Local support and index bound for the different of a curve map

Given: A finite surjective morphism $f:C\to D$ of smooth proper geometrically integral curves over $k$ with $k(C)\mathrm{/}k(D)$ finite separable, a closed point $p\in C$ with image $q=f(p)$, and the relative differential sheaf ${\mathrm{\Omega }}_{C\mathrm{/}D}$.

The morphism $f$ is finite and surjective, the extension $k(C)\mathrm{/}k(D)$ is finite and separable of degree $\mathrm{deg}(f)$, and the local rings ${\mathcal{O}}_{C,p}$, ${\mathcal{O}}_{D,q}$ are discrete valuation rings with uniformizers $t_p$, $t_q$; write ${\mathrm{\Omega }}_{C\mathrm{/}D}$ for the relative differential sheaf. (Finite morphisms of schemes, Ramification index of a morphism of curves, Local rings at closed points of smooth curves are discrete valuation rings, Sheaf of relative Kähler differentials)

Kähler differentials commute with base change: for $C^{\mathrm{\prime }}=C{\times }_D{D}^{\mathrm{\prime }}$ the canonical map $g^{\ast }{\mathrm{\Omega }}_{C\mathrm{/}D}\to {\mathrm{\Omega }}_{C^{\mathrm{\prime }}\mathrm{/}{D}^{\mathrm{\prime }}}$ is an isomorphism; in particular ${\mathrm{\Omega }}_{C\mathrm{/}D,p}{\otimes }_{{\mathcal{O}}_{C,p}}\widehat{{\mathcal{O}}_{C,p}}\cong {\mathrm{\Omega }}_{\widehat{B}\mathrm{/}\widehat{A}}$ for the completed local rings $\widehat{A}=\widehat{{\mathcal{O}}_{D,q}}$, $\widehat{B}=\widehat{{\mathcal{O}}_{C,p}}$; completions are flat, so the local length $l_p$ is not changed. (Relative differentials commute with scheme base change)

Separable function fields have no differentials: if $L\mathrm{/}K$ is a finite separable field extension, then ${\mathrm{\Omega }}_{L\mathrm{/}K}=0$. Indeed $L=K(a)$ for some $a$ (A finite extension generated by elements all but possibly one of which are separable is simple), with minimal polynomial $m$ separable, so its derivative satisfies $m^{\mathrm{\prime }}(a)\mathrm{\ne }0$ (An irreducible polynomial over a field is separable exactly when its derivative is nonzero); the presentation ${\mathrm{\Omega }}_{K[x]\mathrm{/}(m)\mathrm{/}K}\cong (K[x]\mathrm{/}(m))\mathrm{d}x\mathrm{/}(m^{\mathrm{\prime }}(a)\mathrm{d}x)$ from Existence and generators of Kähler differentials and Jacobian presentation of Ω then vanishes because $m^{\mathrm{\prime }}(a)$ is a unit of $L$.

If ${\mathrm{\Omega }}_{C\mathrm{/}D}$ vanishes at the generic point of $C$ then it is a torsion sheaf on the integral curve $C$; a nonzero coherent torsion sheaf on $C$ is supported in a proper closed subset, which is a finite set of closed points. At a closed point $p$ the stalk ${\mathrm{\Omega }}_{C\mathrm{/}D,p}$ is then a finite-length ${\mathcal{O}}_{C,p}$-module, and $l_p$ is its length. (Proper closed subsets of a curve are finite, Composition series and length of a module, Coherent module sheaves)

