Complex spaces and coherence on them

Intuition Beginner

A single curve or surface cut out by holomorphic equations is a local thing: it lives inside one polydisc, defined by a finite list of equations. But the spaces geometers actually care about — a whole compact surface, a quotient by a group, the place where some moduli problem lives — are glued together from many such local pieces. A complex space is what you get when you carefully glue analytic-set pieces along overlaps, the same way a globe is built from flat map charts.

Two pieces of bookkeeping ride along with each chart. First, the points of the piece. Second, and just as important, the holomorphic functions that live on it. You cannot keep only the points: the same point set can come from genuinely different equations, and the functions remember the difference.

The aim of this unit is to make "glued from analytic pieces, with their functions" into a precise object, and to record the one structural fact that makes the whole theory work: the functions form a well-behaved, finitely-controlled system.

Visual Beginner

Picture a shape covered by two round patches. On each patch the shape is an honest analytic set — say a curve in a small disc, drawn by its equation. Where the two patches overlap, the two descriptions must agree: a point seen from the left patch is the same point seen from the right, and a holomorphic function written in the left chart matches the one written in the right.

Now add a second layer of information to each patch: a little tag at every point recording which functions count as holomorphic there. Most of the time the tag is the plain ring of convergent power series. But at a special point the tag can be fatter — it can carry a "ghost" direction that no honest function sees but the algebra still tracks. That fattening is what the picture below marks with a doubled outline at one point.

Worked example Beginner

Take the simplest fattening. In the line $\mathbb{C}$ with coordinate $z$, look at the single point $z=0$ described two different ways. The honest way: the equation $z=0$, whose functions on the point are just the numbers. The fat way: the equation $z\times z=0$, the double point.

On the double point a function is a convergent series in $z$ read modulo $z\times z=0$, so only the constant term and the first $z$-term survive: a function looks like $a+bz$ with two numbers $a$ and $b$. The number $a$ is the value at the point. The number $b$ is a ghost: the function $z$ itself is not the zero function on the fat point, yet $z\times z$ is. So $z$ is a nonzero thing whose square vanishes.

Compare the honest point $z=0$: there a function is just its value $a$, with no ghost.

What this tells us: the same single point carries two different rings of functions. Keeping the points alone would lose the ghost $b$, which is exactly the extra structure a complex space is built to remember.

Check your understanding Beginner

Formal definition Intermediate+

A ringed space is a pair $(X,{\mathcal{O}}_X)$ with $X$ a topological space and ${\mathcal{O}}_X$ a sheaf of rings on $X$, the structure sheaf. A morphism $(f,f^{♯}):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ is a continuous map $f:X\to Y$ together with a sheaf homomorphism $f^{♯}:{\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$. The ringed space is locally ringed when every stalk ${\mathcal{O}}_{X,p}$ is a local ring; the morphism is local when each induced stalk map $f_p^{♯}:{\mathcal{O}}_{Y,f(p)}\to {\mathcal{O}}_{X,p}$ sends the maximal ideal into the maximal ideal. Here all rings are $\mathbb{C}$-algebras and all morphisms are $\mathbb{C}$-algebra morphisms.

The local model is built from data inside a polydisc. Let $\Delta \subseteq {\mathbb{C}}^n$ be an open polydisc, ${\mathcal{O}}_{\Delta }={}_n\mathcal{O}{|}_{\Delta }$ the sheaf of holomorphic functions, and $\mathcal{I}\subseteq {\mathcal{O}}_{\Delta }$ a coherent ideal sheaf 06.10.20. Let $V=\mathrm{supp}({\mathcal{O}}_{\Delta }/\mathcal{I})$ be the analytic set it cuts out, $\iota :V\hookrightarrow \Delta$ the inclusion. The model space is $$ (V, \mathcal{O}V), \qquad \mathcal{O}V := \iota^{-1}(\mathcal{O}\Delta/\mathcal{I}), $$ a locally ringed $\mathbb{C}$-space on the point set $V$ whose stalk at $p$ is the analytic local ring $\mathcal{O}{V,p} = {}_n\mathcal{O}_p/\mathcal{I}_p$.When$\mathcal{I}$isa\ast radical\ast idealsheafthemodelis\ast reduced\ast and$\mathcal{O}_V$isthesheafofgermsofcontinuousfunctionson$V$obtainedbyrestrictingambientholomorphicgerms[\mathrm{06.10.19}];when$\mathcal{I}$isnotradicalthemodelis\ast non-reduced\ast and$\mathcal{O}_V$ carries nilpotents.

