Serre's comparison theorems and Chow's theorem

Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Analytic prerequisite integration by GPT-6 Astra (OpenAI), Codex, Ultra. Independently authored material: CC0.

Locally, analytification preserves exact sequences but adds holomorphic functions. On a projective complex scheme, the three comparison theorems say much more: coherent cohomology agrees, every analytic morphism of coherent sheaves is algebraic, and every coherent analytic sheaf is algebraic. We prove these assertions in that order, including schemes with nilpotents. The proof of the last assertion needs a finite-dimensional analytic cohomology theorem; using the comparison theorem to assert finiteness for an arbitrary analytic sheaf at this stage would be circular.

Throughout, schemes are separated and of finite type over $\mathbf C$. A variety means a reduced such scheme; a complex space may have nilpotents. Analytification retains the full defining ideal, as constructed in Complex analytic spaces and analytification. Write $P=\mathbf P^n_{\mathbf C}$, $P^{\mathrm{an}}=\mathbf P^n(\mathbf C)$, and $\mathcal O(d)^{\mathrm{an}}$ for the analytification of a twist. We use the generation, vanishing and projective cohomology proved in Serre's theorems and Cohomology of projective space.

1. The analytic cohomology foundation

Two analytic theorems are required for the cohomology arguments below. Their programme proofs are in Complex analytic spaces and coherent sheaves. Read the vanishing and finiteness lessons before this lesson: neither obtains analytic cohomology from GAGA or from algebraization. Jean-Pierre Demailly's freely available Complex Analytic and Differential Geometry is further reading. The prerequisite and proof guide identifies the exact statements, proof locations and remaining dependencies.

Analytic vanishing theorem. If a complex manifold admits a smooth strictly plurisubharmonic exhaustion, every coherent analytic sheaf on it has zero cohomology in positive degrees. An exhaustion has relatively compact sublevel sets; strict plurisubharmonicity means its complex Hessian is positive definite. Theorems A and B on Stein manifolds, Theorem 4.1, supplies the proof through local coherent resolutions, approximation and exhaustion. The assertion applies to every coherent sheaf, not only vector bundles; it gives vanishing in every positive degree, not merely finite dimension.

Analytic finiteness theorem (Cartan–Serre). On a compact complex analytic space, the cohomology of every coherent analytic sheaf is finite-dimensional in every degree. Finiteness on compact complex spaces, Theorem 2.1, gives the proof using finite nested acyclic covers. Restriction of coherent sections from a larger open set to a relatively compact smaller one is a compact operator, by the coherent version of Montel's theorem. The induced map between their Čech complexes is an isomorphism on cohomology. The compact-operator criterion is developed in the preceding lesson, Fréchet spaces of sections and Schwartz's theorem. It yields finite dimension and closed coboundaries. These arguments apply to coherent analytic sheaves before any algebraization is known.

For our standard projective cover $U_i=\{T_i\ne0\}$, every nonempty intersection is biholomorphic to

(\mathbf C^*)^r\times\mathbf C^{n-r}.

The function

\psi(z)=\sum_{j=1}^n|z_j|^2+\sum_{j=1}^r|z_j|^{-2}

is a strictly plurisubharmonic exhaustion there. The first sum makes the Hessian positive, while the second prevents escape toward a deleted coordinate hyperplane. The vanishing theorem therefore makes this cover acyclic for every coherent analytic sheaf. By the Čech comparison proved earlier in the course, its ordered complex computes cohomology. There are $n+1$ opens, so

H^q(P^{\mathrm{an}},M)=0\qquad(q>n) \tag{1}

for every coherent analytic module $M$. This bound concerns all coherent analytic modules, including those not yet known to be algebraic.

2. Starting the comparison: the structure sheaf and twists

Lemma 2.1. The analytic structure sheaf of projective space has cohomology $\mathbf C$ in degree zero and zero in every positive degree. The natural comparison from algebraic cohomology is an isomorphism.

Proof. For $n=0$, the space is a point. For general $n$, use the acyclic cover of Section 1. Put $\Omega_S=\{T\in\mathbf C^{n+1}:T_i\ne0\text{ for }i\in S\}$. A function on $U_S=\bigcap_{i\in S}U_i$ pulls back to a homogeneous holomorphic function of weight zero on this product of punctured planes and planes. Conversely a weight-zero function descends by setting any $T_j=1$, $j\in S$; homogeneity makes this independent of the representative. Work directly on $\Omega_S$, without choosing projective ratios for the expansion. Laurent series and homogeneous projections, Theorems 1–4, proves the unique, normally convergent product expansion and its homogeneous decomposition. The allowed exponents are

\alpha\in\mathbf Z^{n+1},\qquad \sum_i\alpha_i=0,\qquad \alpha_j\geq0\quad(j\notin S).

Indeed uniqueness applied to $f(2T)=f(T)$ makes the coefficient of $T^\alpha$ vanish unless its total exponent is zero. Group the expansion by $N=\{j:\alpha_j<0\}$. There are only finitely many such sets. Theorem 3 of the Laurent unit proves that each part extends to the whole $\Omega_N$: on any compact subset, choose smaller positive contour radii in the negative-power coordinates and larger contour radii in the others. The product geometric estimate proves normal convergence there, including along the newly allowed zero coordinates. The extension still has weight zero. Uniqueness of coefficients on $(\mathbf C^*)^{n+1}$ makes every such projection compatible with restriction between chart intersections. Thus each fixed $N$ has one coefficient space, independent of the containing face $S$, rather than a separate choice for every chart.

For fixed $N$, the allowed faces of the Čech complex are exactly the nonempty $S$ containing $N$. If $N$ is proper and nonempty, choose $j\notin N$. Insert $j$ with the sign of its ordered position, extending the component on $\Omega_{S\cup\{j\}}$ first to $\Omega_N$, then restricting to $\Omega_S$. If $j$ is already in the target face, set the contraction to zero. Theorem 5 gives the signed formula and proves $dh+hd=1$, including degree zero. The term removing the inserted $j$ is the identity; every other pair differs by one in its insertion/removal sign and cancels. This is an operation on normally convergent holomorphic components, not only a formal monomial calculation.