Local structure after completion. Choose an affine neighborhood $V=\mathrm{Spec}(A_0)$ of $q$; because $f$ is finite, $f^{-1}(V)=\mathrm{Spec}(R_0)$ with $R_0$ finite over $A_0$. Put $A=(A_0{)}_q$ and $R=R_0{\otimes }_{A_0}A$. The map $A\to R$ is injective and $R$ is a domain, so $R$ is torsion-free over the DVR $A$ and therefore finite free. After the flat completion $A\to \widehat{A}$, the finite algebra $R{\otimes }_A\widehat{A}$ is a product of its local factors $B_{p^{\mathrm{\prime }}}=\widehat{{\mathcal{O}}_{C,p^{\mathrm{\prime }}}}$ indexed by the points $p^{\mathrm{\prime }}$ above $q$: its special fibre is an Artinian ring whose local idempotents lift uniquely in the complete algebra. Each factor is a direct summand, hence finite free over $\widehat{A}$, and is a complete DVR. For the chosen factor write $A=\widehat{{\mathcal{O}}_{D,q}}$, $B=\widehat{{\mathcal{O}}_{C,p}}$, $K_0=\mathrm{Frac}(A)$, $L_0=\mathrm{Frac}(B)$, $K=\kappa (q)$, and $L=\kappa (p)$. Then $L_0\mathrm{/}{K}_0$ is finite separable (it is a factor after base change of the generically separable field extension), and $t_q=u{s}^{e_p}$ for a unit $u\in B$ and a uniformizer $s=t_p$ of $B$. The special fibre $C_0=B\mathrm{/}{t}_{q}B$ has $e_p$ successive quotients isomorphic to $L$ over $K$, so ${\dim }_K{C}_0=e_p[L:K]$. (A finite flat module over a local ring is free, Every nonzero fraction is a unit times a power of a uniformiser, Ramification index of a morphism of curves)