A complex space is a locally ringed $\mathbb{C}$-space $(X,{\mathcal{O}}_X)$ with $X$ Hausdorff such that every point has an open neighbourhood $U$ for which $(U,{\mathcal{O}}_X{|}_U)$ is isomorphic, as a locally ringed $\mathbb{C}$-space, to a model space $(V,{\mathcal{O}}_V)$. A morphism of complex spaces is a morphism of locally ringed $\mathbb{C}$-spaces; it is also called a holomorphic map. The reduction $X_{\mathrm{red}}=(X,{\mathcal{O}}_X/\mathcal{N})$, where $\mathcal{N}\subseteq {\mathcal{O}}_X$ is the nilradical sheaf of locally nilpotent sections, is a reduced complex space with the same point set; $X$ is reduced exactly when $\mathcal{N}=0$.

Counterexamples to common slips

• A reduced complex space is not the same as a complex manifold: the cusp and node of 06.10.19 are reduced complex spaces with singular points, yet ${\mathcal{O}}_X$ has no nilpotents there.
• The point set does not determine ${\mathcal{O}}_X$: the simple point $\{z=0\}$ and the double point $\{z^2=0\}$ have equal supports but ${\mathcal{O}}_{X,0}=\mathbb{C}$ versus $\mathbb{C}[z]/(z^2)$. Non-reducedness is invisible to the topology.
• A holomorphic map is not determined by its effect on points: on a non-reduced space, two distinct maps can agree on supports and differ on the nilpotent part of ${\mathcal{O}}_X$.

Key theorem with proof Intermediate+

Theorem (coherence of the structure sheaf — Oka). Let $(X,{\mathcal{O}}_X)$ be a complex space. Then ${\mathcal{O}}_X$ is a coherent sheaf of rings: for every open $U\subseteq X$ and every homomorphism $\phi :{\mathcal{O}}_X^p{|}_U\to {\mathcal{O}}_X{|}_U$ of ${\mathcal{O}}_X$-modules, the kernel sheaf $\ker \phi$ is locally finitely generated. Consequently every finitely generated ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$, every coherent ideal sheaf, and every quotient ${\mathcal{O}}_X/\mathcal{I}$ by such an ideal is a coherent ${\mathcal{O}}_X$-module. ^{[Grauert–Remmert]}

Proof. Coherence is a local property, so it suffices to verify it on a model space $(V,{\mathcal{O}}_V)$ with ${\mathcal{O}}_V={\iota }^{-1}({\mathcal{O}}_{\Delta }/\mathcal{I})$, $\mathcal{I}\subseteq {\mathcal{O}}_{\Delta }$ coherent. The ambient sheaf ${\mathcal{O}}_{\Delta }={}_n\mathcal{O}{|}_{\Delta }$ is coherent: this is Oka's coherence theorem 06.10.20, whose content is that the sheaf of relations among finitely many holomorphic germs is itself finitely generated, proved by Weierstrass division reducing the number of variables one at a time 06.10.14.

A coherent ideal $\mathcal{I}$ in a coherent ring sheaf has coherent quotient ${\mathcal{O}}_{\Delta }/\mathcal{I}$, because coherence is stable under cokernels of maps between coherent modules. Pushing forward along the closed embedding $\iota :V\hookrightarrow \Delta$ preserves coherence: $\iota$ is finite, and ${\iota }_{\ast }{\mathcal{O}}_V={\mathcal{O}}_{\Delta }/\mathcal{I}$ as ${\mathcal{O}}_{\Delta }$-modules. Thus ${\mathcal{O}}_V$ is coherent as a sheaf of ${\mathcal{O}}_V$-modules, since the property of having finitely generated relation sheaves is inherited under the change of rings ${\mathcal{O}}_{\Delta }\to {\mathcal{O}}_{\Delta }/\mathcal{I}$: a relation among sections of ${\mathcal{O}}_V$ lifts to a relation among representatives in ${\mathcal{O}}_{\Delta }$ modulo $\mathcal{I}$, and finite generation of the ambient relation module together with finite generation of $\mathcal{I}$ yields finite generation downstairs.

