06.10.18 · riemann-surfaces / several-variables

Analytic sets: local parametrisation, dimension, and irreducible components

shipped3 tiersLean: none

Anchor (Master): Gunning–Rossi *Analytic Functions of Several Complex Variables* Ch. III; Łojasiewicz *Introduction to Complex Analytic Geometry* Ch. IV–V; Grauert–Remmert *Coherent Analytic Sheaves* §3

Intuition Beginner

An analytic set is the common zero locus of a finite collection of holomorphic functions near a point. In one complex variable the zeros of a function are isolated points; in several variables they merge into curves, surfaces, and higher-dimensional shapes sitting inside $C^{n}$ . The basic question is: near the origin, what does such a shape look like?

The answer is that it has a dimension, a count of how many free complex directions it has. A single equation in two variables cuts out a one-dimensional shape, a curve. Two independent equations cut down to a point, dimension zero. The dimension is the number of directions you can still move once the equations are imposed.

The other basic fact is that a shape can break into pieces. The set where the product of two coordinates is zero is the union of two coordinate axes: two separate pieces that meet at the origin. A piece that cannot be split further is called irreducible, and every analytic set near a point is a finite union of such pieces.

Visual Beginner

Two pictures carry the whole idea. First, the union of the two coordinate axes in $C^{2}$ , the set where one coordinate times the other equals zero. It is two lines crossing at the origin: two irreducible pieces, and the crossing point is the one place where the shape is not smooth.

Second, the cusp, the curve where the square of one coordinate equals the cube of the other. It is a single connected piece, irreducible, but it has a sharp point at the origin where it fails to be smooth. Away from that point it is a nice one-dimensional curve.

In both pictures the dimension is one: a curve, one free complex direction. The places where the shape looks locally like a flat copy of $C$ form the regular part, and the bad points form the singular part. The singular part is always much smaller than the whole.

Worked example Beginner

Take the cusp: the set of points in $C^{2}$ where $z_{2}^{2} = z_{1}^{3}$ . Watch how it covers the line of $z_{1}$ -values. Pick a $z_{1}$ value, say $z_{1} = 1$ . Then $z_{2}^{2} = 1$ , so $z_{2} = + 1$ or $z_{2} = - 1$ : two points sitting above $z_{1} = 1$ .

Pick $z_{1} = 4$ . Then $z_{2}^{2} = 64$ , so $z_{2} = + 8$ or $z_{2} = - 8$ : again two points. For most values of $z_{1}$ there are exactly two points of the curve sitting above it. The curve is a two-sheeted cover of the $z_{1}$ -line.

Now pick $z_{1} = 0$ . Then $z_{2}^{2} = 0$ , so $z_{2} = 0$ is the only solution: the two sheets have come together into one point. That collision point, the origin, is exactly the sharp cusp point, and it is where the two-to-one covering breaks down.

What this tells us: a one-dimensional analytic set sits over a one-dimensional disc as a finite-to-one cover, the typical number of sheets is its multiplicity (here two), and the special set where sheets collide is the small bad locus.

Check your understanding Beginner

Formal definition Intermediate+

Fix the origin $0 \in C^{n}$ and the local ring $_{n} O = C {z_{1}, \dots, z_{n}}$ of holomorphic germs there, with maximal ideal $m$ . An analytic-set germ $V$ at $0$ is the germ of a set of the form ${f_{1} = \dots = f_{m} = 0}$ for finitely many $f_{i} \in_{n} O$ ; two such sets define the same germ when they agree on some neighbourhood of $0$ . The analytic-set germs form a lattice under germwise union and intersection.

Two maps connect germs to ideals. For a germ $V$ , the ideal of $V$ is $$ \mathcal{I}(V) = { f \in {}_n\mathcal{O} : f \text{ vanishes on } V \text{ near } 0 }, $$ a radical ideal. For an ideal $I \subseteq_{n} O$ , the zero-germ $V (I)$ is the germ of the common zero set of the elements of $I$ . The local Nullstellensatz 06.10.17 gives $I (V (I)) = I$ , so $I$ and $V$ are inclusion-reversing inverse bijections between analytic-set germs and radical ideals of $_{n} O$ .

