06.10.20 · riemann-surfaces / several-variables

Coherent analytic sheaves and Oka's coherence theorem

shipped3 tiersLean: none

Anchor (Master): Gunning *Introduction to Holomorphic Functions of Several Variables* Vol. III §J; Grauert–Remmert *Coherent Analytic Sheaves* §2–4; Cartan séminaire 1951–52

Intuition Beginner

A sheaf is a bookkeeping device: it attaches to every small region of space the holomorphic functions that live there, and it remembers how those functions restrict from larger regions to smaller ones. Near a point in several complex variables, the holomorphic functions form a ring, and a sheaf of modules lets us track families of functions that vary smoothly from point to point.

The key question this unit answers is a finiteness question. Suppose you have a few holomorphic functions and you ask: what are all the ways to combine them, with holomorphic coefficients, so the result is zero? Each such combination is a relation. There could be infinitely many relations, and you might fear you can never list them all.

Coherence is the promise that you can. Near every point, finitely many relations generate all the rest. Oka's theorem says the holomorphic functions themselves enjoy this promise. That single fact powers the whole sheaf theory of several complex variables.

Visual Beginner

Picture two holomorphic functions $f$ and $g$ defined near a point, and imagine the pairs of coefficient functions $(a, b)$ for which $a f + b g = 0$ . These pairs form a shape sitting inside the space of all coefficient pairs. Coherence says this shape, though it may look complicated, is spanned by finitely many basic pairs near every point.

A concrete picture: if $f = z_{1}$ and $g = z_{2}$ , the one obvious relation is $z_{2} \cdot f - z_{1} \cdot g = 0$ , the pair $(z_{2}, - z_{1})$ . Coherence promises that every other relation is a holomorphic multiple of this single basic one. The infinite cloud of relations collapses to one generator.

The lesson of the picture: coherence converts an infinite, hard-to-survey object, the set of all relations, into a finite, listable one near each point. That local finiteness is what lets sheaf theory compute.

Worked example Beginner

Take $f = z_{1}$ and $g = z_{1} z_{2}$ near the origin of two complex variables. We hunt for all coefficient pairs $(a, b)$ , holomorphic near the origin, with $a z_{1} + b z_{1} z_{2} = 0$ . Factor out $z_{1}$ : the condition becomes $z_{1} (a + b z_{2}) = 0$ .

A holomorphic function near the origin times $z_{1}$ is zero only when the function itself is zero, because $z_{1}$ does not vanish on a whole neighbourhood. So the condition reduces to $a + b z_{2} = 0$ , that is $a = - b z_{2}$ .

Now read off the relations. Choose any holomorphic $b$ ; then $a$ is forced to be $- b z_{2}$ . Every relation has the form $(a, b) = b \cdot (- z_{2}, 1)$ . The single pair $(- z_{2}, 1)$ generates them all, with $b$ ranging over holomorphic functions.

What this tells us: the relations among $z_{1}$ and $z_{1} z_{2}$ , an a-priori infinite family, are all multiples of one basic relation. That is coherence in action at the simplest level, and the engine that guarantees it in general is the ability to divide holomorphic functions.

Check your understanding Beginner

Formal definition Intermediate+

Let $Ω \subseteq C^{n}$ be open and let $_{n} O =_{n} O_{Ω}$ denote the sheaf of germs of holomorphic functions on $Ω$ , a sheaf of commutative rings whose stalk at $p$ is the local ring $C {z_{1} - p_{1}, \dots, z_{n} - p_{n}}$ of convergent power series. A sheaf of $_{n} O$ -modules $F$ assigns to each open $U$ an $_{n} O (U)$ -module $F (U)$ compatibly with restriction; its stalk $F_{p}$ is a module over the stalk ring $_{n} O_{p}$ .