Given $\phi :{\mathcal{O}}_V^p{|}_U\to {\mathcal{O}}_V{|}_U$, lift it to $\tilde{\phi }:{\mathcal{O}}_{\Delta }^p\to {\mathcal{O}}_{\Delta }/\mathcal{I}$ near a point; the kernel of $\phi$ is the image in ${\mathcal{O}}_V^p$ of the kernel of the composite ${\mathcal{O}}_{\Delta }^p\to {\mathcal{O}}_{\Delta }/\mathcal{I}$, which is finitely generated by coherence of ${\mathcal{O}}_{\Delta }/\mathcal{I}$. Hence $\ker \phi$ is locally finitely generated, establishing coherence of ${\mathcal{O}}_V$, and the stated consequences are the standard coherence closure properties applied to ideals and their quotients. $\square$

Bridge. Coherence is the foundational reason a complex space can be studied by sheaf cohomology at all: it is exactly the finiteness that lets local data be assembled globally, and it builds toward the Cartan vanishing theorems that this enables. The result appears again in the normalisation construction below, where the integral closure of ${\mathcal{O}}_X$ is shown to be a coherent ${\mathcal{O}}_X$-module so that it actually defines a complex space. The central insight is that coherence of ${\mathcal{O}}_X$ generalises Oka's theorem on ${\mathbb{C}}^n$ from the ambient sheaf to every analytic model, and the bridge is the finite closed embedding $\iota$, which transports coherence from polydisc to model. This is exactly the analytic shadow of the scheme-theoretic fact that a finitely presented ideal on an affine scheme has coherent quotient; the complex-analytic statement is dual to the algebraic one and putting these together identifies coherence with the local finite-presentation property that both geometries share.

Exercises Intermediate+

Advanced results Master

The structure sheaf of a complex space carries the entire local analytic geometry, and coherence is the finiteness that turns that local data into global theorems. The results below record how reducedness, normality, and the analytic spectrum organise the local model, and how the Stein condition globalises Cartan A and B.

The analytic spectrum viewpoint. A model space is recovered from algebra alone. Given a finitely generated analytic $\mathbb{C}$-algebra $A={}_n\mathcal{O}/\mathcal{I}$ — a quotient of a convergent power-series ring — the analytic spectrum $\mathrm{Specan}A$ is the model space $(V,{\mathcal{O}}_V)$ with $V=\mathrm{supp}({\mathcal{O}}_{\Delta }/\mathcal{I})$ and ${\mathcal{O}}_{V,0}=A$. This is the analytic counterpart of $\mathrm{Spec}$: the functor $A\mapsto \mathrm{Specan}A$ from analytic algebras to germs of model spaces is an anti-equivalence onto its image, so morphisms of model germs are the same data as $\mathbb{C}$-algebra homomorphisms reversed. The non-reduced models correspond exactly to algebras with nilpotents, and the embedding dimension ${\dim }_{\mathbb{C}}\mathfrak{m}/{\mathfrak{m}}^2$ of 06.10.19 is read off the algebra. The global complex space is then a locally ringed space glued from these $\mathrm{Specan}$ charts, the analytic mirror of a scheme glued from affine $\mathrm{Spec}$ charts.

