The local ring of an analytic set; regular points, singular locus, Remmert–Stein

Intuition Beginner

An analytic set is the common zero locus of some holomorphic functions — a curve, a surface, or a more tangled shape sitting inside complex space. Near most of its points it looks clean: a smooth sheet, flat like a piece of paper when you zoom in far enough. Those are the regular (or smooth) points. At a regular point the set near you is indistinguishable from ordinary flat space of some fixed number of dimensions.

But not every point is so well behaved. A curve can cross itself, forming an X, or it can come to a sharp cusp like the tip of a teardrop. No matter how far you zoom in at such a point, the X stays an X and the cusp stays a cusp. These are the singular points: the corners, the crossings, the pinches. They never flatten out.

The aim of this unit is to detect singular points by algebra rather than by eye, and then to show that singularities, being rare, cannot block a set from being patched across a thin gap.

Visual Beginner

Picture three small pieces of an analytic curve. The first is a gently curving arc: zoom in and it becomes a straight line, so its centre point is smooth. The second is two arcs meeting in an X — the node. The third is a cusp, where the curve sweeps in, pinches to a point, and sweeps back out.

At the node, two directions of travel pass through one point, so the curve there cannot be flattened to a single line. At the cusp, the curve has a built-in spike that no amount of zooming removes. The smooth arc has neither defect. The shaded dot marks the bad point in each of the last two pictures; the first picture has no shaded dot because every point on it is smooth.

Worked example Beginner

Take the cusp curve given by $z_2\times z_2=z_1\times z_1\times z_1$ in two complex variables — the set of points where $z_2$ squared equals $z_1$ cubed. You can trace it with one travelling parameter $t$: put $z_1=t\times t$ and $z_2=t\times t\times t$. Check it: $z_2$ squared is $t^6$, and $z_1$ cubed is also $t^6$. They match, so every such point is on the curve.

Now watch the origin, where $t=0$. As $t$ passes through $0$, the point slows down, pinches to the origin, and speeds back out. That pinch is the cusp. Near every other point the curve is a smooth arc, but at the origin it has a genuine spike.

Compare the node, the set where $z_1\times z_2=0$. This is the two axes together: the $z_1$-axis and the $z_2$-axis, crossing at the origin. Away from the origin you sit on one axis only — smooth. At the origin both axes meet, so it is a crossing, a singular point.

What this tells us: the same set can be smooth almost everywhere yet carry a single bad point, and we want a test that flags exactly the origin in both pictures.

Check your understanding Beginner

Formal definition Intermediate+

Let $V$ be the germ at $p$ of an analytic set in an open set $\Omega \subseteq {\mathbb{C}}^n$; without loss of generality $p=0$. Write ${}_n\mathcal{O}=\mathbb{C}\{z_1,\dots ,z_n\}$ for the local ring of holomorphic germs at $0$, with maximal ideal ${\mathfrak{m}}_n$, and let $\mathcal{I}(V)\subseteq {}_n\mathcal{O}$ be the ideal of germs vanishing on $V$ 06.10.17. By the analytic Nullstellensatz of that unit, $\mathcal{I}(V)$ is a radical ideal.

The local ring of $V$ at $p$ is the quotient $$ \mathcal{O}{V,p} ;=; {}n\mathcal{O}\big/\mathcal{I}(V). $$ It is a local ring with maximal ideal $\mathfrak{m}{V,p} = \mathfrak{m}n/\mathcal{I}(V)$andresiduefield$\mathcal{O}{V,p}/\mathfrak{m}{V,p} = \mathbb{C}$.Itselementsaregermsofholomorphicfunctionson$V$at$p$;twoambientgermsthatagreeon$V$definethesameelement.TheKrulldimensionof$\mathcal{O}_{V,p}$equalsthelocaldimension$\dim_p V$ defined in 06.10.18.

The Zariski cotangent space of $V$ at $p$ is the finite-dimensional complex vector space ${\mathfrak{m}}_{V,p}/{\mathfrak{m}}_{V,p}^2$, and its dimension is the embedding dimension $$ \operatorname{embdim}p V ;=; \dim{\mathbb{C}} \mathfrak{m}{V,p}/\mathfrak{m}{V,p}^2 . $$ The embedding dimension is the least number of ambient coordinates into which the germ can be embedded: it equals $n$ minus the rank at $p$ of the Jacobian of any generating set of $\mathcal{I}(V)$, and one always has ${\mathrm{embdim}}_{p}V\ge {\dim }_{p}V$.