If $N$ is empty, all exponents are nonnegative and sum to zero, so the only monomial is 1. This is the ordinary simplex complex of constants, with cohomology $\mathbf C$ in degree zero. If $N$ were all indices, their sum would be negative, so this part is absent. These contractions prove the analytic assertion. Algebraically the same cohomology was computed in our projective-space lesson, and comparison sends a constant to the identical constant. $\square$

For any algebraic module $F$, the map on global sections to those of $F^{\mathrm{an}}$ extends canonically to maps

c_F^q:H^q(X,F)\longrightarrow H^q(X^{\mathrm{an}},F^{\mathrm{an}}). \tag{2}

Indeed analytification is an exact functor on all modules, by local flatness. The target groups therefore form a cohomological delta functor, and the universal derived functors of algebraic global sections extend the degree-zero map uniquely. Consequently (2) is natural and commutes with connecting maps in long exact sequences. For coherent modules, it also commutes with closed-immersion pushforward, whose stalkwise quotient description was proved in the preceding lesson.

Lemma 2.2. Comparison is an isomorphism for $\mathcal O(d)$ on $\mathbf P^n$, for every integer $d$ and every degree.

Proof. Induct on $n$. On $\mathbf P^0$ every twist is a one-dimensional module, including negative twists. For $n>0$, let $i:E\hookrightarrow P$ be a hyperplane. Its equation gives

0\to\mathcal O(d-1)\to\mathcal O(d) \to i_*\mathcal O_E(d)\to0. \tag{3}

Analytification preserves this sequence. The induction hypothesis supplies comparison on $E$. The two long exact sequences show that comparison for $\mathcal O(d)$ in all degrees is equivalent to comparison for $\mathcal O(d-1)$: for either unknown term, take five successive terms with that term in the middle; all four other comparison maps are isomorphisms. Begin with $d=0$, proved in Lemma 2.1, and move upward and downward. $\square$

3. The first comparison theorem

Theorem 3.1 (cohomological GAGA). If $X$ is projective over $\mathbf C$ and $F$ is coherent, (2) is an isomorphism for every $q\geq0$.

Proof. Embed $X$ as a closed subscheme of $P$, with its full coherent defining ideal. Closed-immersion pushforward is exact, preserves coherence and cohomology, and commutes with analytification: locally a module over a quotient ring is viewed as a module over the ambient ring, and both analytic constructions tensor its presentation with the same flat analytic ring. None of these operations replaces the ideal by its radical. Thus it suffices to prove the result on $P$.

Descend on $q$, simultaneously for all coherent sheaves. For $q>n$, both groups vanish: algebraically by the standard affine cover, analytically by (1). Choose a presentation

0\to R\to L\to F\to0, \tag{4}

where $L$ is a finite sum of twists and $R$ is coherent. Suppose comparison in degree $q+1$ is already known for every coherent sheaf. Lemma 2.2 gives comparison for $L$ in every degree.

To prove surjectivity for $F$ in degree $q$, start with an analytic class. Its boundary in $H^{q+1}(R^{\mathrm{an}})$ lifts uniquely to an algebraic class. That lift maps to zero in $H^{q+1}(L)$, since its analytic image does and comparison for $L$ is injective. Exactness lifts it to $H^q(F)$. The difference between this class's analytic image and the original class has zero boundary, so comes from $H^q(L^{\mathrm{an}})$. Comparison for $L$ lifts the difference, establishing surjectivity.

This proves surjectivity in degree $q$ for every coherent sheaf, in particular for $R$. If an algebraic class in $H^q(F)$ has zero analytic image, its boundary is zero by injectivity in degree $q+1$ for $R$, so it comes from $H^q(L)$. That lift maps analytically into the image of $H^q(R^{\mathrm{an}})$. Surjectivity for $R$ lifts this correcting class. Subtract it in $H^q(L)$; comparison for $L$ forces the corrected lift to be zero. Thus the original class was zero. The induction proves injectivity and surjectivity in all degrees. $\square$

The simultaneous induction is essential: we first establish surjectivity for all coherent sheaves before using it for the kernel $R$. No finite global locally free resolution of $F$ has been assumed.

4. The second comparison theorem

Theorem 4.1 (full faithfulness). For coherent $F,G$ on a projective complex scheme,

\operatorname{Hom}_X(F,G) \xrightarrow{\sim} \operatorname{Hom}_{X^{\mathrm{an}}}(F^{\mathrm{an}},G^{\mathrm{an}}).

Proof. The internal Hom $A=\mathcal Hom(F,G)$ is coherent. Finite presentations and local flatness give

A^{\mathrm{an}}\cong\mathcal Hom(F^{\mathrm{an}},G^{\mathrm{an}}),

as proved in Proposition 5.1 of the preceding lesson. Apply Theorem 3.1 in degree zero to $A$. These global sections are exactly the two morphism groups, and the comparison takes a morphism to its analytification. $\square$

In particular, an analytic isomorphism between algebraic coherent sheaves lifts uniquely to an algebraic isomorphism: lift it and its inverse, then use faithfulness to identify their compositions with the identities.

5. The third comparison theorem

Lemma 5.1 (analytic generation). Every coherent analytic module $M$ on $P^{\mathrm{an}}$ has $M(d)$ generated by finitely many global sections for all sufficiently large $d$.

Proof. We prove this together with essential surjectivity, inducting on $n$. At $n=0$ a coherent module is a finite-dimensional vector space. Assume essential surjectivity on every hyperplane $E\cong\mathbf P^{n-1}$. On $E$, Theorem 3.1 and algebraic Serre vanishing now imply vanishing of all positive cohomology of sufficiently large twists of every coherent analytic module.

Fix $x\in P^{\mathrm{an}}$ and choose a hyperplane through $x$, with equation $t$. Multiplication has coherent kernel and cokernel:

0\to C\to M(-1)\xrightarrow{t}M\to B\to0.

Both $C$ and $B$ are annihilated by $t$, so are coherent modules on $E^{\mathrm{an}}$. This remains true when multiplication by $t$ is not injective. Let $L_d$ be the image of $M(d-1)\to M(d)$. The two short exact sequences give, for large $d$, surjections

H^1(M(d-1))\twoheadrightarrow H^1(L_d) \twoheadrightarrow H^1(M(d)), \tag{5}

because $H^2(C(d))=H^1(B(d))=0$. By the Cartan–Serre finiteness theorem in Section 1, these dimensions are finite. Thus the nonnegative integers $h^1(M(d))$ eventually stop decreasing. In the stable range both arrows in (5) are isomorphisms. The second short exact sequence then shows that

H^0(M(d))\twoheadrightarrow H^0(B(d)). \tag{6}

By induction $B$ is algebraic on $E$; its large twists are generated by global sections. Lift generators using (6). At $x$, these generate $M(d)_x/tM(d)_x$; Nakayama makes them generate $M(d)_x$, since $t$ belongs to the local maximal ideal.

