Coherent sheaves and Oka's coherence theorem

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The stalks of the sheaf of holomorphic functions are Noetherian rings, so relations among finitely many germs at one point are finitely generated. Oka's theorem says much more: finitely many relations can be chosen so that they generate the relations at every point of a neighbourhood. In the language of sheaves, the sheaf of holomorphic functions is coherent. This lesson proves Oka's theorem and draws the consequences that make coherent analytic sheaves a workable category: kernels, cokernels, images, extensions, tensor products and internal homomorphisms of coherent sheaves are coherent, coherent sheaves have analytic supports, and a coherent sheaf with a free stalk is free near that point.

We use The local ring of holomorphic germs, in particular the Weierstrass division theorem in the form of its Theorem 2.1 and Lemmas 2.3 and 3.2. Sheaves of modules on a ringed space and their stalks are as in Cohomology of sheaves on ringed spaces; the general algebra of coherent modules on a ringed space is taken from the AI Integrated Stacks Project, with exact tags.

Basic references are [Demailly], [Oka 1950] and [Cartan 1950].

1. Coherent modules on a ringed space

Let \(\mathcal A\) be a sheaf of rings on a topological space \(X\) and \(\mathcal S\) a sheaf of \(\mathcal A\)-modules. Sections \(s_1,\ldots,s_q\in\mathcal S(U)\) define a morphism

\[ \sigma:\mathcal A^q|_U\to\mathcal S|_U,\qquad (g_1,\ldots,g_q)\mapsto\sum_j g_js_j , \tag{1.1} \]

and its kernel \(\mathcal R(s_1,\ldots,s_q)\subset\mathcal A^q|_U\) is the sheaf of relations among the \(s_j\). The module \(\mathcal S\) is of finite type (locally finitely generated) if every point has a neighbourhood \(U\) and sections \(s_1,\ldots,s_q\in\mathcal S(U)\) for which \(\sigma\) is surjective, that is, whose germs generate every stalk \(\mathcal S_x\), \(x\in U\).

Definition 1.1. \(\mathcal S\) is coherent if it is of finite type and, for every open \(U\) and all finitely many \(s_1,\ldots,s_q\in\mathcal S(U)\), the sheaf of relations \(\mathcal R(s_1,\ldots,s_q)\) is of finite type. The sheaf of rings \(\mathcal A\) is coherent if it is coherent as a module over itself.

This is the definition of Stacks, Tag 01BV. We use the following general facts, valid on any ringed space.

Proposition 1.2. Let \((X,\mathcal A)\) be a ringed space.

  1. A finite type submodule of a coherent module is coherent. If \(\varphi:\mathcal F\to\mathcal G\) is a morphism of coherent modules, then \(\ker\varphi\), \(\operatorname{im}\varphi\) and \(\operatorname{coker}\varphi\) are coherent. If two of the three modules in a short exact sequence are coherent, so is the third. Stacks, Tag 01BY.
  2. If \(\mathcal A\) is coherent, a module is coherent if and only if it is of finite presentation, that is, locally the cokernel of a morphism \(\mathcal A^p\to\mathcal A^q\). Stacks, Tag 01BZ.
  3. If \(\mathcal G\) is of finite type, \(\mathcal F\) is coherent, and \(\varphi:\mathcal G\to\mathcal F\) is injective on the stalk at \(x\), then \(\varphi\) is injective on a neighbourhood of \(x\). Stacks, Tag 01C0.

By part 1 applied to the split exact sequences \(0\to\mathcal A^{q-1}\to\mathcal A^q\to\mathcal A\to0\), if \(\mathcal A\) is coherent then so is every free module \(\mathcal A^q\). Consequently the relations among finitely many sections of \(\mathcal A^q\) form a coherent submodule of a free module.

Lemma 1.3. Let \(\mathcal S\) be of finite type and \(x\in X\).