A point $p\in V$ is regular (or smooth) of dimension $d$ when $V$ is, near $p$, a $d$-dimensional complex submanifold of $\Omega$ — equivalently, when there are local holomorphic coordinates in which $V$ is the linear subspace $\{z_{d+1}=\cdots =z_n=0\}$. Otherwise $p$ is singular. The singular locus $\mathrm{Sing}V$ is the set of singular points; its complement $\mathrm{Reg}V=V\setminus \mathrm{Sing}V$ is the regular locus. By 06.10.18, $\mathrm{Sing}V$ is itself an analytic subset of $V$, of dimension strictly smaller than ${\dim }_{p}V$ at each point, so the regular locus is open and dense.

Counterexamples to common slips

• Embedding dimension is not the ambient dimension $n$: for the smooth line $\{z_2=0\}$ in ${\mathbb{C}}^2$ the embedding dimension is $1$, not $2$. It measures the minimal embedding of the germ, read off from $\mathfrak{m}/{\mathfrak{m}}^2$.
• "Regular" is a property of the germ, not of the chosen equations. The set $\{z_2^2=0\}$ has the same reduced germ as the smooth line $\{z_2=0\}$, because $\mathcal{I}$ is the radical $(z_2)$; the doubled equation does not create a singularity.
• A point can fail to be regular even when $V$ is topologically a manifold there: the cusp is homeomorphic to a line near $0$, yet $0$ is singular because no holomorphic coordinate change flattens it. Regularity is analytic, not merely topological.

Key theorem with proof Intermediate+

Theorem (Jacobian criterion for regularity). Let $V$ be the germ at $p=0$ of an analytic set of pure dimension $d$ in ${\mathbb{C}}^n$, with $\mathcal{I}(V)=(f_1,\dots ,f_r)$. Then the following are equivalent.

1. $p$ is a regular point of $V$ of dimension $d$.
2. ${\mathrm{embdim}}_{p}V={\dim }_{\mathbb{C}}{\mathfrak{m}}_{V,p}/{\mathfrak{m}}_{V,p}^2=d$.
3. The local ring ${\mathcal{O}}_{V,p}$ is isomorphic, as a $\mathbb{C}$-algebra, to the convergent power-series ring ${}_d\mathcal{O}=\mathbb{C}\{w_1,\dots ,w_d\}$; equivalently ${\mathcal{O}}_{V,p}$ is a regular local ring of dimension $d$.

Otherwise ${\mathrm{embdim}}_{p}V>d$ and $p$ is singular. ^{[Gunning–Rossi]}

Proof. $(1)\Rightarrow (3)$. If $p$ is regular of dimension $d$, there are local coordinates in which $V=\{z_{d+1}=\cdots =z_n=0\}$. Restriction of an ambient germ to $V$ then forgets the variables $z_{d+1},\dots ,z_n$ and retains a convergent power series in $z_1,\dots ,z_d$; the kernel of restriction is exactly $(z_{d+1},\dots ,z_n)=\mathcal{I}(V)$. Hence ${\mathcal{O}}_{V,p}={}_n\mathcal{O}/(z_{d+1},\dots ,z_n)\cong {}_d\mathcal{O}$, a regular local ring of dimension $d$.

$(3)\Rightarrow (2)$. For the power-series ring ${}_d\mathcal{O}$ the maximal ideal is $(w_1,\dots ,w_d)$ and the quotient $\mathfrak{m}/{\mathfrak{m}}^2$ has the residues of $w_1,\dots ,w_d$ as a basis, so its dimension is $d$. A $\mathbb{C}$-algebra isomorphism carries maximal ideal to maximal ideal and so identifies the two cotangent spaces; thus ${\mathrm{embdim}}_{p}V=d$. Since ${\dim }_{p}V=d$ is the Krull dimension, the embedding dimension meets its lower bound.