Generation by finitely many fixed sections holds on a neighbourhood, because their coherent cokernel vanishes there. Once generation holds at a point in degree $d$, multiplying these sections by a homogeneous coordinate nonzero at that point proves generation in every degree $d'\geq d$. Compactness supplies finitely many neighbourhoods and a common bound on $d$. At each such degree take the union of their finite generating families. This proves the lemma. $\square$

Theorem 5.2 (essential surjectivity). Every coherent analytic module on $X^{\mathrm{an}}$, for projective $X$, is isomorphic to $F^{\mathrm{an}}$ for some coherent algebraic $F$. Together with Theorem 4.1, analytification is an equivalence of coherent categories.

Proof. First take $X=P$, continuing the induction of Lemma 5.1. Global generation gives a surjection $L_0^{\mathrm{an}}\to M$, where $L_0$ is a finite sum of equal negative twists. Its kernel $R$ is coherent. Generate a twist of $R$ as well, to obtain

L_1^{\mathrm{an}}\xrightarrow{v}L_0^{\mathrm{an}}\to M\to0.

Full faithfulness lifts $v$ uniquely to an algebraic map $u:L_1\to L_0$. Put $F=\operatorname{coker}u$. Exactness of analytification gives $F^{\mathrm{an}}\cong M$, completing the induction on $n$.

For $i:X\hookrightarrow P$, algebraize $i_*M$ to $G$ on $P$. If $I$ is the ideal of $X$, then $(IG)^{\mathrm{an}}=I^{\mathrm{an}}G^{\mathrm{an}}=0$. Exactness and faithfulness give $IG=0$. Hence $G=i_*F$ for a unique coherent module $F$ on $X$. Compatibility with closed-immersion pushforward identifies $F^{\mathrm{an}}$ with $M$. $\square$

The correct uniqueness assertion fixes the identification. If $(F,\phi)$ and $(G,\psi)$ are algebraizations equipped with isomorphisms to $M$, there is a unique isomorphism $u:F\to G$ satisfying $\psi\circ u^{\mathrm{an}}=\phi$. Without specified identifications, an algebraic sheaf can have many automorphisms: every nonzero scalar acts on $\mathcal O_X$. Thus its bare isomorphism class is unique, but its possible isomorphisms are not.

6. Chow's theorem and geometric consequences

Theorem 6.1 (Chow). Every closed analytic subset of a projective complex scheme is the analytification of a unique reduced closed algebraic subscheme.

Proof. Its analytic vanishing ideal $J\subset\mathcal O_{X^{\mathrm{an}}}$ is coherent by Cartan coherence. Algebraize $J$ as a coherent module by Theorem 5.2, and lift its inclusion into the structure sheaf by Theorem 4.1. Exactness and faithfulness make the lift injective, with image an algebraic coherent ideal $I$ whose analytification is $J$.

The ideal $I$ is radical. At every closed stalk, if $a^m\in I_x$, the image of $a$ belongs to the radical ideal $J_x=I_x\mathcal O_{X^{\mathrm{an}},x}$; faithful flatness contracts this to $a\in I_x$. Radicality at closed stalks suffices: a nonzero nilpotent ideal in the coherent quotient would have support containing a closed point. The reduced closed scheme defined by $I$ therefore analytifies to the given analytic subset. If two algebraic ideals have the same analytic extension, faithful flatness gives equal closed stalks, and coherent support detects equality globally. This proves uniqueness. $\square$

Theorem 6.2. Every holomorphic morphism between the analytifications of projective complex schemes is the analytification of a unique algebraic morphism.

Proof. Analytification commutes with products, by the polynomial chart and quotient constructions in the preceding lesson. The graph is a closed complex subspace of $(X\times Y)^{\mathrm{an}}$: locally its ideal is generated by the target coordinate functions minus their holomorphic pullbacks, together with the target equations. Its structure sheaf is that of $X^{\mathrm{an}}$, including any nilpotents. Its ideal $J$ is coherent. Theorem 5.2 algebraizes $J$, and Theorem 4.1 algebraizes its inclusion in the structure sheaf. Exactness and faithfulness make the image a coherent algebraic ideal $I$ with $I^{\mathrm{an}}=J$. This gives a closed subscheme $Z\subset X\times Y$ whose full analytification is the graph; reduced Chow alone would lose its infinitesimal structure.

The first projection $p:Z\to X$ is proper and its analytification is an isomorphism. Each closed fiber has one complex point and is zero-dimensional. Upper semicontinuity of fiber dimension for proper maps shows that all fibers are zero-dimensional: any nonempty closed locus of positive-dimensional fibers would contain a closed point. Thus $p$ has finite fibers and is finite by Formal functions, Theorem 5.2.

At a closed point $x$, the finite algebra $(p_*\mathcal O_Z)_x=B$ over $A=\mathcal O_{X,x}$ has only one maximal ideal, since the closed fiber has one point. Its maximal-ideal topology is cofinal with its $\mathfrak m_A$-adic topology. Thus $B\otimes_A\widehat A=\widehat B$. Local analytic comparison and the analytic graph isomorphism identify the map $\widehat A\to\widehat B$ with an isomorphism. Faithful flatness of completion makes $A\to B$ an isomorphism. The kernel and cokernel of $\mathcal O_X\to p_*\mathcal O_Z$ are coherent and vanish at every closed point, so they vanish everywhere. A finite map is the relative spectrum of this algebra, hence $p$ is an isomorphism. The second projection composed with $p^{-1}$ is the required morphism. Two algebraic graph ideals with the same analytic extension agree by faithful flatness at closed points and coherent support. Their first projections and second projections therefore give the same algebraic morphism, proving uniqueness. $\square$

Proposition 6.3. For $n\geq1$, every analytic line bundle on $\mathbf P^n(\mathbf C)$ is $\mathcal O(d)^{\mathrm{an}}$ for a unique $d\in\mathbf Z$.