Normality and normalisation. A reduced complex space $X$ is normal at $p$ when ${\mathcal{O}}_{X,p}$ is integrally closed in its total quotient ring; $X$ is normal when it is normal at every point. Normal complex spaces are the well-behaved ones: by Oka's normality criterion their singular locus has codimension at least two, holomorphic functions extend across analytic sets of codimension at least two (a Hartogs-type continuation), and the space is locally irreducible. Every reduced complex space $X$ admits a normalisation: a finite surjective holomorphic map $\nu :\tilde{X}\to X$ with $\tilde{X}$ normal, $\nu$ biholomorphic over the normal locus, and ${\nu }_{\ast }{\mathcal{O}}_{\tilde{X}}$ equal to the integral closure $\overline{{\mathcal{O}}_X}$ of ${\mathcal{O}}_X$ in its sheaf of meromorphic functions. The existence rests on the fact that $\overline{{\mathcal{O}}_X}$ is a coherent ${\mathcal{O}}_X$-module — the Oka–Grauert coherence of the normalisation sheaf — so that $\mathrm{Specan}$ of it is again a complex space. For the cusp, $\nu$ replaces $\mathbb{C}\{t^2,t^3\}$ by $\mathbb{C}\{t\}$ and unpinches the singular point to a smooth disc, exactly as forecast at 06.10.19.

Cartan A and B on Stein spaces. A complex space $X$ is Stein when it is holomorphically separable, holomorphically convex, and holomorphic functions give local coordinates at each point — equivalently, $X$ is biholomorphic to a closed complex subspace of some ${\mathbb{C}}^N$. On a Stein complex space $X$ and for every coherent analytic sheaf $\mathcal{F}$, Cartan's Theorem A states that $\Gamma (X,\mathcal{F})$ generates each stalk ${\mathcal{F}}_p$ over ${\mathcal{O}}_{X,p}$, and Theorem B states $H^q(X,\mathcal{F})=0$ for all $q\ge 1$ ^{[Grauert–Remmert Theory of Stein Spaces]}. These were first proved on Stein manifolds and domains in ${\mathbb{C}}^n$ 06.10.21; the passage to singular Stein spaces is what coherence of ${\mathcal{O}}_X$ makes possible, because the proofs run by coherent resolutions ${\mathcal{O}}_X^p\to \mathcal{F}$ that exist precisely because ${\mathcal{O}}_X$ is coherent. Cousin I and II, the Levi problem, the Oka principle for holomorphic vector bundles, and the solvability of $\bar{\partial }$ with coherent coefficients all follow as corollaries.

Grauert's direct-image theorem. Coherence is preserved by proper holomorphic maps: if $f:X\to Y$ is a proper morphism of complex spaces and $\mathcal{F}$ is a coherent sheaf on $X$, then every higher direct image $R^q{f}_{\ast }\mathcal{F}$ is coherent on $Y$ ^{[Grauert 1960]}. This is the analytic counterpart of Grothendieck's finiteness for proper morphisms of schemes, and it is the engine behind the finiteness of cohomology of coherent sheaves on compact complex spaces (the case $Y=\{\text{pt}\}$), itself the gateway to the analytic Riemann–Roch and to the construction of moduli spaces.

Synthesis. The structure sheaf ${\mathcal{O}}_X$ is the foundational reason a complex space is a single coherent object rather than a heap of charts: coherence is exactly the finiteness that lets local generators glue, and this is dual to the algebraic finite-presentation property that defines coherent sheaves on schemes. The analytic spectrum identifies a model germ with its analytic local algebra, so the reduced/non-reduced split is dual to the presence or absence of nilpotents, and the regular/singular split of 06.10.19 generalises to the normality of ${\mathcal{O}}_{X,p}$. Putting these together, normalisation is the global realisation of taking integral closure stalkwise, and it is coherent precisely because ${\mathcal{O}}_X$ is — the central insight that the geometry of resolving singularities is the algebra of integral closure made into a complex space. Cartan A and B then globalise Oka's coherence to Stein spaces: the vanishing $H^q(X,\mathcal{F})=0$ is the statement that on a Stein space coherence has no cohomological obstruction, and Grauert's direct-image theorem is dual to this, propagating coherence forward under proper maps. This pattern recurs wherever a local analytic algebra is globalised, and it builds toward the cohomological theory of compact complex spaces and the analytic side of GAGA.