$(2)\Rightarrow (1)$. Suppose ${\dim }_{\mathbb{C}}{\mathfrak{m}}_{V,p}/{\mathfrak{m}}_{V,p}^2=d$. The differentials $d{f}_1(0),\dots ,d{f}_r(0)$ span the conormal space, the annihilator of $T_{0}V$ inside $({\mathbb{C}}^n{)}^{\ast }$; the cotangent space ${\mathfrak{m}}_{V,p}/{\mathfrak{m}}_{V,p}^2$ is the quotient of ${\mathfrak{m}}_n/{\mathfrak{m}}_n^2\cong ({\mathbb{C}}^n{)}^{\ast }$ by that span. So $$ \dim_{\mathbb{C}} \mathfrak{m}{V,p}/\mathfrak{m}{V,p}^2 ;=; n - \operatorname{rank}\big(\partial f_i/\partial z_j(0)\big). $$ The hypothesis forces the Jacobian rank to equal $n-d$. After reordering, $f_1,\dots ,f_{n-d}$ have independent differentials at $0$, and the holomorphic implicit function theorem makes $\{f_1=\cdots =f_{n-d}=0\}$ a $d$-dimensional submanifold $M$ through $0$ with $V\subseteq M$. Both $V$ and $M$ are pure of dimension $d$ and $V\subseteq M$ with $M$ irreducible as a germ, so $V=M$ near $0$; hence $p$ is regular of dimension $d$. The final clause follows because the general inequality ${\mathrm{embdim}}_{p}V\ge {\dim }_{p}V$ is strict precisely when $(2)$ fails. $\square$

Bridge. This criterion builds toward the global stratification of an analytic set into its regular and singular parts, and the embedding-dimension test appears again in the normalisation theory forward-linked at 06.10.22, where the failure of ${\mathcal{O}}_{V,p}$ to be regular is repaired by passing to its integral closure. The foundational reason a singularity is invisible to dimension alone is exactly this: dimension is the Krull dimension of ${\mathcal{O}}_{V,p}$, while smoothness is the regularity of that local ring, and the two coincide only when the cotangent space $\mathfrak{m}/{\mathfrak{m}}^2$ has no excess. This is exactly the local-algebra shadow of the Jacobian rank drop, and the criterion generalises the one-variable fact that a plane curve is smooth where its gradient is nonzero. The central insight is that the analytic-geometric question "is $V$ a manifold here?" is dual to the purely algebraic question "is ${\mathcal{O}}_{V,p}$ a power-series ring?", and the bridge is the cotangent space, which both the geometry (tangent directions) and the algebra ($\mathfrak{m}/{\mathfrak{m}}^2$) compute. Putting these together, the singular locus is the locus where this duality breaks, and locating it is the first step toward resolving it.

Exercises Intermediate+

Advanced results Master

The structure of ${\mathcal{O}}_{V,p}$ organises the local analytic geometry of $V$, and the regular/singular split is its first geometric reading. The results below sharpen that split and prove the removable-singularity theorem that controls how analytic sets close up across thin gaps.

The singular locus is analytic of lower dimension. For a reduced analytic-set germ $V$ of pure dimension $d$, the singular locus $\mathrm{Sing}V$ is an analytic subset with ${\dim }_p\mathrm{Sing}V<d$ for every $p$. For a hypersurface this is the gradient locus $\{f=0,df=0\}$ of Exercise 5; in general one uses that the rank of the Jacobian of generators is lower-semicontinuous, so the locus where it drops below $n-d$ is analytic, and the local parametrisation theorem of 06.10.18 bounds its dimension. Consequently $\mathrm{Reg}V$ is a connected (on each component) complex manifold of dimension $d$, dense in $V$, and $V$ is the closure of its regular locus.