Proof. GAGA gives a coherent algebraic module $L$. It is a line bundle: at a closed point choose one element lifting an analytic fiber basis. Nakayama and faithful flatness make the resulting map $A\to L_x$ onto; after analytic tensor it is a map between rank-one free modules with nonzero residue, hence an isomorphism. Faithfulness makes its algebraic kernel zero. The locally free locus is open, and its closed complement contains no closed point, so is empty.

Here is the divisor argument, including why it classifies the bundle. Let $K$ be the rational function field of $P$ and choose a nonzero rational section $s$ of $L$. In a local frame $e_a$, write $s=g_a e_a$, with $g_a\in K^*$. On overlaps the quotients $g_a/g_b$ are units. Thus the orders of the $g_a$ along irreducible hypersurfaces define a finite divisor $D$; finiteness follows from a finite trivializing cover and from the finite numerator and denominator factorizations of each $g_a$. The sheaf $\mathcal O(D)$ consists locally of rational functions $h$ such that $hg_a$ is regular, and $h\mapsto hs$ identifies it with $L$.

The affine charts of $P$ are polynomial rings, and their local rings are factorial. One can see the polynomial assertion inductively from Gauss's argument: in a factorial ring the product of two primitive polynomials is primitive, since reduction modulo each prime factor gives a product of two nonzero polynomials over a domain; extract the greatest common divisor of the coefficients and factor over the fraction field. Clearing denominators and this primitive-product assertion give existence and uniqueness of polynomial factorization. Localization simply discards factors made invertible. Consequently the orders along prime hypersurfaces determine a local Cartier divisor: a rational function with zero order at every prime of a factorial local ring is a unit, by cancelling its numerator and denominator factors.

Every prime hypersurface of $P$ has an irreducible homogeneous equation $f$. To justify the assertion, its cone has a homogeneous prime ideal of height one: on any nonempty coordinate chart the punctured cone is the hypersurface times $\mathbf G_m$, so its codimension is one. In a factorial ring a height-one prime is principal: factor a nonzero element of the prime, choose an irreducible factor in it, and use the chain $(0)\subset(f)\subseteq\mathfrak p$ and height one. A generator of a homogeneous principal ideal is homogeneous. Indeed take its highest and lowest nonzero homogeneous components; divisibility of either component by the generator forces the two degrees to agree, because highest and lowest degrees add under multiplication in a polynomial domain.

If $\deg f=e$ and $H=V(T_0)$, the degree-zero rational function $f/T_0^e$ has divisor $V(f)-eH$: in a local frame the numerator has order one on its prime hypersurface and the denominator has order $e$ on $H$. For $D=\sum m_iV(f_i)$, multiply these rational functions to see that $D$ differs from $dH$, $d=\sum m_i\deg f_i$, by a principal divisor. Multiplication by that rational function gives an isomorphism $\mathcal O(D)\cong\mathcal O(dH)=\mathcal O(d)$. The last equality follows directly from the transition functions $(T_b/T_a)^d$ on the standard charts. Therefore $L\cong\mathcal O(d)$.

If two integers give isomorphic twists, their difference gives a trivial twist. Its global-section dimension would be one. Our projective-space calculation gives zero for a negative degree and $\binom{d+n}{n}$ for a nonnegative degree, which is one only for $d=0$ when $n\geq1$. This proves uniqueness. On $\mathbf P^0$, there is just the trivial line bundle and every twist represents it. $\square$

A compact Riemann surface holomorphically embedded in projective space consequently has algebraic image. It is an algebraic smooth projective curve: dimensions and regularity agree at complex points by local comparison. The hypothesis includes an embedding; this conclusion does not assert that an arbitrary compact analytic space is projectively embeddable.

From a projective modification to a proper scheme

Lemma 6.4. A reduced proper complex scheme $X$ admits a projective surjection $p:Y\to X$, with $Y$ reduced and projective, which is an isomorphism over an open $U$ containing every generic point of $X$. Every proper complex scheme has compact analytification.

Proof. For each reduced irreducible component $X_i$, use the Chow construction proved in Coherence of higher direct images under proper morphisms, Lemma 4.0. It gives a projective modification $Y_i\to X_i$, an isomorphism over a nonempty open, with $Y_i$ integral and projective over $\mathbf C$. Set $Y=\coprod_iY_i$. A finite disjoint union is projective: embed the ambient projective spaces as pairwise disjoint coordinate subspaces of one larger projective space. The same construction over $X$ makes $p$ projective. Remove the exceptional closed subsets and the intersections between distinct components. The remaining open $U$ contains every generic point, and $p^{-1}U\to U$ is an isomorphism.

Every nonempty finite-type fibre over a complex point has a complex point, by the earlier algebraic Nullstellensatz, so $p^{\mathrm{an}}$ is surjective. The space $Y^{\mathrm{an}}$ is compact: a projective closed embedding identifies it with a closed subspace of complex projective space, the continuous image of the unit sphere. Hence $X^{\mathrm{an}}$ is compact. An arbitrary $X$ and its reduction have the same analytic underlying space by the full quotient construction, so compactness follows in that case too. $\square$

Comparing projective direct images

This comparison is relative; its base need not be proper. It precedes the use of a projective modification over a nonprojective proper scheme.

Lemma 6.5 (twists with holomorphic parameters). Let $D\subset\mathbf C^m$ be a polydisc and let $r:\mathbf P^n(\mathbf C)\times D\to D$ be the projection. For every $d\in\mathbf Z$, $H^q(\mathbf P^n(\mathbf C)\times D,\mathcal O(d)) =V_d^q\otimes_{\mathbf C}\mathcal O(D), \tag{7}$ where $V_d^q=H^q(\mathbf P^n_{\mathbf C},\mathcal O(d))$, with the same monomial bases as in the algebraic projective-space computation. These identifications commute with restriction in $D$ and with the comparison maps.

Proof. Use the $n+1$ standard projective charts, multiplied by $D$. Every finite intersection is $(\mathbf C^*)^a\times\mathbf C^{n-a}\times D$. It has a smooth strictly plurisubharmonic exhaustion: add $\sum|w_j|^2$, $\sum_{j\le a}|w_j|^{-2}$, and $\sum_k(R_k^2-|z_k-b_k|^2)^{-1}$, where $D=\prod_k\{|z_k-b_k|<R_k\}$. The first and last terms give a positive complex Hessian, and these terms tend to infinity on every escape from a compact subset. Cartan's Theorem B on Stein manifolds therefore makes this finite cover acyclic for every coherent module, in particular each twist. Its ordered Čech complex computes the cohomology.