For sections $s_{1}, \dots, s_{p} \in F (U)$ , the relation sheaf (or sheaf of syzygies) is the subsheaf $$ \mathcal{R}(s_1, \dots, s_p) \subseteq {}_n\mathcal{O}_U^{,p}, \qquad \mathcal{R}(s_1, \dots, s_p)_q = \Big{ (a_1, \dots, a_p) \in {}_n\mathcal{O}q^{,p} : \sum{j} a_j (s_j)_q = 0 \Big}. $$ A sheaf $F$ is of finite type if every point has a neighbourhood $U$ and finitely many sections generating $F_{q}$ over $_{n} O_{q}$ for all $q \in U$ . It is coherent if it is of finite type and, for every open $U$ and every choice of finitely many sections $s_{1}, \dots, s_{p} \in F (U)$ , the relation sheaf $R (s_{1}, \dots, s_{p})$ is itself of finite type. A ring sheaf $A$ is coherent over itself when $A$ , viewed as a module over itself, is coherent: this amounts to the relation sheaf of any finite tuple of sections of $A$ being locally finitely generated. ^{[Grauert–Remmert Ch. 2]}

The ideal sheaf of a closed analytic subset $V \subseteq Ω$ is the subsheaf $I ⟨ V ⟩ \subseteq_{n} O$ whose stalk at $p$ is the ideal $I (V)_{p}$ of germs vanishing on $V$ near $p$ ; by the local Nullstellensatz it is a radical ideal in each stalk 06.10.18. The convention here is that "coherent analytic sheaf" always means coherent as an $_{n} O$ -module sheaf in the analytic category, never the looser algebraic sense.

Counterexamples to common slips

Finite type is weaker than coherent. A sheaf can be locally finitely generated while having a relation sheaf that fails to be finitely generated. Coherence demands the second condition for every finite tuple of sections over every open set, not merely the existence of finitely many generators.
Coherence is local but not pointwise. A single stalk being a finitely generated module says nothing; coherence requires generators valid on a whole neighbourhood and relation sheaves finitely generated near each point. The neighbourhood uniformity is the entire difficulty.
Radical matters for ideal sheaves. The sheaf with stalks $(z_{1}^{2})$ is a coherent $_{n} O$ -module, but it is not the ideal sheaf of the analytic set ${z_{1} = 0}$ ; that ideal sheaf has stalks $(z_{1})$ . Cartan's theorem concerns the radical ideal sheaf $I ⟨ V ⟩$ attached to the reduced set $V$ .

Key theorem with proof Intermediate+

Theorem (Oka's coherence theorem). The structure sheaf $_{n} O$ on an open $Ω \subseteq C^{n}$ is coherent over itself. Equivalently: for any open $U$ and any finite tuple $s_{1}, \dots, s_{p} \in_{n} O (U)$ , the relation sheaf $R (s_{1}, \dots, s_{p}) \subseteq_{n} O_{U}^{p}$ is of finite type. ^{[Oka 1950]}

Proof. The statement is local, so fix $p \in U$ , take $p = 0$ , and work with stalks at the origin; write $O =_{n} O_{0} = C {z_{1}, \dots, z_{n}}$ . It suffices to show that the stalk $R (s_{1}, \dots, s_{p})_{0}$ is generated by finitely many relations that extend holomorphically and continue to generate the relation stalks at all nearby points; the argument produces generators over a full polydisc neighbourhood. Proceed by induction on the number of variables $n$ .

For $n = 0$ the ring is the field $C$ , every submodule of $C^{p}$ is a finite-dimensional vector space, and relation modules are finitely generated. Assume the theorem holds for $C {z_{1}, \dots, z_{n - 1}}$ and write $O^{'} = C {z_{1}, \dots, z_{n - 1}}$ , $z = z_{n}$ , so $O = O^{'} {z}$ . After discarding any $s_{j}$ that is the zero germ, a generic linear change makes each nonzero $s_{j}$ regular in $z$ of some order; multiply through by units so that, by Weierstrass preparation 06.10.14, we may replace each $s_{j}$ by a Weierstrass polynomial $P_{j} (z) \in O^{'} [z]$ , monic in $z$ of degree $d_{j}$ . Relations among the $s_{j}$ and relations among the $P_{j}$ correspond under multiplication by the unit factors, so it suffices to finitely generate the relation module of the Weierstrass polynomials $P_{1}, \dots, P_{p}$ .