The completed map $A\to B$ is finite flat and a local complete intersection. The graph $C\to C{\times }_{k}D$ is a section of the smooth projection $C{\times }_{k}D\to C$, hence a regular immersion (Stacks, Lemma 31.23.8, tag 067R); composing with the smooth projection $C{\times }_{k}D\to D$ gives an lci morphism (Stacks, Lemma 37.62.7, tag 069J). Finite flatness follows locally because the finite algebra over the target DVR is torsion-free, hence free, and this property is preserved by completion and by taking a direct factor. Since $f$ is finite, it is quasi-finite. The local quasi-finite flat lci criterion gives, after shrinking, a presentation $B=A[x_1,\dots ,x_n,1\mathrm{/}h]\mathrm{/}(f_1,\dots ,f_n)$ with a regular sequence of $n$ equations (Stacks, Lemma 49.10.1, tag 0BWE). If $J=(\mathrm{\partial }{f}_i\mathrm{/}\mathrm{\partial }{x}_j)$, the conormal presentation is $B^n\stackrel{J}{\to }{B}^n\to {\mathrm{\Omega }}_{B\mathrm{/}A}\to 0$, so ${\mathrm{Fitt}}_0({\mathrm{\Omega }}_{B\mathrm{/}A})=(\det J)$ by the definition of the zeroth Fitting ideal. The determinant is nonzero because the generic field extension is separable and thus ${\mathrm{\Omega }}_{L_0\mathrm{/}{K}_0}=0$. The same determinant generates the Noether different. Put $P=A[x_1,\dots ,x_n,1\mathrm{/}h]$, let $I=\ker (B{\otimes }_{A}B\to B)$, and write $g_j=x_j\otimes 1-1\otimes x_j$ in $P{\otimes }_{A}B$. These $g_j$ generate the kernel of multiplication. The Koszul complex on $(f_i)$ in $P$ resolves $B$ because the presentation is a regular sequence. For the diagonal sequence, write $P{\otimes }_{A}B=B[y_1,\dots ,y_n,1\mathrm{/}h(y)]$ and let ${\bar{x}}_j$ be the image of $x_j$ in $B$. Before localization the sequence $y_j-{\bar{x}}_j$ is regular: successively quotienting by its first $r$ terms gives the polynomial ring in the remaining variables over $B$, and the next monic linear polynomial is a nonzerodivisor even when $B$ has zero divisors. Localizing at $h(y)$ preserves regularity, and the final quotient is $B[1\mathrm{/}h(\bar{x})]=B$ because $h(\bar{x})$ is already a unit in $B$. Thus the Koszul complex on $(g_j)$ resolves $B$ over $P{\otimes }_{A}B$. Expanding each polynomial difference gives $f_i(x\otimes 1)-f_i(1\otimes x)={\sum }_j(\mathrm{\partial }{f}_i\mathrm{/}\mathrm{\partial }{x}_j)(1\otimes x)g_j+{\sum }_{j,r}{a}_{ijr}{g}_j{g}_r\mathrm{.}$ The comparison map between these Koszul resolutions sends the degree-one generator for $f_i$ to the linear combination of the $g_j$ with coefficients $1\otimes (\mathrm{\partial }{f}_i\mathrm{/}\mathrm{\partial }{x}_j)$ plus terms in $(g_1,\dots ,g_n)$; its top component is the determinant of that coefficient matrix. Both complexes compute ${\mathrm{Tor}}_{\ast }^P(B,B)$: the first is a free $P$-resolution, and the second is a flat $P$-resolution because $B$ is flat over $A$. After tensoring with $B$, the top homology of the second complex is the kernel of the map with entries $g_j$ on $B{\otimes }_{A}B$, namely $\mathrm{Ann}(I)$. The comparison map carries the generator of the top homology of the first complex to an element of this annihilator; multiplying its image in $B{\otimes }_{A}B$ gives the determinant of the coefficient matrix modulo $I$, which is $\det J$. Thus the Noether different, the image of $\mathrm{Ann}(I)\to B$, is $(\det J)$ (Stacks, Lemma 49.12.2, tag 0BWD). Now let $W={\mathrm{Hom}}_A(B,A)$ and let $\tau \in W$ be the trace functional $b\mapsto {\mathrm{Tr}}_{B\mathrm{/}A}(b)$. The diagonal-annihilator pairing identifies $\mathrm{Ann}(I)$ with ${\mathrm{Hom}}_B(W,B)$ (Stacks, Lemma 49.6.6, tag 0BVS): for a finite $A$-basis $b_i$ and dual basis $b_i^{\vee }$, an element ${\sum }_i{b}_i\otimes c_i$ maps to the functional $b_i^{\vee }\mapsto c_i$. Conversely a $B$-linear functional $\varphi :W\to B$ maps to ${\sum }_i{b}_i\otimes \varphi (b_i^{\vee })$; its $B$-linearity is exactly the relation placing this tensor in $\mathrm{Ann}(I)$. Under this pairing, multiplication on $\mathrm{Ann}(I)$ agrees with evaluation at $\tau$. Indeed, if $b_i{b}_j={\sum }_r{a}_{ij}^r{b}_r$ and $\xi ={\sum }_i{b}_i\otimes c_i\in \mathrm{Ann}(I)$, then the coefficient of $b_i$ in $(b_j\otimes 1)\xi =(1\otimes b_j)\xi$ gives $b_j{c}_i={\sum }_r{a}_{jr}^i{c}_r$. Summing over $i=j$ shows ${\sum }_i{b}_i{c}_i={\sum }_i{\mathrm{Tr}}_{B\mathrm{/}A}(b_i)c_i,$ which is precisely $\mu (\xi )={\varphi }_{\xi }(\tau )$ (Stacks, Lemma 49.6.7, tag 0BVT). Hence the Noether different is the image of evaluation at $\tau$. By the socle argument in [F7], $W=B\lambda$ for a generator $\lambda$ and $\tau =h\lambda$. The image of ${\mathrm{Hom}}_B(W,B)\to B$, $\varphi \mapsto \varphi (\tau )$, is then $(h)$. Stacks, Lemma 49.9.3 (tag 0BW6) identifies this image with the different because $W$ is invertible; Lemma 49.12.3 (tag 0BWG) identifies the different for this quasi-finite syntomic map with the Kähler different; and Lemma 49.7.4 (tag 0BVZ) computes that ideal from the Jacobian presentation. Consequently ${\mathrm{Fitt}}_0({\mathrm{\Omega }}_{B\mathrm{/}A})=(\det J)=(h)\mathrm{.}$ This proves the Jacobian/Koszul/trace-different bridge under the stated finite-flat-lci hypotheses; it uses neither a monogenic extension nor a residue-field perfectness assumption. (Fitting ideal sheaves, Jacobian presentation of Ω, Existence and generators of Kähler differentials)

