Residue of a rational differential at a separable closed point

Assume the Axiom of Choice as inherited from the differential, completion, Hensel and power-series suppliers (The Axiom of Choice). Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field), let $p\in C$ be a closed point whose residue field $\kappa (p)$ is finite separable over $k$ (Separable algebraic elements and separable extensions), and write $K=k(C)$ for the function field of $C$. Put $A={\mathcal{O}}_{C,p}$, a Noetherian discrete valuation ring (Local rings at closed points of smooth curves are discrete valuation rings), and let $\widehat{A}={\widehat{\mathcal{O}}}_{C,p}$ be its maximal-adic completion (The $I$-adic completion of a module, Completion of a Noetherian local ring is local with the same residue field).

The coefficient field. Since $\kappa (p)\mathrm{/}k$ is finite separable it is simple (A finite extension generated by elements all but possibly one of which are separable is simple), so choose $\alpha \in \kappa (p)$ with $\kappa (p)=k(\alpha )$, and let $P\in k[X]$ be its monic minimal polynomial, which is separable; thus $P^{\mathrm{\prime }}(\alpha )\mathrm{\ne }0$ (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element, A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$). The completion $\widehat{A}$ is a complete Noetherian local ring with residue field $\kappa (p)$ (Completion of a Noetherian local ring is local with the same residue field) and is regular (hence a domain) because $A$ is regular (completion preserves regular local rings); being complete and separated in the maximal-adic topology, it is a Henselian pair (Complete separated adic pairs are Henselian). View $P$ in $\widehat{A}[X]$ through $k\to \widehat{A}$: its reduction to $\kappa (p)[X]$ is $P$, and $\alpha$ is a simple root, so Hensel's lemma lifts $\alpha$ to a unique root $\widehat{\alpha }\in \widehat{A}$ of $P$ (Factor lifting implies simple-root lifting). The $k$-algebra map $\iota :\kappa (p)=k(\alpha )⟶\widehat{A},\alpha ⟼\widehat{\alpha },$ is therefore well defined, and it is a section of the residue map $\widehat{A}\to \kappa (p)$ because $\widehat{\alpha }$ reduces to $\alpha$; it is the unique such section, since any section must send $\alpha$ to a root of $P$ lifting $\alpha$ and the lift is unique. This is the $k$-compatible coefficient field $\kappa (p)\hookrightarrow {\widehat{\mathcal{O}}}_{C,p}$ of the statement.

Uniformizers and the identification with $\kappa (p)[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$. Let $t$ be a uniformizer of $A$, so that the maximal ideal of $A$ is $(t)$ and the same element generates the maximal ideal $\mathfrak{m}\widehat{A}$ of $\widehat{A}$ (Local rings at closed points of smooth curves are discrete valuation rings, Completion of a Noetherian local ring is local with the same residue field). Then $\widehat{A}$ is a complete equicharacteristic Noetherian local domain of dimension one, equicharacteristic with coefficient field $\kappa (p)\subseteq \widehat{A}$ via $\iota$, and $t$ is a system of parameters. The continuous map $\phi :\kappa (p)[\text{\hspace{0.17em}⁣}[T]\text{\hspace{0.17em}⁣}]⟶\widehat{A},T⟼t,$ is injective (The parameter power-series map is injective by dimension) and makes $\widehat{A}$ finite over its image (Parameters make a complete local domain finite over the image of a power-series map); since $\widehat{A}\mathrm{/}t\widehat{A}=\kappa (p)$ is generated over the image and $\widehat{A}$ is complete, Nakayama's lemma gives that $\phi$ is surjective (Complete Nakayama lemma). Hence $\phi$ is an isomorphism ${\widehat{\mathcal{O}}}_{C,p}\cong \kappa (p)[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}],$ and this isomorphism is compatible with $\iota$ and sends $T$ to $t$. The dependence on the two auxiliary choices is addressed in the independence statement below.

The residue. Let $\omega \in {\mathrm{\Omega }}_{K\mathrm{/}k}^1$ be a rational differential. By A uniformizer differential generates the module of differentials the differential $\mathrm{d}t$ is a $K$-basis of ${\mathrm{\Omega }}_{K\mathrm{/}k}^1$, so there is a unique $a\in K$ with $\omega =a\mathrm{d}t$. Since $K=\mathrm{Frac}(A)$ embeds in $\mathrm{Frac}(\widehat{A})=\kappa (p)((t))$, the element $a$ has a formal Laurent expansion $a={\sum }_{n\ge N}{a}_n{t}^n$ with all $a_n\in \kappa (p)$ (Formal Laurent series $K((x))$, their order, derivative, and residue), and its $-1$-coefficient $a_{-1}={\mathrm{res}}_t(a)$ is defined. Define the residue of $\omega$ at $p$ by ${\mathrm{res}}_p(\omega ):={\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}(a_{-1})\in k,$ the field trace of the coefficient of $t^{-1}$ (The norm $N_{K\mathrm{/}F}$ and trace ${\mathrm{Tr}}_{K\mathrm{/}F}$ of a finite field extension). The value does not depend on the choice of uniformizer $t$: if $s$ is another uniformizer and $\omega =b\mathrm{d}s$ with $b\in K$, then the formal residues of the two Laurent series are related by the change-of-uniformizer formula, and the coefficient traces agree (The residue is independent of the uniformizer ↗). Consequently the assignment $(\omega ,p)\mapsto {\mathrm{res}}_p(\omega )$ is well defined on ${\mathrm{\Omega }}_{K\mathrm{/}k}^1$ at every closed point $p$ with $\kappa (p)$ finite separable over $k$; it is $k$-linear because the trace is $k$-linear (The norm $N_{K\mathrm{/}F}$ and trace ${\mathrm{Tr}}_{K\mathrm{/}F}$ of a finite field extension).

Perfect fields. If $k$ is perfect, then every algebraic extension of $k$ is separable (Every algebraic extension of a perfect field is separable, Perfect fields: every irreducible polynomial is separable), so $\kappa (p)$ is finite separable over $k$ at every closed point $p$ and ${\mathrm{res}}_p$ is defined everywhere. Over an imperfect field no coefficient-trace formula is asserted here at a closed point with inseparable residue field.