A germ $V$ is irreducible when $V = V_{1} \cup V_{2}$ with $V_{1}, V_{2}$ analytic-set germs forces $V = V_{1}$ or $V = V_{2}$ . Under the Nullstellensatz this matches $I (V)$ being prime. The local ring of $V$ is $_{V} O =_{n} O / I (V)$ , and the dimension of $V$ is the Krull dimension $$ \dim V = \dim {}V\mathcal{O} = \dim {}n\mathcal{O}/\mathcal{I}(V). $$ A point $p \in V$ near $0$ is regular of dimension $d$ when $V$ is, near $p$ , the germ of a $d$ -dimensional complex submanifold of $C^{n}$ ; otherwise $p$ is singular. Write $V{\mathrm{reg}} $an d$ V{\mathrm{sing}} $f or t h e tw o l oc i . T h eco n v e n t i o n t h r o ug h o u t i s t ha t a$ d $- d im e n s i o na l g er m p r o j ec t s t o a$ d $- d im e n s i o na l coor d ina t e p o l y d i sc, w i t h t h e * e x cess *$ n - d$ variables eliminated.

Counterexamples to common slips

Reducible versus connected. The germ ${z_{1} z_{2} = 0}$ is connected (the two axes meet at $0$ ) yet reducible. Irreducibility is an algebraic condition on $I (V)$ being prime, not a topological connectedness condition.
Radical is mandatory. The ideal $(z_{1}^{2}) \subset_{2} O$ is not radical; its zero germ is the line ${z_{1} = 0}$ , whose ideal is $(z_{1}) = (z_{1}^{2})$ . Dimension and components are read off the radical ideal, never the original generators.
Dimension is not the number of equations. The cusp ${z_{2}^{2} - z_{1}^{3} = 0}$ is one equation in two variables and has dimension one, but ${z_{1} = 0, z_{1} - z_{2} = 0}$ is two equations that still cut out the one-point germ of dimension zero only because the equations are independent; counting equations bounds codimension only when the generators form a regular sequence.

Key theorem with proof Intermediate+

Theorem (local parametrisation). Let $V$ be an irreducible analytic-set germ of dimension $d$ at $0 \in C^{n}$ . After a generic linear change of coordinates $(z^{'}, z^{''}) = (z_{1}, \dots, z_{d}, z_{d + 1}, \dots, z_{n})$ , there are polydiscs $Δ^{'} \subset C_{z^{'}}^{d}$ and $Δ^{''} \subset C_{z^{''}}^{n - d}$ so that the projection $π : V \cap (Δ^{'} \times Δ^{''}) \to Δ^{'}$ , $(z^{'}, z^{''}) \mapsto z^{'}$ , is a finite, surjective, proper holomorphic map. There is a proper analytic subset $D \subset Δ^{'}$ (the branch locus) such that over $Δ^{'} ∖ D$ the map $π$ is an unbranched covering of some fixed degree $b$ , and $V_{reg} \supseteq π^{- 1} (Δ^{'} ∖ D)$ . The integer $b$ is the local multiplicity of $V$ . ^{[Gunning Vol. II]}

Proof. Set $p = I (V)$ , a prime of $_{n} O$ with $dim_{n} O / p = d$ . Choose coordinates so that the projection to $z^{'} = (z_{1}, \dots, z_{d})$ is finite on $V$ ; genericity of the linear change guarantees that the $n - d$ excess coordinates $z_{d + 1}, \dots, z_{n}$ are each integral over $C {z^{'}}$ modulo $p$ , which is possible because $dim_{n} O / p = d$ forces a system of parameters supported on $z^{'}$ .

The elimination is iterated Weierstrass preparation 06.10.14. Pick $f \in p$ vanishing on $V$ but not in the smaller ring $C {z_{1}, \dots, z_{n - 1}}$ ; after the generic rotation $f$ is regular in $z_{n}$ , so preparation writes $f = u \cdot W_{n}$ with $W_{n} \in C {z_{1}, \dots, z_{n - 1}} [z_{n}]$ a Weierstrass polynomial. Then $z_{n}$ is integral over $C {z_{1}, \dots, z_{n - 1}}$ on $V$ , and $C {z_{1}, \dots, z_{n - 1}} / (p \cap C {z_{1}, \dots, z_{n - 1}}) ↪_{V} O$ is a finite extension. Repeating for $z_{n - 1}, \dots, z_{d + 1}$ exhibits $_{V} O$ as a finite module over $A := C {z^{'}} / (p \cap C {z^{'}})$ . Because $dim_{V} O = d = dim C {z^{'}}$ and the extension is finite, $p \cap C {z^{'}} = 0$ , so $A = C {z^{'}}$ injects into $_{V} O$ as a Noether normalisation.