  1. If \(t_1,\ldots,t_N\in\mathcal S(U)\) have germs generating \(\mathcal S_x\), their germs generate \(\mathcal S_y\) for all \(y\) in a neighbourhood of \(x\).
  2. If \(\mathcal S_x=0\), then \(\mathcal S\) vanishes on a neighbourhood of \(x\). Hence the support \(\operatorname{Supp}\mathcal S=\{y:\ \mathcal S_y\neq0\}\) is closed.

Proof. (1) Choose generators \(s_1,\ldots,s_q\) of \(\mathcal S\) near \(x\). Each \(s_i\) is, at \(x\), a combination of the \(t_k\) with coefficients in \(\mathcal A_x\); these coefficients and the identity they satisfy extend to a neighbourhood, on which every \(s_i\), and hence every stalk, lies in the span of the \(t_k\). (2) Apply (1) to the empty family. \(\square\)

2. Oka's coherence theorem

An analytic sheaf on a complex manifold \(M\) is a sheaf of \(\mathcal O_M\)-modules. A coherent analytic sheaf on an open subset of \(\mathbf C^n\) is called coherent for short.

Theorem 2.1 (Oka). For every complex manifold \(M\) the sheaf of rings \(\mathcal O_M\) is coherent.

The stalks \(\mathcal O_{M,x}\cong\mathcal O_n\) are Noetherian, so each stalk of a sheaf of relations is finitely generated. The content of the theorem is that one finite family of relations generates the stalks of the relation sheaf at all points of a neighbourhood. The proof is by induction on \(n=\dim M\), and the following lemma reduces relations among Weierstrass polynomials to relations among polynomials of bounded degree, which are governed by one fewer variable.

Let \(\Delta=\Delta'\times\Delta_n\) be a polydisc in \(\mathbf C^{n-1}\times\mathbf C\), and let \(P_1,\ldots,P_q\in\mathcal O(\Delta')[z_n]\) be monic polynomials in \(z_n\), of degrees at most \(\mu\). For \(x=(x',x_n)\in\Delta\) we call an element of \(\mathcal R(P_1,\ldots,P_q)_x\) polynomial of degree at most \(\mu\) if all its components lie in \(\mathcal O_{\mathbf C^{n-1},x'}[z_n]\) and have degree at most \(\mu\) in \(z_n\).

Lemma 2.2. For every \(x\in\Delta\), the \(\mathcal O_{\mathbf C^n,x}\)-module \(\mathcal R(P_1,\ldots,P_q)_x\) is generated by its elements that are polynomial of degree at most \(\mu\).

Proof. We may suppose that \(P_q\) has the maximal degree \(\mu\). The elements

\[ \rho_j=P_q\,e_j-P_j\,e_q\qquad(1\leq j<q), \tag{2.1} \]

where \(e_1,\ldots,e_q\) is the standard basis, are relations that are polynomial of degree at most \(\mu\). Split \(P_q=f'f''\) at \(x\) as in The local ring of holomorphic germs, Lemma 3.2: \(f'\) is a Weierstrass polynomial of some degree \(\mu'\) in \(z_n-x_n\), and \(f''\) is monic of degree \(\mu-\mu'\) with \(f''(x)\neq0\), so \(f''\) is a unit at \(x\).

Let \(g=(g_1,\ldots,g_q)\in\mathcal R(P_1,\ldots,P_q)_x\). For \(j<q\), divide \(g_j\) by \(f'\) at \(x\) (Theorem 2.1 of the same lesson): \(g_j=s_jf'+r_j\) with \(r_j\in\mathcal O_{\mathbf C^{n-1},x'}[z_n]\) of degree less than \(\mu'\). Put \(t_j=s_j/f''\), so that \(g_j=t_jP_q+r_j\), and put \(r_q=g_q+\sum_{j<q}t_jP_j\). Then

\[ g=\sum_{j<q}t_j\rho_j+(r_1,\ldots,r_q), \tag{2.2} \]