Set $d = max_{j} d_{j}$ and let $P = P_{1}$ have degree $d_{1}$ . Given a relation $\sum_{j} a_{j} P_{j} = 0$ with $a_{j} \in O$ , apply Weierstrass division 06.10.14 of each coefficient $a_{j}$ by the monic $P_{1}$ : write $a_{j} = q_{j} P_{1} + r_{j}$ with $q_{j} \in O$ and $r_{j} \in O^{'} [z]$ a polynomial of degree $< d_{1}$ . Substituting, $$ \sum_j a_j P_j = \Big( \sum_j q_j P_j \Big) P_1 + \sum_j r_j P_j = 0 . $$ The term $\sum_{j} q_{j} P_{j}$ is generated by the Koszul relations $P_{i} \cdot e_{j} - P_{j} \cdot e_{i}$ (where $e_{i}$ is the $i$ -th standard generator), which lie in the relation module; subtracting a combination of these reduces every relation modulo Koszul relations to one with all coefficients $a_{j} = r_{j}$ of bounded $z$ -degree $< d_{1}$ . Thus the relation module is generated by the finitely many Koszul relations together with the relations $\sum_{j} r_{j} P_{j} = 0$ in which each $r_{j}$ has degree $< d_{1}$ in $z$ .

It remains to finitely generate this bounded-degree part. A relation $\sum_{j} r_{j} P_{j} = 0$ with $de g_{z} r_{j} < d_{1}$ is an identity in $O^{'} [z]$ between polynomials of bounded degree; collecting coefficients of each power $z^{k}$ for $0 \leq k < d_{1} + d$ turns it into a finite system of $O^{'}$ -linear equations in the $O^{'}$ -coefficients of the $r_{j}$ . The solution module of such a finite linear system over $O^{'}$ is, by the inductive hypothesis applied to relations over $O^{'} = C {z_{1}, \dots, z_{n - 1}}$ , finitely generated. Lifting these generators back to relations among the $P_{j}$ and adjoining the Koszul relations yields a finite generating set for $R (P_{1}, \dots, P_{p})_{0}$ , valid over a polydisc on which all the Weierstrass data converge. This completes the induction. $□$

Bridge. Oka's theorem builds toward the entire sheaf-cohomological machine of several complex variables, and it appears again in 06.10.21 as the standing finiteness hypothesis under which Cartan's Theorems A and B hold. The foundational reason coherence is hard in the analytic category, and a free theorem in the algebraic one, is exactly that $_{n} O$ is not Noetherian as a global ring while its stalks are; the proof above recovers finiteness one variable at a time by Weierstrass division, which generalises one-variable polynomial division into the bounded-degree reduction that tames the syzygies. Putting these together, the relation sheaf is dual to a finite linear system over the smaller ring, and the central insight is that division by a monic Weierstrass polynomial identifies relations among power series with relations among polynomials of bounded degree, where induction can finish the job; the bridge is that this same finite-free-resolution structure is what supports coherence of the ideal sheaf of an analytic set in the Advanced results.

Exercises Intermediate+

Exercise 3 (medium, short-answer).

Show that the relation sheaf of $s_{1} = z_{1}$ , $s_{2} = z_{1} z_{2}$ in $_{2} O$ is generated by the single relation $(z_{2}, - 1)$ , and identify it as a multiple of the Koszul relation modified by a unit.

Hint

Set $a z_{1} + b z_{1} z_{2} = 0$ , cancel $z_{1}$ , and solve for $a$ in terms of $b$ .