Finiteness of the module $_{V} O$ over $C {z^{'}}$ makes $π$ a finite proper map on representatives over a small polydisc $Δ^{'}$ ; properness comes from the monic Weierstrass equations, whose roots stay bounded as $z^{'}$ varies. Let $Frac (A) = M$ be the field of fractions of $C {z^{'}}$ . Since $p$ is prime and the extension is finite, the fraction field $K$ of $_{V} O$ is a finite field extension of $M$ , say of degree $b = [K : M]$ . By the primitive element theorem some linear combination $w$ of $z_{d + 1}, \dots, z_{n}$ generates $K$ over $M$ and satisfies a monic minimal polynomial $P (z^{'}, w) = w^{b} + c_{1} (z^{'}) w^{b - 1} + \dots + c_{b} (z^{'})$ with $c_{i} \in C {z^{'}}$ .

Let $δ (z^{'}) \in C {z^{'}}$ be the discriminant of $P$ in $w$ , a nonzero germ because $K / M$ is separable in characteristic zero. Set $D = {δ = 0}$ , a proper analytic subset of $Δ^{'}$ . Over $Δ^{'} ∖ D$ the polynomial $P (z^{'}, \cdot)$ has $b$ distinct simple roots depending holomorphically on $z^{'}$ , and each excess coordinate $z_{j}$ is a holomorphic function of $(z^{'}, w)$ there; so $π^{- 1} (z^{'})$ consists of exactly $b$ points and $π$ restricts to an unbranched $b$ -sheeted covering whose local sections are graphs of holomorphic maps. A graph of a holomorphic map is a complex submanifold, so these points lie in $V_{reg}$ . Surjectivity of $π$ onto $Δ^{'}$ follows from properness together with the monic equations having a root over every $z^{'}$ . This is the finite branched covering, with branch locus $D$ and degree $b$ . $□$

Bridge. The local parametrisation theorem builds toward the definition of a complex space, where this branched-covering normal form is the local chart, and it appears again in 06.10.19 when the structure sheaf of an analytic set is assembled stalk by stalk. The foundational reason a $d$ -dimensional germ is governed by a $d$ -variable polydisc is exactly the Noether normalisation produced here: iterated Weierstrass preparation 06.10.14 generalises the one-variable factorisation into a finite integral extension, and this is exactly the algebraic shadow of the geometric covering. Putting these together, the branch locus is dual to the discriminant ideal and the covering degree is the central insight that ties local multiplicity to field-extension degree; the bridge is that everything analytic about $V$ over $Δ^{'} ∖ D$ is the graph of a holomorphic section, which is what hands the regular locus its manifold structure.

Exercises Intermediate+

Exercise 6 (hard, short-answer).

Let $V$ be an irreducible $d$ -dimensional germ with parametrisation $π : V \to Δ^{'}$ of degree $b$ and branch locus $D$ . Show that $V_{reg}$ is dense in $V$ by exhibiting an open dense subset of $V$ consisting of regular points.

Hint

Over $Δ^{'} ∖ D$ the covering is unbranched; use that this is dense in $Δ^{'}$ and that $π$ is finite.

Answer

The set $π^{- 1} (Δ^{'} ∖ D)$ is open in $V$ and consists of regular points, since over $Δ^{'} ∖ D$ the map $π$ is an unbranched covering whose local inverses are holomorphic graphs, hence manifold charts. Because $D$ is a proper analytic subset of the irreducible $Δ^{'}$ , its complement is dense; and $π$ being finite and surjective makes $π^{- 1} (Δ^{'} ∖ D)$ dense in $V = π^{- 1} (Δ^{'})$ (the fibre over any branch point is a limit of nearby unbranched fibres). So $V_{reg}$ contains an open dense subset, hence is dense. Rubric: full credit for openness of the unbranched preimage, density of $Δ^{'} ∖ D$ , and the finiteness step.

Exercise 7 (hard, short-answer).

Using that $_{n} O$ is Noetherian (Rückert basis theorem, 06.10.14), prove that every analytic-set germ $V$ is a finite union of irreducible components, and that this decomposition is unique up to order once redundant pieces are removed.