as one checks componentwise, so \(r=(r_1,\ldots,r_q)\) is a relation: \(\sum_{j<q}r_jP_j+r_qf'f''=0\). The polynomial \(S=-\sum_{j<q}r_jP_j\) has degree less than \(\mu'+\mu\) and is divisible by the Weierstrass polynomial \(f'\) in \(\mathcal O_{\mathbf C^n,x}\), with quotient \(f''r_q\). By Lemma 2.3 of the same lesson, \(S=f'T\) with a polynomial \(T\), and uniqueness of division gives \(f''r_q=T\), a polynomial of degree less than \(\mu\). Hence

\[ r=\frac1{f''}\bigl(f''r_1,\ldots,f''r_{q-1},\,T\bigr), \]

where \(f''r_j\) has degree less than \((\mu-\mu')+\mu'=\mu\). So \(r\) is a unit multiple of a relation that is polynomial of degree less than \(\mu\), and by (2.2) \(g\) lies in the submodule generated by such relations and the \(\rho_j\). \(\square\)

Proof of Theorem 2.1. The statement is local, so let \(M\) be an open subset of \(\mathbf C^n\); we argue by induction on \(n\). For \(n=0\) the stalks are \(\mathbf C\) and there is nothing to prove. Let \(n\geq1\), let \(F_1,\ldots,F_q\in\mathcal O(U)\) and \(a\in U\); we show that \(\mathcal R(F_1,\ldots,F_q)\) is generated near \(a\) by finitely many sections. Take \(a=0\).

Reductions. If some \(F_j\) vanishes identically near \(0\), then near \(0\) the relation sheaf is the direct sum of \(\mathcal O\,e_j\) and the relation sheaf of the remaining functions; so we may assume that no germ \(F_{j,0}\) is zero. If some \(F_j(0)\neq0\), say \(F_q(0)\neq0\), then near \(0\) the relations are generated by \(F_qe_j-F_je_q\), \(j<q\): a relation \(g\) equals \(\sum_{j<q}(g_j/F_q)(F_qe_j-F_je_q)\). So we may assume that all \(F_j\) vanish at \(0\). After one linear change of coordinates all \(F_j\) are regular in \(z_n\) (apply Lemma 4.1 of Holomorphic functions of several variables to the product \(F_1\cdots F_q\)), and \(F_j=u_jP_j\) with units \(u_j\) and Weierstrass polynomials \(P_j\). On a polydisc \(\Delta=\Delta'\times\Delta_n\) about \(0\) on which the \(u_j\) have no zeros and the coefficients of the \(P_j\) are holomorphic, the isomorphism \((g_j)\mapsto(g_ju_j)\) of \(\mathcal O^q\) carries \(\mathcal R(F_1,\ldots,F_q)\) onto \(\mathcal R(P_1,\ldots,P_q)\). So we may assume \(F_j=P_j\).

Polynomial relations. A polynomial element \(g\) of degree at most \(\mu\) at \(x\in\Delta\) has components \(g_j=\sum_{k=0}^\mu u_{jk}z_n^k\) with \(u_{jk}\in\mathcal O_{\mathbf C^{n-1},x'}\). Write \(P_j=\sum_{l=0}^{\mu}p_{jl}z_n^l\) with \(p_{jl}\in\mathcal O(\Delta')\). The expression \(\sum_jg_jP_j\) is a polynomial in \(z_n\) of degree at most \(2\mu\) with coefficients in \(\mathcal O_{\mathbf C^{n-1},x'}\), and it is zero as a germ at \(x\) if and only if all its coefficients are zero: for fixed \(z'\) near \(x'\), a polynomial in \(z_n\) that vanishes for all \(z_n\) near \(x_n\) has zero coefficients. So \(g\) is a relation exactly when

\[ \sum_{j=1}^q\ \sum_{k+l=m}p_{jl}\,u_{jk}=0\qquad(m=0,1,\ldots,2\mu). \tag{2.3} \]

This says that the vector \(u=(u_{jk})\in\mathcal O_{x'}^{q(\mu+1)}\) is a relation among the \(q(\mu+1)\) sections \(c_{jk}=(p_{j,m-k})_{0\leq m\leq2\mu}\) of the free module \(\mathcal O_{\Delta'}^{2\mu+1}\) (with \(p_{j,l}=0\) for \(l<0\) or \(l>\mu\)). By the induction hypothesis \(\mathcal O_{\Delta'}\) is coherent, hence so is \(\mathcal O_{\Delta'}^{2\mu+1}\) (Section 1), and the relation sheaf \(\mathcal R(c_{jk})\) is generated on a neighbourhood \(\Omega'\) of \(0\) by finitely many sections \(U_1,\ldots,U_N\in\mathcal O(\Omega')^{q(\mu+1)}\).