Answer

From $a z_{1} + b z_{1} z_{2} = z_{1} (a + b z_{2}) = 0$ and $z_{1}$ a non-zero-divisor we get $a = - b z_{2}$ , so every relation is $b \cdot (- z_{2}, 1)$ ; one generator $(- z_{2}, 1)$ . The Koszul relation of $(z_{1}, z_{1} z_{2})$ is $(z_{1} z_{2}) e_{1} - (z_{1}) e_{2} = z_{1} \cdot (z_{2}, - 1)$ , which is $z_{1}$ times the generator $(z_{2}, - 1)$ ; dividing the Koszul relation by the common factor $z_{1}$ yields the actual generator, reflecting that $z_{1}, z_{1} z_{2}$ is not a regular sequence. Rubric: full credit for the generator $(- z_{2}, 1)$ (equivalently $(z_{2}, - 1)$ ) and the regular-sequence-failure observation.

Exercise 4 (medium, short-answer).

Explain why the bounded-degree reduction in the proof of Oka's theorem requires the $P_{j}$ to be Weierstrass polynomials (monic in $z_{n}$ ), not merely holomorphic in $z_{n}$ .

Hint

Weierstrass division by a divisor needs that divisor to be monic so the remainder has strictly smaller, bounded degree.

Answer

Weierstrass division of $a_{j}$ by $P_{1}$ produces a remainder $r_{j} \in O^{'} [z]$ of degree strictly below $de g_{z} P_{1}$ precisely because $P_{1}$ is monic; without monicity the division algorithm does not terminate with a polynomial remainder of controlled degree, and the reduction to a finite $O^{'}$ -linear system collapses. Preparation supplies the monic representative after a generic rotation makes each $s_{j}$ regular in $z_{n}$ . Rubric: full credit for naming monicity as the condition that bounds the remainder degree.

Exercise 6 (hard, short-answer).

Prove that coherence is preserved under taking the kernel of a sheaf morphism between coherent sheaves: if $φ : F \to G$ is a morphism of coherent $_{n} O$ -modules, then $ker φ$ is coherent.

Hint

Locally $F$ is generated by finitely many sections $s_{i}$ ; the kernel stalks are relations among the $φ (s_{i})$ inside $G$ , plus relations among the $s_{i}$ .

Answer

Work near a point where $F$ is generated by $s_{1}, \dots, s_{p}$ . A germ $\sum a_{i} s_{i}$ lies in $ker φ$ iff $\sum a_{i} φ (s_{i}) = 0$ , i.e. $(a_{i}) \in R (φ (s_{1}), \dots, φ (s_{p}))$ , the relation sheaf of finitely many sections of the coherent $G$ , hence of finite type. Choose finitely many generators $(a_{i}^{(k)})$ of that relation sheaf; the sections $\sum_{i} a_{i}^{(k)} s_{i}$ then generate $ker φ$ over a neighbourhood, so $ker φ$ is of finite type. Its relation sheaves are sub-relations of those of $F$ , again finitely generated by coherence of $F$ . Therefore $ker φ$ is coherent. Rubric: full credit for reducing kernel generation to the relation sheaf of $φ (s_{i})$ and invoking finite type of relations.

Exercise 7 (hard, short-answer).

Using Oka's theorem, prove that for any finitely generated ideal sheaf $J \subseteq_{n} O$ (stalkwise generated by fixed global sections $g_{1}, \dots, g_{m}$ ), the sheaf $J$ is coherent.

Hint

$J$ is the image of the map $_{n} O^{m} \to_{n} O$ , $(a_{j}) \mapsto \sum a_{j} g_{j}$ ; use that images of coherent sheaves under maps to coherent sheaves are coherent, which reduces to finite generation of the relation sheaf.