Hint

Translate to a primary/radical decomposition of $I (V)$ via the Nullstellensatz, and use minimal primes.

Answer

Let $a = I (V)$ , a radical ideal of the Noetherian ring $_{n} O$ . A radical ideal in a Noetherian ring is the intersection of its finitely many minimal primes, $a = p_{1} \cap \dots \cap p_{r}$ . Applying $V$ and reversing inclusions, $V = V (p_{1}) \cup \dots \cup V (p_{r})$ , each $V (p_{i})$ irreducible because $p_{i}$ is prime. Discard any $V (p_{i}) \subseteq V (p_{j})$ , $i \neq = j$ ; the survivors are the irreducible components. Uniqueness: the minimal primes of a radical ideal are uniquely determined, and an irreducible component corresponds to a minimal prime, so no irredundant decomposition can have a different set of maximal irreducible pieces. Rubric: full credit for the minimal-prime intersection, the irreducibility of each piece, and the uniqueness argument.

Lean formalization Intermediate+

Mathlib does not carry analytic-set germs or the local parametrisation theorem, so no compiling formalisation is attached; lean_status: none. The statement below is a schematic Lean-flavoured signature recording the target, not a checked artifact. It names the objects a future formal layer would need: the germ lattice, the dimension as Krull dimension of the quotient local ring, and the branched-covering conclusion. The realisation depends first on formalising $_{n} O$ and its Weierstrass theory, as catalogued in 06.10.14.

-- Schematic only; not part of any lake build (lean_status: none).
-- Targets: AnalyticGermSet n, its ideal I, and dimension via Krull dim.
structure AnalyticGermSet (n : ℕ) where
  ideal : Ideal (ConvergentPowerSeries ℂ n)   -- radical; = 𝓘(V)
  isRadical : ideal.IsRadical

noncomputable def dim {n} (V : AnalyticGermSet n) : ℕ :=
  ringKrullDim ((ConvergentPowerSeries ℂ n) ⧸ V.ideal)   -- placeholder

-- Local parametrisation (statement target, proof unavailable):
-- if V.ideal.IsPrime then ∃ (d := dim V) and a generic projection π
-- making V a finite branched covering of a d-dim polydisc, branch locus
-- a proper analytic subset, covering degree = local multiplicity.

Advanced results Master

The parametrisation theorem promotes immediately to the structure of the singular locus. With $V$ irreducible of dimension $d$ and $π : V \to Δ^{'}$ the degree- $b$ covering branched over $D = {δ = 0}$ , the regular locus contains $π^{- 1} (Δ^{'} ∖ D)$ and is therefore open and dense, while the points lying over $D$ are the only candidates for singularities ^{[Gunning Vol. II]}. The singular locus is itself an analytic set: $V_{sing}$ is the germ cut out by $I (V)$ together with the vanishing of the $(n - d) \times (n - d)$ minors of the Jacobian of a generating set, a condition that is closed and analytic. Hence $V_{sing}$ is a proper analytic subset of $V$ .

The singular locus drops dimension. $V_{sing}$ is a proper analytic subset of the irreducible $V$ , so $I (V) ⊊ I (V_{sing})$ strictly; since $I (V)$ is prime, the strict inclusion forces $dim V_{sing} < dim V$ . The mechanism is the Active Lemma of Grauert–Remmert: a holomorphic function not vanishing on an irreducible germ cuts its dimension by exactly one, and $δ \circ π$ is such an active element on $V$ , vanishing precisely on the over-branch set ^{[Grauert–Remmert §3.3]}. So the bad set is not merely small in measure; it is analytically of strictly smaller dimension, and the regular locus is a connected $d$ -dimensional complex manifold, dense in $V$ .

Dimension as a chain of integral extensions. The Krull dimension $dim_{V} O = d$ agrees with three other counts: the maximal length of a chain of prime ideals in $_{V} O$ ; the transcendence degree over $C$ of the fraction field $K$ of $_{V} O$ ; and the dimension of the polydisc $Δ^{'}$ the parametrisation covers. The Noether normalisation $C {z^{'}} ↪_{V} O$ realises all three: it is finite, so it preserves Krull dimension; $K$ is finite over $Frac (C {z^{'}})$ , which has transcendence degree $d$ ; and $Δ^{'} \subset C^{d}$ is the image polydisc. The agreement is the geometric content of the algebraic dimension theory of $_{n} O$ .