Conclusion. Put \(G_\nu=\bigl(\sum_kU_\nu^{jk}(z')z_n^k\bigr)_{1\leq j\leq q}\), a section of \(\mathcal R(P_1,\ldots,P_q)\) over \(\Omega=\Omega'\times\Delta_n\). At every \(x\in\Omega\), the germs \(U_{\nu,x'}\) generate the module of solutions of (2.3), so the germs \(G_{\nu,x}\) generate the polynomial relations of degree at most \(\mu\) at \(x\). By Lemma 2.2 these generate \(\mathcal R(P_1,\ldots,P_q)_x\). Thus \(G_1,\ldots,G_N\) generate the relation sheaf on \(\Omega\). \(\square\)

Reference: [Oka 1950] proves this theorem; the proof above follows the arrangement in [Demailly], which credits it.

3. The category of coherent analytic sheaves

From now on, \(M\) is a complex manifold and "coherent" means coherent over \(\mathcal O_M\). By Theorem 2.1 and Proposition 1.2(2), an analytic sheaf is coherent if and only if it has local finite presentations

\[ \mathcal O^p|_U\xrightarrow{\ A\ }\mathcal O^q|_U\longrightarrow\mathcal S|_U\longrightarrow0, \tag{3.1} \]

where \(A\) is a \(q\times p\) matrix of holomorphic functions on \(U\).

Theorem 3.1. Let \(M\) be a complex manifold.

  1. Kernels, images and cokernels of morphisms of coherent sheaves are coherent; extensions of coherent sheaves are coherent; finitely generated subsheaves of coherent sheaves are coherent. The intersection and the sum of two coherent subsheaves of a coherent sheaf are coherent.
  2. If \(\mathcal F\) and \(\mathcal G\) are coherent, so are \(\mathcal F\otimes_{\mathcal O}\mathcal G\) and \(\mathcal Hom_{\mathcal O}(\mathcal F,\mathcal G)\), and the stalk of the latter at \(x\) is \(\operatorname{Hom}_{\mathcal O_x}(\mathcal F_x,\mathcal G_x)\).
  3. The annihilator ideal of a coherent sheaf is coherent, and the support of a coherent sheaf is an analytic subset of \(M\).
  4. Every coherent sheaf has, near each point and for every \(m\), an exact sequence \(\mathcal O^{p_m}\to\cdots\to\mathcal O^{p_1}\to\mathcal O^{p_0}\to\mathcal S\to0\).

Proof. (1) The first three statements are Proposition 1.2(1). The intersection of \(\mathcal F,\mathcal G\subset\mathcal S\) is the kernel of \(\mathcal F\to\mathcal S/\mathcal G\), and the sum is the image of \(\mathcal F\oplus\mathcal G\to\mathcal S\).

(2) Tensor products: from presentations (3.1) of \(\mathcal F\) and \(\mathcal G\), right exactness of \(\otimes\) presents \(\mathcal F\otimes\mathcal G\) as a cokernel of a morphism of free modules of finite rank. Homomorphisms: from \(\mathcal O^p\to\mathcal O^q\to\mathcal F\to0\) on \(U\), left exactness of \(\mathcal Hom(-,\mathcal G)\) gives an exact sequence \(0\to\mathcal Hom(\mathcal F,\mathcal G)\to\mathcal G^q\to\mathcal G^p\), so \(\mathcal Hom(\mathcal F,\mathcal G)\) is a kernel of a morphism of coherent sheaves. Applying the same reasoning to stalks, where \(\operatorname{Hom}_{\mathcal O_x}(-,\mathcal G_x)\) is also left exact, identifies the stalks.