The local ring detects the singularity type. The isomorphism class of ${\mathcal{O}}_{V,p}$ as an analytic local algebra is a complete invariant of the germ $(V,p)$ up to biholomorphism — this is the analytic analogue of the fact that an affine variety is determined by its coordinate ring. The cusp and the node have non-isomorphic local rings: ${\mathcal{O}}_{\text{cusp},0}=\mathbb{C}\{t^2,t^3\}$ is a domain (the cusp is irreducible), whereas ${\mathcal{O}}_{\text{node},0}=\mathbb{C}\{z_1,z_2\}/(z_1{z}_2)$ has zero divisors ${\bar{z}}_1,{\bar{z}}_2$ (the node has two branches), so it is not even an integral domain. Both have embedding dimension $2$ and dimension $1$, yet they are distinguished by whether the local ring is reduced-and-irreducible or merely reduced.

The Remmert–Stein extension theorem. Let $\Omega \subseteq {\mathbb{C}}^n$ be open, $A\subset \Omega$ an analytic subset with $\dim A<d$, and $V\subseteq \Omega \setminus A$ a purely $d$-dimensional analytic subset of $\Omega \setminus A$. Then the closure $\bar{V}$ in $\Omega$ is an analytic subset of $\Omega$, purely of dimension $d$ ^{[Remmert–Stein]}.

The dimension hypothesis $\dim A<d$ is sharp: the punctured-disc image of $z\mapsto (z,e^{1/z})$ has an essential singularity at $0$ whose closure is not analytic, but there the relevant codimension count fails. The force of the theorem is that a lower-dimensional set is too thin to be the support of new limiting structure: the volume of $V$ near $A$ is finite, and a finite-volume positive analytic chain has no choice but to extend.

Normalisation as the sequel. When ${\mathcal{O}}_{V,p}$ is not normal — not integrally closed in its total ring of fractions — the singularity is, at least partly, a failure of the ring to absorb the algebraic functions bounded near $p$. The cusp is the model: $\mathbb{C}\{t^2,t^3\}$ is not integrally closed because $t=t^3/t^2$ is integral over it (a root of $X^2-t^2$) yet not in it. The normalisation $\tilde{V}\to V$ replaces ${\mathcal{O}}_{V,p}$ by its integral closure, resolving such non-normal singular points; for a curve it separates branches and unpinches cusps, producing a manifold. This forward-links to where complex spaces and their normalisations appear at 06.10.22.

Synthesis. The local ring ${\mathcal{O}}_{V,p}$ is the foundational reason the geometry of an analytic set is governed by commutative algebra one point at a time. The regular/singular split is dual to the regular/non-regular split of local rings: $p$ is smooth exactly when ${\mathcal{O}}_{V,p}$ is a power-series ring, and this is exactly the equality of embedding dimension with dimension that the Jacobian criterion computes. Putting these together, the singular locus is the analytic subset where the cotangent space $\mathfrak{m}/{\mathfrak{m}}^2$ carries excess directions, and the central insight is that this excess is what normalisation removes — the failure of ${\mathcal{O}}_{V,p}$ to be integrally closed is the algebraic name for a non-normal singularity, and the integral closure is the algebraic name for resolving it. The Remmert–Stein theorem then generalises the one-variable removable-singularity theorem from functions to whole analytic sets: a lower-dimensional gap is dual to a removable point, and the bridge realising both is a finite-volume bound that forbids essential limiting behaviour. This pattern recurs whenever a local analytic object is reconstructed from its behaviour off a thin set, and it builds toward the theory of coherent ideal sheaves and complex spaces.

Full proof set Master

Proposition 1 (embedding dimension via the Jacobian). Let $V$ be a germ at $0$ in ${\mathbb{C}}^n$ with $\mathcal{I}(V)=(f_1,\dots ,f_r)$. Then ${\dim }_{\mathbb{C}}{\mathfrak{m}}_{V,0}/{\mathfrak{m}}_{V,0}^2=n-\mathrm{rank}(\partial f_i/\partial z_j(0))$.