Decomposition and the local multiplicity. For a general (not necessarily irreducible) germ $V$ with components $V_{1}, \dots, V_{r}$ , the dimension is $dim V = max_{i} dim V_{i}$ , and each component carries its own covering degree. Components of strictly smaller dimension than the maximum are the embedded-free lower-dimensional pieces; the geometry near a generic point of a top-dimensional component is the unbranched cover of that component alone. The local multiplicity of $V$ at $0$ , in the sense of the parametrisation, is the sum of the covering degrees of the top-dimensional components passing through $0$ , weighted by the order to which the others meet them — a refinement made precise by the analytic intersection theory built on this normal form.

Normality and the branch locus. When $_{V} O$ is integrally closed in its fraction field — the normal case — the singular locus has codimension at least two in $V$ , so for a normal surface germ ( $d = 2$ ) the singularities are isolated. The parametrisation's branch locus $D$ need not coincide with $π (V_{sing})$ : branching can occur over points where $V$ is still regular (the cusp is regular along its unbranched part but the projection branches at the cusp point, where $V$ is also singular; the node ${z_{1} z_{2} = 0}$ shows the converse can fail to be an equality after normalisation). Normalisation separates these by replacing $V$ with the analytic spectrum of the integral closure, a finite modification that resolves the codimension-one part of the branching.

Synthesis. The local parametrisation theorem is the foundational reason the local geometry of an analytic set is controlled by a finite branched cover of a coordinate polydisc, and every later structural fact is dual to a feature of that cover. Putting these together: dimension is the dimension of the base polydisc, which is exactly the Krull dimension of the local ring and the transcendence degree of its fraction field; the irreducible decomposition is the central insight that the geometry is the disjoint reading of the minimal primes over $I (V)$ ; and the regular-locus density together with the strictly-lower-dimensional singular locus generalises the one-variable fact that zeros are isolated, now read as: degeneracies sit on a proper analytic subset. This is exactly the package — finite cover, dimension, components, regular versus singular — that builds toward the definition of a complex space and appears again in the coherence theory of analytic sheaves, where the branched cover supplies the finite free resolutions and the discriminant supplies the singular support.

Full proof set Master

Proposition (Noetherian decomposition into irreducible components). Every analytic-set germ $V$ at $0 \in C^{n}$ is a finite union $V = V_{1} \cup \dots \cup V_{r}$ of irreducible germs with no $V_{i} \subseteq V_{j}$ for $i \neq = j$ , and this irredundant decomposition is unique up to reordering. The $V_{i}$ correspond to the minimal primes over $I (V)$ .

Proof. By the Nullstellensatz 06.10.17, $a := I (V)$ is a radical ideal and $V = V (a)$ . The ring $_{n} O$ is Noetherian (Rückert, 06.10.14), so $a$ has finitely many minimal primes $p_{1}, \dots, p_{r}$ , and being radical it equals their intersection $a = ⋂_{i} p_{i}$ . Applying the inclusion-reversing bijection $V$ gives $V = ⋃_{i} V (p_{i})$ , with each $V_{i} := V (p_{i})$ irreducible because $p_{i}$ is prime. Removing any $V_{i}$ contained in another $V_{j}$ corresponds to removing a non-minimal prime; since the $p_{i}$ are exactly the minimal primes, no removals are forced and the decomposition is already irredundant after discarding repeats. For uniqueness, suppose $V = ⋃_{j} W_{j}$ is another irredundant decomposition into irreducibles. Each $W_{j} = V (q_{j})$ for a prime $q_{j} \supseteq a$ , and $W_{j} \subseteq V$ forces $q_{j} \supseteq p_{i}$ for some $i$ with, by irredundancy, equality; conversely each $p_{i}$ appears, so the two families of primes coincide. $□$

Proposition (regular locus dense and open; singular locus a proper analytic subset of lower dimension). Let $V$ be an irreducible germ of dimension $d$ at $0$ . Then $V_{reg}$ is open and dense in $V$ , $V_{sing}$ is an analytic-set germ properly contained in $V$ , and $dim V_{sing} < d$ .

