteq W$, and only finitely many $Z$ meeting any fixed affine chart receive a nonzero order: on that chart $W$ is a Noetherian domain and $s$ is represented by a fraction whose numerator and denominator vanish on only finitely many height-one primes, exactly as in the finiteness discussion of Rational equivalence and the Chow group of cycles. The class $[{\mathrm{div}}_L(s)]$ is independent of $s$ because the ratio of two rational sections is a rational function of $k(W{)}^{\ast }$ and principal divisors lie in ${\mathrm{Rat}}_{n-1}(W)$; this is the definition of rational equivalence. In the general case the operation is extended by proper pushforward along $i$ and by linearity; that it descends through rational equivalence on $X$ is the content of Tame symbol reciprocity in dimension two (the Key Lemma) (the tame symbol reciprocity that controls the difference of two iterated Cartier intersections) together with the facts that proper pushforward and flat pullback each descend to Chow groups (Flat pullback of cycles and of rational equivalence, Proper pushforward of cycles and the norm formula). Property (ii) is the cycle computation of Cycles of coherent sheaves and of closed subschemes, with flat pullback. For (iii), the flat-pullback identity is the generic-length calculation in Flat pullback of cycles and of rational equivalence applied to the divisor cycle; for proper pushforward, trivializing the line bundle at codimension-one generic points reduces the identity to the norm-order formula in Proper pushforward of cycles and the norm formula. Both identities are checked on integral cycles and extend linearly. Property (iv) follows because an isomorphism of invertible sheaves identifies their local generators and hence the resulting divisor coefficients.