For a nonempty set $S\subset\{0,\ldots,n\}$, put $\Omega_S=\{T\in\mathbf C^{n+1}:T_i\ne0\ (i\in S)\}.$ A section of $\mathcal O(d)$ on the corresponding intersection, pulled back to $\Omega_S\times D$, is a holomorphic function of homogeneous weight $d$ in $T$. Its product Laurent expansion is $\sum_{\substack{\alpha\in\mathbf Z^{n+1}\\ \sum\alpha_i=d,\ \alpha_j\ge0\ (j\notin S)}} c_\alpha(z)T^\alpha . \tag{8}$ Indeed multitorus Cauchy integrals give the coefficients; the integrands are jointly holomorphic, so the coefficients are holomorphic in $z$. Uniqueness and the identity $f(2T,z)=2^df(T,z)$ impose the weight condition. The same Cauchy estimates are uniform for $z$ in a compact subset of $D$; hence this Laurent series converges normally on every compact subset of $\Omega_S\times D$.

Group its terms by the negative-coordinate set $N=\{i:\alpha_i<0\}$. Each group extends normally to $\Omega_N\times D$: on a given compact set choose inner contour radii for coordinates in $N$ and outer radii for all other coordinates. The inner-to-compact and compact-to-outer radius ratios are strictly less than one. Cauchy's estimate bounds the resulting sum by products of convergent geometric series, uniformly on that compact set and on its compact set of $z$-parameters. This also proves extension across newly allowed zero coordinates. The coefficient projections and extensions commute with restriction in both the projective charts and $D$.

For each nonempty proper $N$, choose $j\notin N$. Extend every $N$-component to the common domain $\Omega_N\times D$, so that restriction becomes the identity on its coefficient space. Regard cochains as alternating functions of distinct vertices, zero on repeated vertices and on faces not containing $N$. Set $(hc)_{i_0\ldots i_{p-1}}=c_{j i_0\ldots i_{p-1}}.$ After inserting $j$, reorder with its sign and restrict to the desired face. The term deleting the inserted vertex is the identity; each other insertion-deletion pair has opposite signs. Thus $dh+hd=1$. In degree zero the would-be augmented value is zero, because the singleton $\{j\}$ does not contain $N$; the same identity therefore contracts the unaugmented complex. These operators act on normally convergent series, by the preceding estimates, rather than on formal expressions alone. For $N=\varnothing$, the finitely many nonnegative exponents of total weight $d$ give the ordinary simplex complex, with its coefficient module in degree zero. For $N=\{0,\ldots,n\}$, the finitely many strictly negative exponents of weight $d$ occur only on the full face, in degree $n$. Empty exponent sets contribute zero. When $n=0$, these two possibilities both occur in degree zero and together give the single basis element for the trivial line bundle on a point. This is precisely the algebraic monomial computation with coefficient ring $\mathcal O(D)$, proving (7). $\square$

Proposition 6.6. Let $f:Y\to X$ be a projective morphism of complex schemes, and let $F$ be coherent. Then the natural maps $(R^qf_*F)^{\mathrm{an}} \longrightarrow R^qf^{\mathrm{an}}_*F^{\mathrm{an}} \tag{9}$ are isomorphisms for every $q\ge0$. In particular the sheaves on the right are coherent.

Proof. First take $X=\mathbf A^m_{\mathbf C}$, $Y=\mathbf P^n_X$, and write $r$ for the projection. By Lemma 6.5 and the local description of higher direct images as the sheafifications of cohomology on inverse images of opens, comparison is an isomorphism for every $\mathcal O(d)$. Algebraically $R^qr_*\mathcal O(d)=V_d^q\otimes\mathcal O_X$, by the earlier relative projective-space calculation. Analytically the same equality follows on the polydisc basis from (7).

For any coherent $H$ on $\mathbf P^n_X$, both sides of (9) vanish for $q>n$. Algebraically use the finite standard affine cover. Analytically use the same standard cover over a small polydisc; all its intersections are the Stein manifolds just described, and $H^{\mathrm{an}}$ is coherent on them. Choose a finite twist presentation $0\to K\to E\to H\to0, \qquad E=\bigoplus_j\mathcal O(d_j), \tag{10}$ with $K$ coherent, by the earlier Serre generation theorem over the affine base.

Descend on $q$, simultaneously for every coherent $H$. Assume comparison in degree $q+1$ is an isomorphism for all coherent modules. In the two long exact direct-image sequences from (10), comparison for $E$ is already an isomorphism. At any analytic base stalk, lift the boundary of an analytic degree-$q$ element uniquely through degree $q+1$ for $K$. Its image in degree $q+1$ for $E$ is zero, so algebraic exactness lifts it to degree $q$ for $H$. The difference has zero boundary and can be corrected using degree $q$ for $E$. This proves surjectivity for every $H$. In particular it proves surjectivity for $K$. If an algebraic element for $H$ has zero analytic image, its boundary is zero by the degree-$(q+1)$ injectivity for $K$. Lift it from $E$, correct that lift with the just-proved surjectivity for $K$, and use injectivity for $E$ to conclude it is zero. This proves injectivity. Analytification is exact on the base, so these are two exact sequences of stalk modules. The induction proves (9) in this ambient case.

For a general affine base $X=\operatorname{Spec}A$, embed it as a closed subscheme $k:X\hookrightarrow\mathbf A^m$. A projective embedding gives a closed immersion $i:Y\hookrightarrow\mathbf P^n_X\hookrightarrow\mathbf P^n_{\mathbf A^m}$. Put $H=i_*F$. Exact closed-immersion pushforward and its compatibility with analytification identify the ambient comparison just proved with $k_*$ applied to (9). Closed-immersion pushforward detects isomorphisms on its source, so (9) follows. Covering an arbitrary base by affine opens proves the result: the natural comparison maps agree on overlaps, and higher direct images commute with open restriction. This proof never replaces a coherent module by a finite globally free resolution. $\square$

Cohomological GAGA for every proper scheme

Theorem 6.7. If $X$ is proper over $\mathbf C$ and $F$ is coherent, then $H^q(X,F)\longrightarrow H^q(X^{\mathrm{an}},F^{\mathrm{an}}) \tag{11}$ is an isomorphism for every $q\ge0$.