Full proof set Master

Proposition 1 (reduction is a reduced complex space). Let $(X,{\mathcal{O}}_X)$ be a complex space with nilradical sheaf $\mathcal{N}\subseteq {\mathcal{O}}_X$. Then $\mathcal{N}$ is a coherent ideal sheaf and $X_{\mathrm{red}}=(X,{\mathcal{O}}_X/\mathcal{N})$ is a reduced complex space with $(X_{\mathrm{red}}{)}_{\mathrm{red}}=X_{\mathrm{red}}$.

Proof. Coherence is local, so work on a model $(V,{\mathcal{O}}_V)$ with ${\mathcal{O}}_V={\iota }^{-1}({\mathcal{O}}_{\Delta }/\mathcal{I})$. The nilradical of ${\mathcal{O}}_V$ corresponds to $\sqrt{\mathcal{I}}/\mathcal{I}$, where $\sqrt{\mathcal{I}}$ is the radical ideal sheaf. By the analytic Nullstellensatz 06.10.19 and the coherence of the radical of a coherent ideal (a theorem of Cartan: the radical of a coherent ideal sheaf is coherent), $\sqrt{\mathcal{I}}$ is coherent; hence the quotient $\sqrt{\mathcal{I}}/\mathcal{I}$ is coherent, and this is $\mathcal{N}$ on the model. Thus $\mathcal{N}$ is coherent on all of $X$. The quotient ${\mathcal{O}}_X/\mathcal{N}$ is locally ${\mathcal{O}}_{\Delta }/\sqrt{\mathcal{I}}$, the structure sheaf of the reduced model with radical ideal, so $X_{\mathrm{red}}$ is again locally a model space, hence a complex space, and it is reduced because $\sqrt{\mathcal{I}}$ is its own radical. Idempotence of reduction is then immediate from $\sqrt{\sqrt{\mathcal{I}}}=\sqrt{\mathcal{I}}$. $\square$

Proposition 2 (universal property of the reduction). For every reduced complex space $Z$ and morphism $f:Z\to X$, there is a unique morphism $\bar{f}:Z\to X_{\mathrm{red}}$ with $f=j\circ \bar{f}$, where $j:X_{\mathrm{red}}\hookrightarrow X$ is the canonical closed embedding.

Proof. The embedding $j$ has $j^{♯}:{\mathcal{O}}_X\twoheadrightarrow {\mathcal{O}}_X/\mathcal{N}$ the quotient. Given $f=(f_0,f^{♯})$ with $Z$ reduced, the pullback $f^{♯}:{\mathcal{O}}_X\to (f_0{)}_{\ast }{\mathcal{O}}_Z$ sends nilpotent sections to nilpotent sections, and ${\mathcal{O}}_Z$ has no nonzero nilpotents, so $f^{♯}(\mathcal{N})=0$. Hence $f^{♯}$ factors uniquely through ${\mathcal{O}}_X/\mathcal{N}$, defining ${\bar{f}}^{♯}:{\mathcal{O}}_X/\mathcal{N}\to (f_0{)}_{\ast }{\mathcal{O}}_Z$ with the same continuous map $f_0$ (whose image lies in the common support). This $\bar{f}=(f_0,{\bar{f}}^{♯})$ is the required factorisation, and uniqueness is forced because $j^{♯}$ is surjective, so a factoring ${\bar{f}}^{♯}$ is determined by $f^{♯}$. $\square$

Proposition 3 (coherence of the integral closure / normalisation sheaf). Let $X$ be a reduced complex space. The integral closure $\overline{{\mathcal{O}}_X}$ of ${\mathcal{O}}_X$ in its sheaf ${\mathcal{M}}_X$ of meromorphic functions is a coherent ${\mathcal{O}}_X$-module, and $\mathrm{Specan}$ of it is a normal complex space $\tilde{X}$ with a finite morphism $\nu :\tilde{X}\to X$.

