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Residues of differentials on curves. (English) Zbl 0159.22702
In order to obtain “abstract residues” of differentials, the author generalizes the notion of trace thus: An endomorphism $$\vartheta$$ of a vector space $$V$$ over a field $$k$$ is called finite potent if $$\vartheta^nV$$ is finite dimensional for some $$n$$; for such $$\vartheta$$ a trace $$\text{Tr}(\vartheta)\in k$$ is defined by the properties: (T$$_1$$) If $$V$$ is finite dimensional, then $$\text{Tr}_V(\vartheta)$$ is the ordinary trace; (T$$_2$$) If $$W$$ is a subspace of $$V$$, and $$\vartheta W\subset W$$, then $$\text{Tr}_V(\vartheta)= \text{Tr}_W(\vartheta)+\text{Tr}_{V/W}(\vartheta)$$; (T$$_3$$) If $$\vartheta$$ is nilpotent, then $$\text{Tr}_V(\vartheta)=0$$. It follows: (T$$_4$$) If $$F$$ is a finite potent subspace of $$\text{End}(V)$$ (i.e., if there exists an $$n$$ such that for any family of $$n$$ elements $$\vartheta_1,\ldots,\vartheta_n\in F$$ the space $$\vartheta_1\cdots\vartheta_nV$$ is finite dimensional), then $$\text{Tr}_V: F\to k$$ is $$k$$-linear. (The author doubts whether the rule $$\text{Tr}_V\vartheta_1+\text{Tr}_V\vartheta_2=\text{Tr}_V(\vartheta_1+\vartheta_2)$$ holds in general, although he has not a counter example.
The author defines “abstract residues” for the situation: $$k$$ is a field, $$K$$ is a commutative $$k$$-algebra (with 1), $$V$$ is a $$K$$-module, and $$A$$ a $$k$$-subspace of $$V$$ such that $$(fA+A)/A$$ is finite dimensional for all $$f$$ in $$K$$; then there exists a $$k$$-linear “residue map” $$\text{res}_A^V: \Omega_{K/k}^1 \to k$$ such that for each pair of elements $$f$$ and $$g$$ in $$K$$ we have $$\mathrm{res}_A^V(fdg)=\mathrm{Tr}_V([\pi f,g])$$, where $$\pi: V\to V$$ is the $$k$$-linear projection of $$V$$ onto $$A$$. ($$\Omega_{K/k}^1$$ is the module of $$k$$-differentials in $$K$$; the commutator $$[\pi f,g]=\pi fg-g\pi f$$ is finite potent, obviously.) It follows that the residue map is continuous, i.e. if $$fA\subset A$$ and $$gA\subset A$$, then $$\text{res}_A^Vf dg=0$$) and is additive in $$A$$ (i.e. $$\text{res}_A^V+\text{res}_B^V=\text{res}_{A\cap B}^V+\text{res}_{A+B}^V$$ for $$A,B$$ as above with respect to $$K$$). Also if $$g\in K$$, then $$\text{res}_A^V(g^n dg)=0$$ for all $$n\geq 0$$; if $$g$$ is invertible, $$\text{res}_A^V(g^n dg)=0$$ for all $$n\leq -2$$ and if, in addition $$gA\subset A$$, then $$\text{res}_A^V(g^{-1} dg)=\dim_k(A/gA)$$.
Let $$K$$ be any function field in one variable over $$k$$ and $$X$$ a connected, regular scheme of dimension 1, proper over $$k$$, such that $$K=k(X)$$; the closed points $$p$$ of $$X$$ correspond to the discrete valuation rings $$\mathcal O_p$$ with the field of fractions $$K$$ which contain $$k$$. Denote by $$A_p$$ the completion of $$\mathcal O_p$$ and by $$K_p$$ the field of fractions of $$A_p$$. The author defines $$\text{res}_p: \Omega_{K/k}^1 \to k$$ to be the $$k$$-linear map such that $$\text{res}_p f dg=\text{res}_{A_p}^{K_p}(fdg)$$.
“All the standard theorems on residues of differentials on curves follow easily from this definition by proofs which are natural and independent of the characteristic of $$k$$.” In particular, if $$p$$ is a $$k$$-rational point of $$X$$ so $$A_p\approx k[[t]]$$ and $$f\in K_p\approx k((t))$$, then $$\text{res}_pf dt=$$ coefficient of $$t^{-1}$$ in $$f(t)$$ (which follows easily from the properties of $$\text{res}$$. Hence the definition agrees with the classical one). Also the fact that “the sum of the residues is zero on a complete curve $$X^n$$ results directly, without computation, from the finiteness of $$H^0(X,\mathcal O_X)$$ and $$H^1(X,\mathcal O_X)$$.
Using the theorems on residues the author presents briefly the “duality theorem” (for an arbitrary regular curve $$X$$ proper over $$k$$ there is a “dualizing sheaf” $$\dot\mathcal J_{X/K}$$ and a morphism $$c: \Omega_{X/K}^1\to \dot\mathcal J_{X/K}$$ which is an isomorphism at all points $$p$$ where $$X$$ is smooth over $$k$$).

##### MSC:
 14-XX Algebraic geometry
##### Keywords:
algebraic geometry
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