Proof. Let $\mathcal P(F)$ mean that all maps (11) are isomorphisms. It holds for zero and is invariant under isomorphisms. The natural comparison maps commute with boundaries because analytification is exact and the maps are the morphism of cohomological delta functors extending the map on sections. Two out of three in a short exact sequence follows from its two long exact cohomology sequences.

Use Coherence of higher direct images under proper morphisms, Theorem 3.1, whose full ideal-filtration argument permits one generic-rank-one witness for each integral closed support. For every integral closed subscheme $i:Z\hookrightarrow X$, choose the earlier Chow construction's projective modification $p:Y\to Z$. Both $Y\to\operatorname{Spec}\mathbf C$ and $p$ are projective. A very ample line bundle $L$ for an absolute projective embedding is also $p$-ample: its embedding together with $p$ is a closed embedding into $\mathbf P^N_Z$, by the closed-graph argument. Relative Serre vanishing provides $a$ with $R^jp_*L^a=0\qquad(j>0).$ Set $G=p_*L^a$, a coherent module by projective finiteness. Over the dense isomorphism open it is an invertible module. Consequently $i_*G$ has support exactly $Z$, generic stalk annihilated by the maximal ideal of $\mathcal O_{X,\eta_Z}$, and generic dimension one over $\kappa(\eta_Z)$.

Proposition 6.6 identifies $G^{\mathrm{an}}$ with $p^{\mathrm{an}}_*L^{a,\mathrm{an}}$ and makes all higher analytic direct images zero. The algebraic and analytic Leray sequences therefore collapse to natural isomorphisms $H^q(Z,G)=H^q(Y,L^a),\qquad H^q(Z^{\mathrm{an}},G^{\mathrm{an}})=H^q(Y^{\mathrm{an}},L^{a,\mathrm{an}}).$ Projective cohomological GAGA on $Y$ identifies the right sides. Naturality of Leray and of direct-image comparison identifies this with (11) for $i_*G$, using exact closed-immersion pushforward. The earlier dévissage theorem now proves $\mathcal P(F)$ for every coherent $F$, including modules on a nonreduced $X$. $\square$

Morphisms and extension classes

Lemma 6.8. For coherent $F,G$ on a complex scheme, the sheaves $\mathcal Ext^q_X(F,G)$ are coherent and $\mathcal Ext^q_X(F,G)^{\mathrm{an}} \cong\mathcal Ext^q_{X^{\mathrm{an}}}(F^{\mathrm{an}},G^{\mathrm{an}}) \tag{12}$ for every $q\ge0$.

Proof. On an algebraic affine open, repeatedly present the finitely generated syzygies of $F$ by finite free modules. Noetherianness ensures that this can be done to any prescribed finite length. For a fixed $q$, the finite free prefix through degrees $q+1$ computes $\operatorname{Ext}^q$ by taking Hom into $G$, then a kernel modulo an image. The sheaf computation uses the same finite prefix: free module sheaves are locally projective, so applying internal Hom to it computes the corresponding sheaf Ext in these degrees. The kernels and images are finite modules, hence coherent. Localization of this finite computation is exact and commutes with its finite Hom terms, so these local sheaves glue.

At a complex point, tensor the prefix with the flat analytic local algebra $B$ over the algebraic local ring $A$. Exactness leaves a free-resolution prefix for $B\otimes_A F_x$. Finite freeness identifies each analytic Hom term with its algebraic Hom term tensored with $B$; flatness preserves the kernels, images and quotient. This proves (12) at every analytic stalk. The construction is natural and agrees on overlaps, giving the sheaf isomorphism. $\square$

Proposition 6.9. For proper $X$ and coherent $F,G$, the natural maps $\operatorname{Ext}^n_X(F,G)\longrightarrow \operatorname{Ext}^n_{X^{\mathrm{an}}}(F^{\mathrm{an}},G^{\mathrm{an}}) \tag{13}$ are isomorphisms for every $n\ge0$. In degree zero this gives full faithfulness; in degree one every analytic extension between algebraic coherent modules is algebraic.

Proof. Internal derived Hom and Ext sheaves, Theorem 3.1 and equations (3.2)–(3.3), identifies global Ext with the hypercohomology of $K=R\mathcal Hom(F,G)$. Its cohomology sheaves are $\mathcal Ext^q(F,G)$, zero for $q<0$. Applying the earlier first-quadrant hypercohomology exact-couple construction to the good truncations of $K$ gives $E_2^{p,q}=H^p(X,\mathcal Ext^q(F,G)) \Longrightarrow\operatorname{Ext}^{p+q}_X(F,G). \tag{14}$ Here is why precisely the same construction applies: represent $G$ by an injective resolution in nonnegative degrees and take internal Hom from $F$. This is a nonnegative complex representing $K$. Its good-truncation triangles have successive terms $\mathcal Ext^q(F,G)[-q]$; their exact couples give (14), exactly as in the earlier Leray proof. In total degree $n$, only $p,q\ge0$ with $p+q=n$ occur, so the abutment filtration has finitely many terms. No finite projective dimension is asserted.

Exact analytification gives the natural derived comparison $K^{\mathrm{an}}\to R\mathcal Hom(F^{\mathrm{an}},G^{\mathrm{an}})$. One construction is to analytify the injective-resolution Hom complex and then map to the Hom complex of an analytic injective resolution; the tensor-Hom evaluation map supplies the comparison, and injective comparison makes it independent of choices. The finite local calculation in Lemma 6.8 identifies the maps on cohomology sheaves. Naturality of the truncation exact couples gives a morphism from (14) to its analytic counterpart. Theorem 6.7 and Lemma 6.8 make every map on the $E_2$ page an isomorphism. Therefore every subsequent page, and every successive quotient of the finite abutment filtration, is an isomorphism. Induction up that finite filtration proves (13).