Answer

$J$ is of finite type by hypothesis, generated by $g_{1}, \dots, g_{m}$ . For coherence we must finitely generate the relation sheaf of any finite tuple of sections of $J$ . Any such section is an $_{n} O$ -combination of the $g_{j}$ , so it suffices to bound relations among the $g_{j}$ themselves; but $R (g_{1}, \dots, g_{m}) \subseteq_{n} O^{m}$ is of finite type precisely by Oka's coherence theorem applied to the tuple $(g_{1}, \dots, g_{m})$ . Relations among arbitrary sections of $J$ pull back to relations among the $g_{j}$ composed with the surjection $_{n} O^{m} ↠ J$ , hence are finitely generated. Therefore $J$ is coherent. Rubric: full credit for citing Oka coherence for $R (g_{1}, \dots, g_{m})$ and reducing general relations to it.

Lean formalization Intermediate+

Mathlib carries coherent modules over a ring in the abstract but no analytic sheaf theory on $C^{n}$ , so no compiling formalisation is attached; lean_status: none. The signature below is a Lean-flavoured target, not a checked artifact. It names the objects a future formal layer would require: the sheaf of convergent-power-series rings, the relation (syzygy) sheaf of a finite tuple, and the coherence predicate. The realisation depends first on formalising $_{n} O$ and its Weierstrass division, catalogued in 06.10.14.

-- Schematic only; not part of any lake build (lean_status: none).
-- Targets: the sheaf of rings 𝓞 on Ω ⊆ ℂⁿ and the coherence predicate.

variable {n : ℕ} (Ω : Opens (Fin n → ℂ))

-- Relation (syzygy) sheaf of a finite tuple of sections of a module sheaf.
-- relationSheaf F s = ker of the map 𝓞^p → F sending (aⱼ) ↦ Σ aⱼ • sⱼ.
def relationSheaf (F : SheafOfModules 𝓞) {p : ℕ}
    (s : Fin p → F.sections Ω) : SheafOfModules 𝓞 := sorry

-- Coherence: finite type + relation sheaf of every finite tuple is finite type.
def IsCoherentSheaf (F : SheafOfModules 𝓞) : Prop := sorry

-- Oka coherence (statement target, proof unavailable):
-- theorem oka_coherence : IsCoherentSheaf (𝓞 : SheafOfModules 𝓞)
-- proof would induct on n using Weierstrass division (see 06.10.14).

Advanced results Master

Coherence is stable under the operations that build sheaf theory, and Oka's theorem is the seed from which every concrete coherent analytic sheaf grows. The first consequence is the three-lemma stability package: in a short exact sequence $0 \to F^{'} \to F \to F^{''} \to 0$ of $_{n} O$ -modules, if two of the three are coherent so is the third; kernels, cokernels, and images of morphisms between coherent sheaves are coherent; and finite direct sums and tensor products of coherent sheaves are coherent ^{[Grauert–Remmert Ch. 2]}. The proofs reduce every assertion to local finite generation of relation sheaves, which the structure-sheaf case supplies. The class of coherent $_{n} O$ -modules is thus an abelian subcategory of all $_{n} O$ -modules, closed under the standard functors.

Cartan's coherence of the ideal sheaf. For a closed analytic subset $V \subseteq Ω$ , the ideal sheaf $I ⟨ V ⟩ \subseteq_{n} O$ is coherent ^{[Cartan 1950]}. The local content is that the radical ideal $I (V)_{p}$ at each point extends to finitely many holomorphic generators on a neighbourhood whose germs generate $I (V)_{q}$ at every nearby $q$ . The proof leverages the local parametrisation of $V$ from 06.10.18: writing $V$ as a finite branched cover of a polydisc, the discriminant and the monic Weierstrass equations of the excess coordinates furnish explicit generators, and Oka coherence applied to those generators bounds the relations. Because $I ⟨ V ⟩$ is a subsheaf of the coherent $_{n} O$ that is locally finitely generated, the stability three-lemma upgrades local finite generation to full coherence.

Coherence of the structure sheaf of an analytic set. With $I ⟨ V ⟩$ coherent, the quotient $O_{V} :=_{n} O / I ⟨ V ⟩$ , supported on $V$ , is coherent as an $_{n} O$ -module by the cokernel stability of the three-lemma. This is the structure sheaf of the reduced analytic set $V$ , and its coherence is what makes $V$ into a complex space with a workable finiteness theory. The stalk $(O_{V})_{p} =_{V} O_{p}$ recovers the local ring of 06.10.18, so coherence of $O_{V}$ is the global-on-a-neighbourhood promotion of the stalkwise local-ring theory.