Proof. The statement is local, so take a reduced model germ with ${\mathcal{O}}_{X,p}$ a reduced analytic local ring of dimension $d$. By the local parametrisation theorem 06.10.19, ${\mathcal{O}}_{X,p}$ is a finite module over a subalgebra ${}_d\mathcal{O}\cong \mathbb{C}\{w_1,\dots ,w_d\}$, which is a normal (regular) domain. The total quotient ring of ${\mathcal{O}}_{X,p}$ is a finite product of finite field extensions of $\mathrm{Frac}({}_d\mathcal{O})$, one per irreducible component. The integral closure $\overline{{\mathcal{O}}_{X,p}}$ is the integral closure of ${}_d\mathcal{O}$ in that total quotient ring; by the finiteness of integral closure for an excellent normal domain in a finite separable extension (the analytic Noether normalisation makes the extension finite, and characteristic zero makes it separable), $\overline{{\mathcal{O}}_{X,p}}$ is a finite ${}_d\mathcal{O}$-module, hence a finite ${\mathcal{O}}_{X,p}$-module. Finiteness over the coherent base sheaf upgrades to coherence of $\overline{{\mathcal{O}}_X}$ as an ${\mathcal{O}}_X$-module (Grauert–Remmert). Therefore $\overline{{\mathcal{O}}_X}$ is a coherent sheaf of ${\mathcal{O}}_X$-algebras, and $\tilde{X}={\mathrm{Specan}}_X\overline{{\mathcal{O}}_X}$ is a complex space; its stalks are integrally closed by construction, so $\tilde{X}$ is normal, and $\nu$ is finite because $\overline{{\mathcal{O}}_X}$ is ${\mathcal{O}}_X$-finite. $\square$

Proposition 4 (Theorem B implies Cousin I on a Stein space). Let $X$ be a Stein complex space. Then the additive Cousin problem is solvable: given an open cover $\{U_i\}$ and meromorphic data $\{(U_i,m_i)\}$ with $m_i-m_j$ holomorphic on $U_i\cap U_j$, there is a global meromorphic $m$ with $m-m_i$ holomorphic on $U_i$.

Proof. The differences $m_i-m_j\in {\mathcal{O}}_X(U_i\cap U_j)$ form a $1$-cocycle for the cover $\{U_i\}$ valued in the coherent sheaf ${\mathcal{O}}_X$, representing a class in $H^1(X,{\mathcal{O}}_X)$. By Cartan's Theorem B on the Stein space $X$, $H^1(X,{\mathcal{O}}_X)=0$, so the cocycle is a coboundary: there exist $h_i\in {\mathcal{O}}_X(U_i)$ with $m_i-m_j=h_i-h_j$ on overlaps. Then $m:=m_i-h_i$ is independent of $i$ on overlaps, so it glues to a global meromorphic function, and $m-m_i=-h_i$ is holomorphic on $U_i$, solving the problem. $\square$

Proposition 5 (finiteness on a compact complex space). Let $X$ be a compact complex space and $\mathcal{F}$ a coherent sheaf on $X$. Then ${\dim }_{\mathbb{C}}{H}^q(X,\mathcal{F})<\infty$ for every $q\ge 0$.

Proof. Apply Grauert's direct-image theorem to the constant map $f:X\to \{\mathrm{pt}\}$, which is proper because $X$ is compact. The higher direct images $R^q{f}_{\ast }\mathcal{F}$ are coherent sheaves on the one-point complex space $\{\mathrm{pt}\}$, whose coherent sheaves are finite-dimensional $\mathbb{C}$-vector spaces. Since $R^q{f}_{\ast }\mathcal{F}=H^q(X,\mathcal{F})$ for the map to a point, each cohomology group is a finite-dimensional $\mathbb{C}$-vector space. $\square$

Connections Master

• The local ring of an analytic set; regular and singular points 06.10.19. The stalk ${\mathcal{O}}_{X,p}$ of a complex space is exactly the analytic local ring studied there; smooth points of $X$ are where ${\mathcal{O}}_{X,p}$ is a power-series ring, and the non-normal singularities detected by the embedding-dimension excess are the points the normalisation $\nu :\tilde{X}\to X$ of this unit repairs. The cusp computation is the shared worked example.