For clarity, $\operatorname{Ext}^1$ here classifies short exact sequences with specified ends. Embed $G$ in an injective $I$, with quotient $Q$. A map $F\to Q$ yields an extension by pulling back $I\to Q$. Two such maps give equivalent extensions exactly when their difference lifts to $I$. Conversely a map from $G$ to $I$ extends across any extension's middle module by injectivity and gives such a map to $Q$. Thus extensions are the cokernel of $\operatorname{Hom}(F,I)\to\operatorname{Hom}(F,Q)$, which is the injective-resolution $\operatorname{Ext}^1$. An extension of coherent modules is coherent, algebraically and analytically. This construction also shows that (13) sends an algebraic extension to its analytification. $\square$

Algebraizing an arbitrary coherent analytic module

Lemma 6.10 (a uniform support power). Let $X$ be proper, $Z\subset X$ a closed reduced subscheme with coherent ideal $J$, and $M$ a coherent analytic module whose support is contained in $Z^{\mathrm{an}}$. Some power $(J^{\mathrm{an}})^r$ annihilates $M$.

Proof. At a point $x$, the finite analytic module $M_x$ has support equal to the zero germ of its annihilator. The analytic Nullstellensatz implies that every germ in $J^{\mathrm{an}}_x$, whose zero germ contains that support, lies in $\sqrt{\operatorname{Ann}(M_x)}$. Choose finitely many generators $g_i$ of that ideal. If $g_i^{a_i}$ annihilates $M_x$, every product of $\sum_i(a_i-1)+1$ generators contains one such power. Thus a power of the whole ideal annihilates the stalk. Its product with $M$ is coherent, so its vanishing at this stalk holds on a neighbourhood. Lemma 6.4 makes $X^{\mathrm{an}}$ compact; choose finitely many of these neighbourhoods and take their maximum exponent. This is the required uniform $r$. $\square$

Theorem 6.11 (proper GAGA, with nilpotents). For every proper complex scheme $X$, analytification is an equivalence $\operatorname{Coh}(X)\simeq\operatorname{Coh}(X^{\mathrm{an}}).$ Together with Theorem 6.7 it preserves all coherent cohomology groups.

Proof. Full faithfulness is Proposition 6.9 in degree zero. Prove essential surjectivity by induction on $\dim X$, proving the reduced case first at each dimension, then all its nilpotent thickenings. The empty case is immediate. Assume algebraization already proved for all proper schemes of smaller dimension, including their nilpotents, and first let $X$ be reduced of the current dimension.

Choose Lemma 6.4's $p:Y\to X$, which is an isomorphism over $U$ containing all generic points. For a coherent analytic module $M$, its analytic pullback $p^{\mathrm{an},*}M$ is coherent, by finite presentations. Since $Y$ is projective, the already proved projective GAGA algebraizes it to a coherent $H$ on $Y$, with a specified isomorphism $H^{\mathrm{an}}\cong p^{\mathrm{an},*}M$. Proposition 6.6 and projective finiteness give $N=p_*H\in\operatorname{Coh}(X),\qquad N^{\mathrm{an}}\cong p^{\mathrm{an}}_*p^{\mathrm{an},*}M.$ The pullback-pushforward adjunction has its canonical unit $u:M\longrightarrow N^{\mathrm{an}}. \tag{15}$ It is an isomorphism over $U^{\mathrm{an}}$. Its coherent kernel $K$ and cokernel $C$ are therefore supported on the analytification of the proper algebraic closed subset $Z=X\setminus U$. Give $Z$ its reduced structure, with ideal $J$. Every component of $Z$ has smaller dimension than $X$, because $U$ contains every generic point. Lemma 6.10 gives a common $r$ with $(J^{\mathrm{an}})^rK=(J^{\mathrm{an}})^rC=0$. Consequently $K,C$ are coherent modules on the analytic nilpotent thickening $Z_r=V(J^r)$, a proper scheme of smaller dimension. Induction algebraizes them; exact closed-immersion pushforward supplies coherent $K_0,C_0$ on $X$ with specified identifications $K_0^{\mathrm{an}}=K$, $C_0^{\mathrm{an}}=C$.

Full faithfulness lifts the analytic quotient map $N^{\mathrm{an}}\to C$ to a map $N\to C_0$. Its cokernel has zero analytification and is therefore zero, so it is surjective. Let $I_0$ be its coherent kernel. Exactness identifies $I_0^{\mathrm{an}}$ with the image of (15). The given analytic short exact sequence $0\to K_0^{\mathrm{an}}\to M\to I_0^{\mathrm{an}}\to0$ has a class in $\operatorname{Ext}^1_{X^{\mathrm{an}}}(I_0^{\mathrm{an}},K_0^{\mathrm{an}})$. Proposition 6.9 lifts that class to an algebraic extension $0\to K_0\to F\to I_0\to0.$ The middle module is coherent. Equality of extension classes gives an analytic isomorphism $F^{\mathrm{an}}\cong M$ compatible with the specified ends. This proves the reduced case at the current dimension.

Finally let $X$ be nonreduced of that dimension. Its coherent nilradical $A$ has $A^s=0$ for some $s$: on each of a finite affine cover choose finitely many nilpotent generators, and take a uniform sufficiently large power. The finite filtration $M\supset A^{\mathrm{an}}M\supset\cdots\supset(A^{\mathrm{an}})^sM=0$ has coherent quotients annihilated by $A^{\mathrm{an}}$. They are coherent modules on $X_{\mathrm{red}}^{\mathrm{an}}$, which algebraize by the reduced case just proved at the same dimension. Regard their algebraizations as modules on $X$ by closed-immersion pushforward. Starting with the bottom quotient, lift each successive extension using Proposition 6.9 for this proper $X$; the finite upward induction algebraizes $M$. This completes the dimension induction for all proper schemes. $\square$

With identifications to the analytic module fixed, the algebraization is unique up to the unique compatible isomorphism, by full faithfulness. This assertion does not erase the automorphisms of a coherent module.

7. Exercises with solutions

Exercise 7.1 (easy). Show why the cohomological comparison theorem fails for $\mathbf A^1$.

Solution. In degree zero it is the inclusion $\mathbf C[z]\to\mathcal O(\mathbf C)$. The entire function $e^z$ is absent from the image, as its derivatives never become zero. Thus even degree-zero surjectivity fails. The projective hypothesis supplies the compactness and twist arguments used above.

Exercise 7.2 (medium). Classify analytic line bundles on projective space, explaining the dimension-zero exception.