Contrast with the algebraic category. On a Noetherian scheme the structure sheaf is coherent as a formal consequence of the Noetherian condition on the affine coordinate rings, with no analysis involved; the algebraic coherent sheaf 04.06.02 inherits its finiteness for free. The analytic statement is genuinely deeper because the ring of convergent power series $_{n} O$ is not Noetherian as a sheaf of global sections over a positive-dimensional base, and the local rings, although Noetherian, do not assemble into a Noetherian global object. Oka's theorem is precisely the replacement: it secures the finiteness with neighbourhood uniformity that the Noetherian hypothesis would supply algebraically, and it does so by hard local analysis, the Weierstrass division, rather than by an abstract chain condition.

Synthesis. Oka's coherence theorem is the foundational reason the analytic category supports a sheaf cohomology as powerful as the algebraic one, and every later finiteness result is dual to the local syzygy-boundedness it provides. Putting these together: the ideal sheaf $I ⟨ V ⟩$ is coherent because Cartan's argument identifies its generators with the Weierstrass data of the branched cover, the structure sheaf $O_{V}$ is coherent because it is the cokernel of a map of coherent sheaves, and the abelian-category stability is the central insight that propagates coherence through every kernel, image, and extension. This generalises the one-variable fact that a holomorphic function has locally finitely many zeros, now read sheaf-theoretically as: the relations among holomorphic data are locally finitely generated; and it builds toward Cartan's Theorems A and B, where coherence is exactly the hypothesis that makes the cohomology of a Stein open vanish. The contrast with 04.06.02 is the bridge between two finiteness regimes: what is automatic on a Noetherian scheme is, in the analytic world, the theorem proved here.

Full proof set Master

Proposition (stability of coherence under cokernels). Let $φ : F \to G$ be a morphism of coherent $_{n} O$ -modules. Then $im φ$ and $coker φ$ are coherent.

Proof. The image $im φ$ is a subsheaf of the coherent $G$ , and it is of finite type because locally $F$ is generated by finitely many sections $s_{i}$ whose images $φ (s_{i})$ generate $im φ$ . A subsheaf of a coherent sheaf that is of finite type is coherent: its relation sheaves are sub-relation-sheaves of those of $G$ , hence finitely generated. So $im φ$ is coherent. For the cokernel, the sequence $G \to coker φ \to 0$ is exact and $coker φ = G / im φ$ . Since $G$ is of finite type, so is its quotient $coker φ$ . For coherence of the quotient, take finitely many sections $t_{1}, \dots, t_{q}$ of $coker φ$ near a point; lift them to sections $\tilde{t}_{k}$ of $G$ . A relation $\sum b_{k} t_{k} = 0$ in $coker φ$ means $\sum b_{k} \tilde{t}_{k} \in im φ$ , i.e. $\sum b_{k} \tilde{t}_{k} = \sum c_{l} φ (s_{l})$ for some germs $c_{l}$ . The tuples $(b_{k}, c_{l})$ form the relation sheaf of $(\tilde{t}_{1}, \dots, \tilde{t}_{q}, φ (s_{1}), \dots, φ (s_{m}))$ inside $G$ , finitely generated by coherence of $G$ ; projecting to the $b_{k}$ -coordinates yields finitely many generators of $R (t_{1}, \dots, t_{q})$ . Hence $coker φ$ is coherent. $□$

Proposition (Oka coherence implies coherence of $O_{V}$ ). Let $V \subseteq Ω$ be a closed analytic subset whose ideal sheaf $I ⟨ V ⟩$ is coherent. Then $O_{V} =_{n} O / I ⟨ V ⟩$ is a coherent $_{n} O$ -module.