Proof. The cotangent space ${\mathfrak{m}}_n/{\mathfrak{m}}_n^2$ of the ambient ring is $n$-dimensional with basis the residues of $z_1,\dots ,z_n$; the linear map sending a germ $g\in {\mathfrak{m}}_n$ to its residue is the first-order Taylor part $g\mapsto dg(0)$, an isomorphism ${\mathfrak{m}}_n/{\mathfrak{m}}_n^2\cong ({\mathbb{C}}^n{)}^{\ast }$. Reduction modulo $\mathcal{I}(V)$ gives a surjection ${\mathfrak{m}}_n/{\mathfrak{m}}_n^2\twoheadrightarrow {\mathfrak{m}}_{V,0}/{\mathfrak{m}}_{V,0}^2$ whose kernel is the image of $\mathcal{I}(V)\cap {\mathfrak{m}}_n$ in ${\mathfrak{m}}_n/{\mathfrak{m}}_n^2$, that is, the span of the differentials $d{f}_i(0)$ of the generators (higher-order parts of elements of $\mathcal{I}(V)$ die in ${\mathfrak{m}}_n^2$, and the linear parts of a generating set span the linear parts of the whole ideal). The dimension of that span is the rank of the Jacobian matrix $(\partial f_i/\partial z_j(0))$. Subtracting from $n$ gives the claim. $\square$

Proposition 2 (Jacobian criterion). With $V$ pure of dimension $d$ at $0$, the point $0$ is regular of dimension $d$ if and only if $\mathrm{rank}(\partial f_i/\partial z_j(0))=n-d$, equivalently ${\mathrm{embdim}}_{0}V=d$.

Proof. If $0$ is regular, choose coordinates with $V=\{z_{d+1}=\cdots =z_n=0\}$; then $\mathcal{I}(V)=(z_{d+1},\dots ,z_n)$, whose Jacobian at $0$ has rank $n-d$, and by Proposition 1 the embedding dimension is $d$. Conversely, if the rank is $n-d$, then after reindexing $f_1,\dots ,f_{n-d}$ have linearly independent differentials at $0$. The holomorphic implicit function theorem makes $M=\{f_1=\cdots =f_{n-d}=0\}$ a complex submanifold of dimension $d$ through $0$, and $V\subseteq M$ since these $f_i\in \mathcal{I}(V)$. The germ $M$ is irreducible (a manifold germ is), of the same dimension $d$ as the pure-dimensional $V$, and a pure-dimensional analytic germ contained in an irreducible germ of equal dimension equals it; so $V=M$ near $0$ and $0$ is regular. The embedding-dimension reformulation is Proposition 1. $\square$

Proposition 3 ($\mathrm{Sing}V$ is analytic, of lower dimension). For a reduced pure-$d$-dimensional germ $V$, the set $\mathrm{Sing}V$ is analytic and ${\dim }_0\mathrm{Sing}V<d$.

Proof. By Proposition 2, a point $q$ near $0$ is singular if and only if the Jacobian $(\partial f_i/\partial z_j(q))$ has rank $<n-d$. The locus where a holomorphic matrix has rank $<n-d$ is cut out by the vanishing of all its $(n-d)\times (n-d)$ minors, holomorphic functions of $q$; intersected with $V$ this exhibits $\mathrm{Sing}V$ as analytic. For the dimension bound, invoke the local parametrisation theorem on each irreducible component: such a component is a branched cover of a $d$-dimensional polydisc, biholomorphic — hence regular — over the complement of a branch set of dimension $<d$, so the singular points of the component sit over that branch set and have dimension $<d$. Points where two components meet form an analytic set of dimension $<d$ as well. Taking the union over finitely many components and intersection loci keeps the dimension below $d$. $\square$

Proposition 4 (Remmert–Stein extension theorem). Let $A\subset \Omega \subseteq {\mathbb{C}}^n$ be analytic with $\dim A<d$, and let $V\subseteq \Omega \setminus A$ be purely $d$-dimensional analytic. Then $\bar{V}$ is analytic in $\Omega$, purely of dimension $d$.

Proof (key estimate). The statement is local at a point $a\in A$; away from $A$ there is nothing to prove since $V$ is already analytic there. Choose coordinates so that the projection $\pi :{\mathbb{C}}^n\to {\mathbb{C}}^d$ onto the first $d$ coordinates is, near $a$, proper and finite on $V$ — possible because $\dim V=d$ and $\dim A<d$, so a generic $d$-plane meets $A$ in dimension $\le \dim A-(n-d)<0$, i.e. not at all near $a$, making the fibres of $\pi {|}_{\bar{V}}$ discrete. The central quantitative input is that $V$ has locally finite $2d$-dimensional Hausdorff measure near $a$: a pure $d$-dimensional analytic set is a positive analytic chain, and Wirtinger's inequality bounds its volume on a compact set by the integral of the $d$-th power of the Kähler form, which is finite. Finite volume forbids $V$ from oscillating wildly toward $A$.

