Residues of differentials on curves.

*(English)*Zbl 0159.22702In 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 \(k\)).

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 \(k\)).

Reviewer: Alexandru Brezuleanu (Bucureşti)

##### MSC:

14-XX | Algebraic geometry |