Delta invariant of a curve singularity

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field and let $X$ be an integral proper finite-type curve over $k$ (Curves over a field). Let $\nu :X^{\mathrm{n}\mathrm{u}}\to X$ be its normalization (Normalization of an integral finite-type curve by gluing affine integral closures), and let $x\in X$ be a closed point. Write ${\mathcal{O}}_{X,x}$ for the local ring at $x$ and $({\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}{)}_x$ for the stalk of the direct image of the normalization's structure sheaf. The delta invariant of $X$ at $x$ is ${\delta }_x(X):={\dim }_k(({\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}{)}_x\mathrm{/}{\mathcal{O}}_{X,x}),$ the $k$-dimension (Vector space over a field, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) of this quotient ${\mathcal{O}}_{X,x}$-module. The total delta invariant is $\delta (X):={\sum }_{x\in X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}}{\delta }_x(X),$ where $X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ is the singular (non-regular) locus (Regular and singular loci), and the sum is over its closed points. The Axiom of Choice is inherited from the cited normalization, coherence, one-dimensional regular-local, regular-locus and curve-topology interfaces; $k$ may be any algebraically closed field in any characteristic.

Each ${\delta }_x(X)$ is a finite nonnegative integer, and ${\delta }_x(X)=0$ if and only if $x$ is regular. The singular locus $X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ is a finite set of closed points, so the total invariant $\delta (X)$ is a finite sum.

The normalization theorem supplies, on each affine open $U=\mathrm{Spec}A$ of $X$, a chart ${\nu }^{-1}(U)=\mathrm{Spec}B$ in which $A\subseteq B\subseteq k(X)$ and $B$ is the integral closure of $A$ in $k(X)$ (Integral closure in an extension ring and integrally closed domains); $B$ is a finite $A$-module. The finite morphism $\nu$ is affine (Finite is affine and local on its target). On this chart the direct image has sections $B_f$ on every principal open $D(f)\subseteq U$, with localization as restriction (Affine pushforward algebra localizes). Thus $({\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}){\mathrm{\mid }}_U\cong \tilde{B}$ is quasi-coherent (Quasi-coherent module on a scheme) and of finite type as an ${\mathcal{O}}_X$-module (Finite type and finitely presented module sheaves).

The scheme $X$ is locally Noetherian: its affine coordinate rings are finite- type algebras over the Noetherian field $k$ (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, Locally Noetherian and Noetherian schemes), and their localizations are Noetherian (Every quotient and every localisation of a Noetherian ring is Noetherian). The structure sheaf and ${\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}$ are therefore coherent by the quasi-coherent finite-type criterion, and their cokernel $\mathcal{Q}:=\mathrm{coker}({\mathcal{O}}_X⟶{\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})$ is coherent as well (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves). The map is injective: on each such chart it is the inclusion $A\hookrightarrow B$ inside the common function field. Consequently, if $x$ corresponds to the maximal ideal $\mathfrak{m}$ of $A$, then ${\mathcal{Q}}_x\cong (B\mathrm{/}A{)}_{\mathfrak{m}}\cong B_{\mathfrak{m}}\mathrm{/}{A}_{\mathfrak{m}},$ using the stalk-localization identification for associated sheaves and the exactness of module localization (The stalk of an associated sheaf is the localisation, The stalk of the affine structure sheaf at a prime is A_p, Localisation of modules is exact).

This stalk description retains every branch over $x$. Indeed, with $S=A\setminus \mathfrak{m}$, the algebra $B_{\mathfrak{m}}=S^{-1}B$ is canonically $B{\otimes }_A{A}_{\mathfrak{m}}$ by localization-as-tensor (Localisation of modules is extension of scalars). Integral closure commutes with localization for this multiplicative set, so $B_{\mathfrak{m}}$ is the integral closure of $A_{\mathfrak{m}}$ in $k(X)$; it is finite over $A_{\mathfrak{m}}$ (Integrality and integral closure commute with localisation). The residue field $\kappa (x)$ is $k$: since $x$ is closed, it is a field finitely generated as a $k$-algebra. Zariski's lemma makes $\kappa (x)\mathrm{/}k$ finite, and algebraic closedness makes it trivial (A field finitely generated as a k-algebra is a finite extension of k). Thus $C:=B_{\mathfrak{m}}\mathrm{/}\mathfrak{m}{B}_{\mathfrak{m}}$ is a finite-dimensional $k$-algebra. The algebra $B_{\mathfrak{m}}$ is integral over the local ring $A_{\mathfrak{m}}$, so each maximal ideal contracts to $\mathfrak{m}$ (Under an integral extension, a prime is maximal if and only if its contraction is maximal); these ideals correspond to the maximal ideals of $C$ (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal). There are only finitely many: for any finite list of $r$ distinct maximal ideals of $C$, the Chinese remainder map onto the product of their nonzero residue fields is surjective, so $r\le {\dim }_{k}C$ (Chinese remainder theorem for pairwise comaximal ideals). Thus $B_{\mathfrak{m}}$ has finitely many maximal ideals. It is a nonzero domain, so AC and the proper-ideal/maximal-ideal theorem give it at least one maximal ideal; it is therefore semilocal and can have several branches over $x$ without selecting one (In a nonzero commutative ring, every proper ideal is contained in a maximal ideal).