• Coherent analytic sheaves and Oka's coherence theorem 06.10.20. Oka coherence of ${\mathcal{O}}_{\Delta }$ on a polydisc is the input that this unit transports across the finite embedding $\iota :V\hookrightarrow \Delta$ to prove ${\mathcal{O}}_X$ coherent on every model; the coherent ideal sheaves of that unit are precisely the data cutting out the local models, and coherence of the normalisation sheaf is what makes $\nu$ exist.

• Cartan Theorems A and B in ${\mathbb{C}}^n$ 06.10.21. The Stein vanishing theorems proved there for domains and manifolds are globalised here to singular Stein complex spaces, the passage being made possible by coherence of ${\mathcal{O}}_X$; Cousin I/II and the Levi problem reappear as corollaries on the singular Stein spaces this unit introduces.

• Scheme 04.02.01. A complex space is the analytic counterpart of a scheme: both are locally ringed spaces glued from local models — analytic-set germs $\mathrm{Specan}A$ here versus affine spectra $\mathrm{Spec}R$ there — and the reduced/non-reduced distinction by the nilradical sheaf is identical, with GAGA linking the two over a complex projective base.

• Coherent sheaf 04.06.02. Coherence of ${\mathcal{O}}_X$ and of ideal and normalisation sheaves on a complex space is the analytic instance of the same finite-presentation notion; the cohomological consequences — finiteness on compact $X$, vanishing on Stein $X$ — mirror the projective and affine theorems on schemes.

Historical & philosophical context Master

The notion of a complex space crystallised in the early 1950s out of Henri Cartan's seminar at the École Normale Supérieure, where the language of sheaves, introduced by Jean Leray and developed by Cartan and Jean-Pierre Serre, was first turned on the local algebra of holomorphic functions ^{[Cartan 1951]}. Cartan's exposés on idéaux de fonctions holomorphes and faisceaux analytiques cohérents established that the structure sheaf of an analytic set is coherent and proved Theorems A and B for domains of holomorphy, recasting the Oka–Cartan solution of the Cousin and Levi problems as cohomology-vanishing statements. The coherence of the structure sheaf is Oka's theorem, published by Kiyoshi Oka in his series Sur les fonctions analytiques de plusieurs variables; Cartan's contribution was to read it as the finiteness condition that makes sheaf cohomology computable.

Hans Grauert and Reinhold Remmert systematised the resulting theory in Coherent Analytic Sheaves and Theory of Stein Spaces ^{[Grauert–Remmert]}, where complex spaces with non-reduced structure sheaves, the normalisation by integral closure, and Cartan A and B on singular Stein spaces receive their definitive treatment. Grauert's 1960 paper in the Publications mathématiques de l'IHÉS proved that higher direct images of coherent sheaves under proper holomorphic maps are coherent ^{[Grauert 1960]}, the analytic finiteness theorem that underwrites the cohomology of compact complex spaces. Serre's 1956 GAGA paper then identified, over a complex projective base, the analytic and algebraic coherent-sheaf categories, joining the complex-space theory of this unit to the scheme theory of 04.02.01.

Bibliography Master

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  author    = {Cartan, Henri},
  title     = {S{\'e}minaire Henri Cartan, {\'E}cole Normale Sup{\'e}rieure: Fonctions analytiques de plusieurs variables complexes},
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}

@article{Grauert1960Direct,
  author  = {Grauert, Hans},
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  year    = {1960},
  pages   = {5--64}
}

@book{GrauertRemmertCAS,
  author    = {Grauert, Hans and Remmert, Reinhold},
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  volume    = {265},
  publisher = {Springer},
  year      = {1984}
}

@book{GrauertRemmertStein,
  author    = {Grauert, Hans and Remmert, Reinhold},
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  publisher = {Springer},
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}

@book{GunningSCVIII,
  author    = {Gunning, Robert C.},
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}

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

@article{Oka1950Coherence,
  author  = {Oka, Kiyoshi},
  title   = {Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithm{\'e}tiques},
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}