Now consider the elementary symmetric functions of the finitely many sheets of $V$ over $\pi$: for each fixed $z^{\prime }\in {\mathbb{C}}^d$ off the (lower-dimensional) image of $A$, the fibre ${\pi }^{-1}(z^{\prime })\cap V$ consists of $k$ points (counted with multiplicity, $k$ constant generically), and the power sums $$ p_m(z') = \sum_{j=1}^{k} \zeta_j(z')^m, \qquad \zeta_j(z') \text{ the fibre coordinates}, $$ are holomorphic on $\pi (\Omega )\setminus \pi (A)$ and bounded near $\pi (A)$ by the finite-volume estimate. By the Riemann removable-singularity theorem in the base — applied across the lower-dimensional analytic set $\pi (A)$, which is removable for bounded holomorphic functions — each $p_m$, hence each elementary symmetric function ${\sigma }_m(z^{\prime })$, extends holomorphically across $\pi (A)$. The Weierstrass polynomial $P(z^{\prime },w)={\prod }_j(w-{\zeta }_j(z^{\prime }))=w^k-{\sigma }_1{w}^{k-1}+\cdots$ then has holomorphic coefficients on all of the base neighbourhood, and $\{P=0\}$ is an analytic set whose intersection with $\Omega \setminus A$ is $V$. Its closure is therefore $\bar{V}$, exhibiting $\bar{V}$ as analytic. Purity of dimension is inherited from $V$ since the extension adds only the limit points over $\pi (A)$, which carry dimension $\le \dim A<d$ and so cannot form a separate component of dimension $\ne d$. $\square$

Proposition 5 (non-normality of the cusp). The local ring ${\mathcal{O}}_{V,0}=\mathbb{C}\{t^2,t^3\}$ of the cusp is a one-dimensional analytic domain that is not integrally closed; its integral closure is $\mathbb{C}\{t\}$.

Proof. The map $z_1\mapsto t^2,z_2\mapsto t^3$ identifies ${\mathcal{O}}_{V,0}=\mathbb{C}\{z_1,z_2\}/(z_2^2-z_1^3)$ with the subring $R=\mathbb{C}\{t^2,t^3\}\subset \mathbb{C}\{t\}$; this is injective because $z_2^2-z_1^3$ generates the full kernel (it is irreducible and the parametrisation is the normalisation), so $R$ is a domain of Krull dimension $1$. The element $t=t^3/t^2$ lies in the fraction field and satisfies $X^2-t^2=0$ with $t^2\in R$, so $t$ is integral over $R$ but $t\notin R$ (it has a nonzero $t^1$-coefficient). Hence $R$ is not integrally closed. Its integral closure is $\mathbb{C}\{t\}$: that ring is finite over $R$ (generated by $1$ and $t$), integrally closed (a regular, hence normal, local ring), and contains $R$, so it is the integral closure. $\square$

Connections Master

• The ideal of an analytic set; the local Nullstellensatz 06.10.17. The local ring ${\mathcal{O}}_{V,p}={}_n\mathcal{O}/\mathcal{I}(V)$ is built directly on the radical ideal $\mathcal{I}(V)$ produced there; the Nullstellensatz $\mathcal{I}(\mathrm{loc}I)=\sqrt{I}$ is what guarantees ${\mathcal{O}}_{V,p}$ is reduced, so that the regular/singular analysis of this unit is not corrupted by nilpotents.

• Dimension and the regular/singular decomposition 06.10.18. The local dimension ${\dim }_{p}V$ used throughout is the Krull dimension of ${\mathcal{O}}_{V,p}$ established there, and the fact that $\mathrm{Sing}V$ is a proper analytic subset is the input that makes the regular locus dense. This unit sharpens that decomposition by giving the embedding-dimension criterion that locates $\mathrm{Sing}V$ pointwise.