Proof. By Oka's theorem $_{n} O$ is coherent over itself. The inclusion $I ⟨ V ⟩ ↪_{n} O$ is a morphism of $_{n} O$ -modules with both source (by hypothesis) and target (by Oka) coherent. Its cokernel is exactly $O_{V}$ . By the cokernel-stability proposition, the cokernel of a morphism of coherent sheaves is coherent, so $O_{V}$ is coherent. $□$

Proposition (coherence of the relation module of two coordinate monomials). In $_{2} O$ the relation sheaf $R (z_{1}, z_{1} z_{2})$ is free of rank one, generated near the origin by $(z_{2}, - 1)$ ; in particular the structure sheaf restriction is coherent in this concrete case.

Proof. A germ relation is $(a, b)$ with $a z_{1} + b z_{1} z_{2} = z_{1} (a + b z_{2}) = 0$ . As $z_{1}$ is a non-zero-divisor in the integral domain $_{2} O$ , this forces $a + b z_{2} = 0$ , i.e. $a = - b z_{2}$ . Thus every relation equals $b \cdot (z_{2}, - 1) \cdot (- 1)$ , that is a multiple of $(z_{2}, - 1)$ (absorbing the sign into $b$ ), so $(z_{2}, - 1)$ generates. The map $_{2} O \to R (z_{1}, z_{1} z_{2})$ , $b \mapsto (- b z_{2}, b)$ , is injective because its second coordinate recovers $b$ ; hence the relation module is free of rank one. This exhibits a finite (single) generating set for the relation sheaf, confirming the coherence conclusion of Oka's theorem in this instance, and shows the module is even free, not merely finitely generated. $□$

Connections Master

Analytic sets and local parametrisation 06.10.18. The branched-covering normal form and the finiteness of the local ring $_{V} O$ over a polydisc base, established there, are the explicit generators that Cartan's proof of ideal-sheaf coherence consumes; this unit promotes that stalkwise local-ring data into coherence of the sheaves $I ⟨ V ⟩$ and $O_{V}$ with neighbourhood uniformity.
Local analytic Nullstellensatz 06.10.17. The radical-ideal description of $I (V)_{p}$ supplied by the Nullstellensatz is what makes $I ⟨ V ⟩$ the correct reduced ideal sheaf whose coherence Cartan proves; without the radical normalisation the ideal sheaf would carry nilpotent ambiguity and the contrast with the algebraic reduced structure sheaf would blur.
Cartan Theorems A and B in $C^{n}$ 06.10.21. Coherence is the standing hypothesis on $F$ under which Theorem B vanishes $H^{q} (Ω, F)$ for $q \geq 1$ on a Stein open and Theorem A makes global sections generate every stalk; Oka's theorem and Cartan's ideal-sheaf coherence are precisely what guarantee the sheaves of interest fall under that hypothesis.
Complex spaces and coherence on them 06.10.22. The coherence of $O_{V}$ proved here is what upgrades a reduced analytic set into a complex space with a finiteness-respecting structure sheaf; the general coherence of the structure sheaf of a complex space, and Grauert's direct-image coherence theorem, build on this local model.
Algebraic coherent sheaf 04.06.02. The same finiteness is automatic on a Noetherian scheme and a theorem in the analytic category; the comparison is the analytic half of the GAGA dictionary, where coherent algebraic and analytic sheaves on a projective variety correspond.

Historical & philosophical context Master

The coherence of the sheaf of holomorphic functions was proved by Kiyoshi Oka in the seventh of his memoirs on analytic functions of several variables, published in 1950 in the Bulletin de la Société Mathématique de France, where he introduced the idéaux de domaines indéterminés that encode the relation sheaf in pre-sheaf-theoretic language ^{[Oka 1950]}. In the same volume Henri Cartan recast Oka's finiteness in the language of ideals and modules of analytic functions and proved the coherence of the ideal sheaf of an analytic set, completing the package that the structure sheaf of an analytic set is coherent ^{[Cartan 1950]}. The word coherent (faisceau cohérent) and the sheaf formalism were fixed in Cartan's Paris séminaire of the early 1950s, which turned Oka's analytic finiteness into the cohomological theory now called the Oka–Cartan theory.