Proof. Take the parametrisation $π : V \cap (Δ^{'} \times Δ^{''}) \to Δ^{'}$ of degree $b$ branched over $D = {δ = 0}$ from the Key Theorem. Over $Δ^{'} ∖ D$ the inverse images are graphs of holomorphic maps, hence $d$ -dimensional submanifolds, so $U := π^{- 1} (Δ^{'} ∖ D) \subseteq V_{reg}$ and $U$ is open in $V$ . Since $Δ^{'}$ is irreducible and $D$ is a proper analytic subset, $Δ^{'} ∖ D$ is dense in $Δ^{'}$ ; finiteness and surjectivity of $π$ then make $U$ dense in $V$ , so $V_{reg}$ is dense and open.

That $V_{sing}$ is analytic: choose generators $f_{1}, \dots, f_{m}$ of $I (V)$ . A point $p \in V_{reg}$ of dimension $d$ is one where the Jacobian $(\partial f_{i} / \partial z_{k})$ has rank $n - d$ ; the locus where the rank drops below $n - d$ is cut out by the vanishing of all $(n - d) \times (n - d)$ minors, an analytic condition. Hence $V_{sing} = V \cap {all (n - d) -minors vanish}$ is an analytic-set germ. It is proper in $V$ because $V_{reg} \neq = \emptyset$ (it is dense). Finally, $V_{sing} ⊊ V$ with $V$ irreducible gives $I (V) ⊊ I (V_{sing})$ ; as $I (V)$ is prime, this strict containment of a prime in the radical ideal $I (V_{sing})$ forces every minimal prime over $I (V_{sing})$ to strictly contain $I (V)$ , dropping Krull dimension: $dim V_{sing} = dim_{n} O / I (V_{sing}) < dim_{n} O / I (V) = d$ . $□$

Proposition (worked decomposition: node and cusp). The germ ${z_{1} z_{2} = 0}$ has two irreducible components, the coordinate axes, with singular locus ${0}$ ; the germ ${z_{2}^{2} - z_{1}^{3} = 0}$ is irreducible of dimension one with singular locus ${0}$ and is a two-sheeted cover of the $z_{1}$ -disc branched at the origin.

Proof. For the node, $(z_{1} z_{2}) = (z_{1}) \cap (z_{2})$ in $_{2} O$ , an intersection of two primes, which are the minimal primes; the components are ${z_{1} = 0}$ and ${z_{2} = 0}$ , two regular lines of dimension one. Off the origin exactly one axis passes through each point, where the germ is a smooth line, so those points are regular; at the origin the germ is reducible, hence singular, giving $V_{sing} = {0}$ , dimension $0 < 1$ . For the cusp, $z_{2}^{2} - z_{1}^{3}$ is irreducible in $C {z_{1}} [z_{2}]$ (it is Eisenstein-type: monic of degree $2$ in $z_{2}$ with the lower coefficient $- z_{1}^{3}$ in the maximal ideal but not its square in a way that admits no monic degree-one factor over $C {z_{1}}$ , since $z_{1}^{3}$ is not a square in $C {z_{1}}$ ), so $I (V) = (z_{2}^{2} - z_{1}^{3})$ is prime and $V$ is irreducible of dimension one. Projection to $z_{1}$ has fibre ${z_{2} : z_{2}^{2} = z_{1}^{3}}$ , generically two points, one point over $z_{1} = 0$ ; the discriminant in $z_{2}$ is $4 z_{1}^{3}$ , so the branch locus is ${z_{1} = 0}$ and the cover is two-sheeted branched at the origin. The Jacobian $(- 3 z_{1}^{2}, 2 z_{2})$ vanishes only at $(0, 0)$ on $V$ , so $V_{sing} = {0}$ . $□$

Connections Master

Local Nullstellensatz 06.10.17. The bijection between analytic-set germs and radical ideals that underlies the whole unit — irreducible germs to primes, components to minimal primes, dimension to Krull dimension — is the Nullstellensatz of the prior unit; this unit is its geometric payoff, turning the radical-ideal dictionary into a structure theorem for germs.
Weierstrass preparation and division 06.10.14. Iterated preparation is the engine of the local parametrisation theorem: each excess coordinate is made integral over the base ring by a Weierstrass polynomial, producing the Noether normalisation and the monic equations that bound the covering. The Noetherian property used in the decomposition is the Rückert basis theorem proved there.
Complex spaces and structure sheaves 06.10.19. The branched-covering normal form is the local model glued to define a complex space; the dimension and regular/singular splitting established here become the dimension theory and smooth-locus of complex-analytic spaces, and the local ring $_{V} O$ is the stalk of the structure sheaf.
Coherence and analytic sheaf modules 06.10.20. The finiteness of $_{V} O$ over $C {z^{'}}$ is the local input to coherence of the ideal sheaf of an analytic set; the finite free resolutions over the polydisc base come directly from the parametrisation's module structure.
Normalisation and singularities 06.10.22. The codimension and analyticity of the singular locus proved here is the starting point for normalisation, which replaces a germ by the analytic spectrum of the integral closure of $_{V} O$ and resolves the codimension-one branching of the parametrisation.