The generic stalk of $\mathcal{Q}$ is zero: localizing $A\subseteq B\subseteq k(X)$ at the generic point gives $k(X)$ on both sides. At a closed point the local ring $A_{\mathfrak{m}}={\mathcal{O}}_{X,x}$ is a one-dimensional Noetherian local domain. The curve has chain dimension one (Chain dimension and the empty-space convention). The affine open $U$ contains both the generic point and $x$, so in $A$ the generic prime $(0)$ is strictly contained in the maximal ideal $\mathfrak{m}$; no longer prime chain is possible in $A$ because $U$ is an open subspace of this one-dimensional integral curve. Thus the only primes of $A_{\mathfrak{m}}$ are $(0)$ and $\mathfrak{m}{A}_{\mathfrak{m}}$, and the support of ${\mathcal{Q}}_x$ is contained in the maximal ideal: its localization at $(0)$ is zero. If $x$ is regular, the one-dimensional regular-local/DVR theorem makes $A_{\mathfrak{m}}$ a DVR, and the DVR characterization makes it integrally closed (one dimensional regular local rings are dvrs, Equivalent characterizations of a DVR). Localization of integral closure then gives $B_{\mathfrak{m}}=A_{\mathfrak{m}}$, so ${\mathcal{Q}}_x=0$.

For every closed $x$, the module $M:={\mathcal{Q}}_x$ is finite over the Noetherian local ring $R:=A_{\mathfrak{m}}$. Its support is contained in $\{\mathfrak{m}R\}$. The support-annihilator theorem and the radical-as-prime- intersection theorem imply $\sqrt{{\mathrm{Ann}}_R(M)}\supseteq \mathfrak{m}R$; if $M=0$ this is immediate, and otherwise $\mathfrak{m}R$ belongs to the support since $M_{\mathfrak{m}R}=M$, so it is the only prime containing the annihilator (For a finite module, support is the set of primes containing the annihilator, The radical of an ideal is the intersection of the prime ideals containing it, Annihilators, torsion elements and the torsion subset of a module). Choose finite generators $u_1,\dots ,u_r$ of $\mathfrak{m}R$: $R$ is Noetherian, so its maximal ideal is a submodule of its Noetherian regular module and is finitely generated (Left and right Noetherian rings, Noetherian modules: every submodule is finitely generated). For each $i$, some $e_i\ge 1$ has $u_i^{e_i}M=0$. Therefore, with $N=1+{\sum }_i(e_i-1)$, every degree-$N$ monomial in these generators contains some $u_i^{e_i}$, so $(\mathfrak{m}R{)}^{N}M=0$. The resulting finite filtration $M\supseteq \mathfrak{m}RM\supseteq \cdots \supseteq (\mathfrak{m}R{)}^{N}M=0$ has finite-dimensional $k$-vector-space quotients: if generators of $M$ and $\mathfrak{m}R$ are fixed, the finitely many products of $j$ maximal-ideal generators with generators of $M$ generate $(\mathfrak{m}R{)}^{j}M$, so each layer is finitely generated; it is killed by $\mathfrak{m}R$ and its residue field is $k$. Thus each ${\delta }_x(X)$ is a finite nonnegative integer.

Finally, ${\delta }_x(X)=0$ exactly when ${\mathcal{O}}_{X,x}$ equals its integral closure in $k(X)$, which is exactly when this one-dimensional Noetherian local domain is integrally closed. By the DVR characterization this is equivalent to being a DVR, and by the one-dimensional regular-local/DVR theorem this is equivalent to regularity. Hence ${\delta }_x(X)=0$ if and only if $x$ is regular. Because $k$ is algebraically closed it is perfect (every irreducible polynomial over $k$ is linear; Perfect fields: every irreducible polynomial is separable), the regular-locus-open theorem makes $X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ closed (Openness of the regular locus over a perfect field). The generic point has local ring $k(X)$, a field and hence regular, so $X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ is proper. Every proper closed subset of a finite-type integral curve is finite and consists of closed points (Proper closed subsets of a curve are finite). Therefore the sum defining $\delta (X)$ is finite and is supported precisely on the singular closed points.