Robert Gunning's third volume of lectures gives the homological development with the Weierstrass-division induction used above, and situates the analytic coherence theorem against its algebraic counterpart on Noetherian schemes ^{[Gunning Vol. III]}. The systematic modern treatment, including the stability calculus for the abelian category of coherent analytic sheaves, is the monograph of Hans Grauert and Reinhold Remmert ^{[Grauert–Remmert Ch. 4]}. The contrast with the algebraic category was made functorial by Serre's GAGA comparison of 1956, which identifies coherent algebraic and analytic sheaves on a projective complex variety.

Bibliography Master

@article{Oka1950,
  author  = {Oka, Kiyoshi},
  title   = {Sur les fonctions analytiques de plusieurs variables, VII: Sur quelques notions arithm{\'e}tiques},
  journal = {Bull. Soc. Math. France},
  volume  = {78},
  year    = {1950},
  pages   = {1--27}
}

@article{Cartan1950,
  author  = {Cartan, Henri},
  title   = {Id{\'e}aux et modules de fonctions analytiques de variables complexes},
  journal = {Bull. Soc. Math. France},
  volume  = {78},
  year    = {1950},
  pages   = {29--64}
}

@book{GunningVolIII,
  author    = {Gunning, Robert C.},
  title     = {Introduction to Holomorphic Functions of Several Variables, Volume III: Homological 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}
}

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

@book{HormanderSCV,
  author    = {H{\"o}rmander, Lars},
  title     = {An Introduction to Complex Analysis in Several Variables},
  edition   = {3rd},
  publisher = {North-Holland},
  year      = {1990}
}

@article{Serre1956GAGA,
  author  = {Serre, Jean-Pierre},
  title   = {G{\'e}om{\'e}trie alg{\'e}brique et g{\'e}om{\'e}trie analytique},
  journal = {Ann. Inst. Fourier},
  volume  = {6},
  year    = {1956},
  pages   = {1--42}
}

Prerequisites

06.10.18
06.10.14

Tier anchors

beginner: Gunning *Introduction to Holomorphic Functions of Several Variables* Vol. III (informal opening on sheaves of relations); Grauert–Remmert *Coherent Analytic Sheaves* intro
intermediate: Gunning–Rossi *Analytic Functions of Several Complex Variables* Ch. IV; Hörmander *An Introduction to Complex Analysis in Several Variables* §7.5
master: Gunning *Introduction to Holomorphic Functions of Several Variables* Vol. III §J; Grauert–Remmert *Coherent Analytic Sheaves* §2–4; Cartan séminaire 1951–52

References

Gunning — Introduction to Holomorphic Functions of Several Variables, Vol. III: Homological Theory · Wadsworth & Brooks/Cole 1990, §J (coherent analytic sheaves; Oka's theorem; ideal sheaf of an analytic set)
Oka — Sur les fonctions analytiques de plusieurs variables, VII: Sur quelques notions arithmétiques · Bull. Soc. Math. France 78 (1950), 1–27 (the coherence of the sheaf of relations; idéaux de domaines indéterminés)
Cartan — Idéaux et modules de fonctions analytiques de variables complexes · Bull. Soc. Math. France 78 (1950), 29–64 (coherence of the ideal sheaf of an analytic set)
Grauert–Remmert — Coherent Analytic Sheaves · Springer Grundlehren 265, 1984, Ch. 2 (sheaves of modules, coherence) and Ch. 4 (Oka coherence)
Hörmander — An Introduction to Complex Analysis in Several Variables · North-Holland 1990 (3rd ed.), §7.5 (coherent analytic sheaves; Oka's theorem via Weierstrass division)

Estimated time

beginner: 16m
intermediate: 42m
master: 75m