Historical & philosophical context Master

The local structure theory of analytic sets grew from Weierstrass's preparation theorem of the 1860s into the systematic local complex-analytic geometry of the twentieth century. Walther Rückert's 1933 paper on power-series ideals supplied the basis theorem and the algebraic description of analytic-set germs through their ideals, making the dictionary between germs and ideals precise ^{[Gunning–Rossi]}. The local parametrisation theorem in the branched-covering form used here was standardised in the texts of the Oka–Cartan era and is presented in detail by Gunning in the second volume of his local-theory lectures ^{[Gunning Vol. II]}.

Hassler Whitney's Complex Analytic Varieties (1972) and Stanisław Łojasiewicz's Introduction to Complex Analytic Geometry (1991) gave the regular/singular stratification and the proper-analytic-subset character of the singular locus their modern form ^{[Łojasiewicz]}; Hans Grauert and Reinhold Remmert systematised the dimension theory and the Active Lemma that controls how a non-vanishing function cuts dimension ^{[Grauert–Remmert §3.3]}. Reinhart Remmert's survey of the local theory of complex spaces traces the lineage from Weierstrass and Rückert through the coherence theorems of Oka and Cartan ^[Remmert].

Bibliography Master

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

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

@incollection{RemmertLocal,
  author    = {Remmert, Reinhold},
  title     = {Local Theory of Complex Spaces},
  booktitle = {Several Complex Variables VII},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {74},
  publisher = {Springer},
  year      = {1994},
  pages     = {7--96}
}

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

@book{LojasiewiczCAG,
  author    = {{\L}ojasiewicz, Stanis{\l}aw},
  title     = {Introduction to Complex Analytic Geometry},
  publisher = {Birkh{\"a}user},
  year      = {1991}
}

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

@article{Ruckert1933,
  author  = {R{\"u}ckert, Walther},
  title   = {Zum Eliminationsproblem der Potenzreihenideale},
  journal = {Math. Ann.},
  volume  = {107},
  year    = {1933},
  pages   = {259--281}
}

Prerequisites

06.10.17
06.10.14

Tier anchors

beginner: Gunning *Introduction to Holomorphic Functions of Several Variables* Vol. II, Ch. on local analytic sets (informal opening); Remmert *Local Theory of Complex Spaces* intro
intermediate: Gunning *Introduction to Holomorphic Functions of Several Variables* Vol. II §M; Whitney *Complex Analytic Varieties* Ch. 4; Grauert–Remmert *Coherent Analytic Sheaves* §3.1
master: Gunning–Rossi *Analytic Functions of Several Complex Variables* Ch. III; Łojasiewicz *Introduction to Complex Analytic Geometry* Ch. IV–V; Grauert–Remmert *Coherent Analytic Sheaves* §3

References

Gunning — Introduction to Holomorphic Functions of Several Variables, Vol. II: Local Theory · Wadsworth & Brooks/Cole 1990, Ch. on local analytic sets (local parametrisation theorem; dimension; irreducible components)
Gunning–Rossi — Analytic Functions of Several Complex Variables · Prentice-Hall 1965, Ch. III (local rings of analytic sets; branched coverings; dimension)
Remmert — Local Theory of Complex Spaces · in Several Complex Variables VII, Springer EMS 74, 1994, §1–2 (analytic-set germs, decomposition, regular and singular loci)
Grauert–Remmert — Coherent Analytic Sheaves · Springer Grundlehren 265, 1984, §3.1–3.4 (Noetherian decomposition; dimension; Active Lemma; regular points)
Łojasiewicz — Introduction to Complex Analytic Geometry · Birkhäuser 1991, Ch. IV–V (Rückert description; normal parametrisation; singular locus)

Estimated time

beginner: 16m
intermediate: 40m
master: 70m