The dual module $W={\mathrm{Hom}}_A(B,A)$ is free of rank one over $B$, but its generator is not generally the trace functional. Since $B$ is finite free over $A$, reduction gives $W\mathrm{/}{t}_{q}W\cong {\mathrm{Hom}}_K(C_0,K)$. The algebra $C_0=B\mathrm{/}(s^{e_p})$ has socle ${\mathrm{Ann}}_{C_0}(s)=(s^{e_p-1})\mathrm{/}(s^{e_p})$, one-dimensional over its residue field $L$; hence it is Artinian Gorenstein. For completeness, choose a $K$-linear functional $\varphi :C_0\to K$ whose restriction to the socle is nonzero. The multiplication map $C_0\to {\mathrm{Hom}}_K(C_0,K)$, $c\mapsto (d\mapsto \varphi (cd))$, is injective: if $c\mathrm{\ne }0$, the nonzero ideal $(c)$ meets the socle, and since the socle is one-dimensional over $L$, multiplying a nonzero element of that intersection by a suitable lift of an element of $L$ makes its $\varphi$-value nonzero. It is therefore an isomorphism by equality of finite $K$-dimensions. Lift its generator to $\lambda \in W$. The map $B\to W$, $b\mapsto b\lambda$, is an isomorphism modulo $t_q$; Nakayama makes it surjective, and both sides are free $A$-modules of the same finite rank, so it is an isomorphism. Write the trace functional as $\tau =h\lambda$ for the resulting generator $\lambda$ and some $h\in B$. By [F6], $(h)={\mathrm{Fitt}}_0({\mathrm{\Omega }}_{B\mathrm{/}A})$, so $l_p={\mathrm{length}}_B(B\mathrm{/}(h))={\mathrm{ord}}_B(h)$. (Assuming the Axiom of Choice, Nakayama's lemma, Length and valuation in a DVR, Composition series and length of a module)

The trace functional on the fibre $C_0$ is ${\tau }_0(c)={\mathrm{Tr}}_{C_0\mathrm{/}K}(c)=e_p{\mathrm{Tr}}_{L\mathrm{/}K}(\bar{c})$. Indeed the filtration by $(s{)}^j$ has $e_p$ quotients isomorphic to $L$, and multiplication by $c$ acts on each quotient as multiplication by $\bar{c}$; summing their traces gives the formula. The field trace map is nonzero exactly for a separable finite field extension, and therefore ${\tau }_0\mathrm{\ne }0$ exactly when $L\mathrm{/}K$ is separable and $e_p$ is invertible in $K$. This is a statement about the trace functional; the trace pairing on $C_0$ may be degenerate when $e_p>1$. Trace commutes with this finite-free base change, so ${\tau }_0$ is the reduction of $\tau$. (Stacks Project, Lemma 49.4.8 (tag 0C13); The degree $[K:F]={\dim }_{F}K$ of a finite field extension)

For a nonzero $h\in B$ with ${\mathrm{ord}}_B(h)=d$, ${\mathrm{length}}_B(B\mathrm{/}(h))=d$. If the reduction of $h$ modulo $s^{e_p}$ is zero, then $h\in (s^{e_p})$ and $d\ge e_p$; if that reduction is a nonzero element of the socle $(s^{e_p-1})\mathrm{/}(s^{e_p})$, then $d=e_p-1$. (Length and valuation in a DVR, Every nonzero fraction is a unit times a power of a uniformiser, Composition series and length of a module)

Perfect residue fields: if $k$ is perfect then every finite extension of $k$ is separable, and both $\kappa (p)$ and $\kappa (q)$ are finite extensions of $k$; hence $L\mathrm{/}K$ is separable for every point $p$. (Every algebraic extension of a perfect field is separable, Ramification index of a morphism of curves)

The sheaf ${\mathrm{\Omega }}_{C\mathrm{/}D}$ is coherent under the stated Axiom of Choice. On affine charts $V=\mathrm{Spec}(A)$ of $D$, the finite morphism has $f^{-1}(V)=\mathrm{Spec}(B)$ with $B$ finite over $A$; since $D$ is a finite-type curve over the Noetherian field $k$, $A$ is Noetherian, so $B$ is finite type and finitely presented over $A$. A finite polynomial presentation of $B$ makes ${\mathrm{\Omega }}_{B\mathrm{/}A}$ a cokernel between finite free $B$-modules by the Jacobian presentation. Affine compatibility identifies ${\mathrm{\Omega }}_{C\mathrm{/}D}$ locally with the associated sheaf of these finite modules, so it is quasi-coherent of finite type. The same finite-type-over-$k$ argument makes $C$ locally Noetherian, and hence ${\mathrm{\Omega }}_{C\mathrm{/}D}$ is coherent. (Curves over a field, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Finite morphisms of schemes, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Every algebra of finite type over a Noetherian ring is finitely presented, Jacobian presentation of Ω, Affine charts recover the algebraic module of differentials, Quasi-coherent module on a scheme, Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme, The Axiom of Choice)