• Forward to complex spaces and normalisation 06.10.22. The non-normal singularities detected here by the strict inequality ${\mathrm{embdim}}_{p}V>{\dim }_{p}V$ are exactly the points the normalisation map repairs; the local ring ${\mathcal{O}}_{V,p}$ and its integral closure are the objects on which normalisation acts, and the cusp computation is the model example carried into that unit.

• Forward to the analytic structure built from local rings 06.10.20. The assignment $p\mapsto {\mathcal{O}}_{V,p}$ globalises to the structure sheaf of $V$ as a complex space; the regularity criterion of this unit becomes the local model for what a smooth point of a complex space is, and Remmert–Stein becomes a theorem about extending closed analytic subspaces.

• Weierstrass preparation and division 06.10.14. The branched-cover / local parametrisation theorem underlying both the dimension bound on $\mathrm{Sing}V$ and the Remmert–Stein proof is a consequence of Weierstrass preparation: the projection $\pi :V\to {\mathbb{C}}^d$ is finite because a defining Weierstrass polynomial makes ${\mathcal{O}}_{V,p}$ a finite module over ${}_d\mathcal{O}$, and the symmetric functions of the sheets are the coefficients of that polynomial.

Historical & philosophical context Master

The local theory of analytic sets — the local ring ${\mathcal{O}}_{V,p}$, the regular locus, and the singular set — was systematised in the 1950s as part of the Behnke–Stein and Oka–Cartan reconstruction of several-variable complex analysis on algebraic foundations. Reinhold Remmert and Karl Stein proved their extension theorem in Über die wesentlichen Singularitäten analytischer Mengen ^{[Remmert–Stein 1953]} (Mathematische Annalen 126, 1953, pp. 263–306), establishing that an analytic set defined off a lower-dimensional analytic obstruction closes up to an analytic set across it. The result was the analytic-geometry counterpart of Riemann's removable-singularity theorem for functions, extended from the zero-dimensional removable point of one-variable theory to obstructions of arbitrary codimension at least one in the dimension.

The structural picture — that a reduced analytic set is the closure of its regular locus, a complex manifold, and that the singular set is a proper analytic subset of strictly lower dimension — was given its definitive treatment by Hassler Whitney in Complex Analytic Varieties ^[Whitney] and in the local-theory volumes of Gunning–Rossi ^{[Gunning–Rossi]} and Gunning's Local Theory ^[Gunning]. The role of the local ring as the carrier of singularity data, and the normalisation as the device for resolving non-normal singularities, were developed by Grauert and Remmert and placed within the theory of coherent analytic sheaves ^{[Grauert–Remmert]}. The cusp $\mathbb{C}\{t^2,t^3\}$ became the standard first example of a non-normal local ring, its failure of integral closure the prototype of the singularity that normalisation removes.

Bibliography Master

@article{RemmertStein1953,
  author  = {Remmert, Reinhold and Stein, Karl},
  title   = {{\"U}ber die wesentlichen Singularit{\"a}ten analytischer Mengen},
  journal = {Math. Ann.},
  volume  = {126},
  year    = {1953},
  pages   = {263--306}
}

@book{GunningRossi,
  author    = {Gunning, Robert C. and Rossi, Hugo},
  title     = {Analytic Functions of Several Complex Variables},
  publisher = {Prentice-Hall},
  year      = {1965}
}

@book{GunningLocal,
  author    = {Gunning, Robert C.},
  title     = {Introduction to Holomorphic Functions of Several Variables, Volume II: Local Theory},
  publisher = {Wadsworth \& Brooks/Cole},
  year      = {1990}
}

@book{WhitneyCAV,
  author    = {Whitney, Hassler},
  title     = {Complex Analytic Varieties},
  publisher = {Addison-Wesley},
  year      = {1972}
}

@book{GrauertRemmertCAS,
  author    = {Grauert, Hans and Remmert, Reinhold},
  title     = {Coherent Analytic Sheaves},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {265},
  publisher = {Springer},
  year      = {1984}
}

@book{GrauertRemmertSteinTheory,
  author    = {Grauert, Hans and Remmert, Reinhold},
  title     = {Theory of Stein Spaces},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {236},
  publisher = {Springer},
  year      = {1979}
}