Solution. Essential surjectivity algebraizes the bundle; local faithful flatness descends its rank-one freeness as in Proposition 6.3. A rational section and homogeneous prime-divisor equations make its divisor a multiple of a hyperplane, so it is $\mathcal O(d)^{\mathrm{an}}$. For $n\geq1$, the global-section dimension distinguishes the trivial twist and hence makes $d$ unique. For $n=0$, every $\mathcal O(d)$ is the same one-dimensional bundle, so there is no unique integer.

Exercise 7.3 (medium). Deduce Chow's theorem for a nonempty analytic hypersurface of $\mathbf P^n$ directly from its ideal, using GAGA.

Solution. An analytic hypersurface in projective space has an invertible coherent ideal: the convergent local rings are factorial, by The local ring of holomorphic germs, Theorem 3.1, so a reduced pure codimension-one vanishing ideal is locally the product of its prime equations. Algebraize this ideal and its inclusion into the structure sheaf. The ideal descends to an algebraic line bundle by the same local argument as in Proposition 6.3, so is $\mathcal O(-d)$. Its inclusion is a nonzero section of $\mathcal O(d)$, hence a homogeneous polynomial of degree $d$. A nonempty zero set requires $d>0$. The ideal's analytic extension is the original vanishing ideal; radicality descends by faithful flatness. Thus this polynomial defines exactly the given reduced hypersurface; the ideal formulation handles reducible hypersurfaces as well.

Exercise 7.4 (medium). Describe holomorphic maps $\mathbf P^1(\mathbf C)\to\mathbf P^1(\mathbf C)$.

Solution. Theorem 6.2 makes the map algebraic. Its pullback of $\mathcal O(1)$ is $\mathcal O(d)$, and the pulled-back coordinate sections generate it, so $d\geq0$. They are homogeneous degree-$d$ polynomials $F_0,F_1$ without a common projective zero. The map is $[T_0:T_1]\mapsto[F_0(T):F_1(T)]$, and in an affine target chart it is their rational quotient. For $d=0$ it is constant; for $d>0$ its degree is $d$.

Exercise 7.5 (medium). Compute analytic cohomology of $\mathcal O(d)$ on $\mathbf P^1$, and verify comparison explicitly.

Solution. Use $z=T_1/T_0$, with trivializations $e_0=T_0^d$, $e_1=T_1^d=z^de_0$. The acyclic two-chart cover gives the degree-one quotient of holomorphic functions on $\mathbf C^*$ by entire functions in $z$ and $z^d$ times entire functions in $z^{-1}$. In the Laurent expansion the first family removes exponents $a\geq0$; the second removes exponents $a\leq d$. The remaining basis is $z^{d+1},\ldots,z^{-1}$ when $d\leq-2$, and is empty otherwise. Thus $h^1=\max(-d-1,0)$. Global sections must have exponents both $a\geq0$ and $a\leq d$, giving $1,z,\ldots,z^d$ for $d\geq0$ and zero otherwise. Therefore $h^0=\max(d+1,0)$; higher cohomology vanishes. The algebraic Laurent Čech calculation has the identical bases, and comparison takes each basis monomial to itself.

Exercise 7.6 (harder). State and prove the uniqueness of algebraization with a fixed analytic identification. Give a counterexample to uniqueness of the bare isomorphism.

Solution. For $\phi:F^{\mathrm{an}}\to M$ and $\psi:G^{\mathrm{an}}\to M$, full faithfulness lifts $\psi^{-1}\phi$ uniquely to $u:F\to G$. Lifting the inverse and using faithfulness proves $u$ is an isomorphism. It is the unique one compatible with the identifications. On a nonempty projective variety, multiplication by any $\lambda\in\mathbf C^*$ is an automorphism of $\mathcal O_X$; there are many isomorphisms of that bare sheaf to itself. The compatibility condition singles out the correct one.

8. Sources and the comparison chain

J.-P. Serre's Géométrie algébrique et géométrie analytique, Annales de l'Institut Fourier 6 (1956), 1–42, is the historical source for these three theorems and their applications. The linked edition is free to read; no permission to copy or translate it is asserted. Chow's theorem is credited to W.-L. Chow. The arguments here are independently written; Section 1 links the analytic programme proofs and Demailly's freely accessible exposition. Component authorship and reuse terms remain recorded in the respective lessons.

Kiran S. Kedlaya's GAGA, updated 30 April 2009, MIT OpenCourseWare 18.726, treats comparison, full faithfulness, algebraization and Chow in Sections 3–5 and 8. Its notes are reusable under CC BY-NC-SA 4.0. They offer useful parallel reading, while this course retains its explicit Laurent contraction, two-stage descending comparison induction and treatment of noninjective hyperplane multiplication.

For a broader framework, Jack Hall's GAGA theorems, arXiv:1804.01976v3, 17 May 2022, Theorem A and Example 9.2, treats analytic GAGA for proper schemes. Theorem A requires coherence, finite total coherent cohomology, detection at closed points and specified local flatness and residue-field conditions. Its non-Noetherian extensions have further hypotheses. This public author draft is further reading; the proofs above are supplied in the programme.

The corresponding verified open AI Integrated Stacks Project treatments are cohomological comparison, full faithfulness, analytic global generation, essential surjectivity, and Chow's theorem. Their reference texts retain GFDL 1.2 and contain AI-written additions not reviewed by the official Stacks maintainers. Section 1 states the analytic finiteness and acyclicity inputs; Section 2 gives the Laurent structure-sheaf calculation assuming those inputs. The referenced chapter also invokes analytic finiteness in its generation argument.

Using the analytic programme results in Section 1 and the full quotient construction in the preceding lesson, Sections 2–6 establish comparison and algebraicity for projective complex schemes, including nonreduced schemes. The chain is local analytic algebra and faithful flatness; analytic Čech cohomology and finiteness; comparison for twists; comparison for coherent sheaves; internal Hom and full faithfulness; analytic generation and two presentations; algebraicity of full ideals and graphs. Section 6 extends the result to every proper complex scheme: relative projective comparison and rank-one dévissage give cohomological comparison, local Ext and its finite abutment filtration give extension comparison, and the modification unit reconstructs a coherent analytic module from its two defects on a smaller closed support. A final finite nilpotent filtration handles nonreduced schemes.