(3) The annihilator of \(\mathcal S\) is the kernel of \(\mathcal O\to\mathcal Hom(\mathcal S,\mathcal S)\), \(f\mapsto f\cdot\mathrm{id}\), hence coherent by (1) and (2); this is also Stacks, Tag 0H2L. For the support, use a presentation (3.1) near \(x\). The stalk \(\mathcal S_y\) is zero exactly when \(A_y:\mathcal O_y^p\to\mathcal O_y^q\) is surjective, which by Nakayama's lemma happens exactly when the constant matrix \(A(y)\) has rank \(q\). So \(\operatorname{Supp}\mathcal S\cap U\) is the common zero set of the \(q\times q\) minors of \(A\), an analytic subset.

(4) Inductively, the kernel of a surjection \(\mathcal O^{p_k}\to\mathcal Z_{k-1}\) onto a coherent sheaf is coherent, hence finitely generated on a smaller neighbourhood. \(\square\)

A locally free sheaf of rank \(r\) is a sheaf locally isomorphic to \(\mathcal O^r\); locally free sheaves are coherent. The locally free sheaves are the sheaves of holomorphic sections of holomorphic vector bundles: transition matrices of local frames are holomorphic maps into \(\mathrm{GL}_r(\mathbf C)\) satisfying the cocycle condition, and conversely such a cocycle glues trivial bundles.

Proposition 3.2. If \(\mathcal S\) is coherent and \(\mathcal S_x\) is a free \(\mathcal O_x\)-module of rank \(r\), then \(\mathcal S\) is free of rank \(r\) on a neighbourhood of \(x\).

Proof. Choose sections \(s_1,\ldots,s_r\) near \(x\) whose germs form a basis of \(\mathcal S_x\). The morphism \(\sigma:\mathcal O^r\to\mathcal S\) of (1.1) is surjective near \(x\) by Lemma 1.3(1) and injective near \(x\) by Proposition 1.2(3), because \(\sigma_x\) is an isomorphism. \(\square\)

Corollary 3.3. Let \(\mathcal S\) be coherent and \(x\in M\). If a finite free resolution \(0\to\mathcal O_x^{p_m}\to\cdots\to\mathcal O_x^{p_0}\to\mathcal S_x\to0\) of the stalk exists, then \(\mathcal S\) has a finite free resolution of the same length on a neighbourhood of \(x\).

Proof. Build the sequence of Theorem 3.1(4) near \(x\) up to the stage \(m-1\), choosing at each stage generators whose germs at \(x\) give the given resolution of \(\mathcal S_x\); this is possible because a surjection onto a stalk lifts to a morphism near \(x\) that is surjective near \(x\) by Lemma 1.3. The kernel \(\mathcal Z_{m-1}\) of the last map is coherent, and its stalk at \(x\) is the kernel at the corresponding stage of the given resolution, which is the free module \(\mathcal O_x^{p_m}\). By Proposition 3.2, \(\mathcal Z_{m-1}\) is free near \(x\). \(\square\)

Every finitely generated \(\mathcal O_n\)-module has a free resolution of length at most \(n\), because \(\mathcal O_n\) is a regular local ring of dimension \(n\) (The local ring of holomorphic germs, Proposition 1.1) and regular local rings have finite global dimension equal to their dimension Stacks, Tag 00O7; over a local ring a finitely generated projective module is free. By Corollary 3.3 we obtain:

Theorem 3.4 (local syzygies). Every coherent analytic sheaf on an \(n\)-dimensional complex manifold has, near each point, a resolution \(0\to\mathcal O^{p_n}\to\cdots\to\mathcal O^{p_0}\to\mathcal S\to0\) by free sheaves of finite rank, of length at most \(n\).

4. Exercises

Exercise 4.1. Let \(F_1=z_1\), \(F_2=z_2\) on \(\mathbf C^2\). Show that \(\mathcal R(F_1,F_2)\) is the free sheaf generated by \((z_2,-z_1)\).

Solution. If \(g_1z_1+g_2z_2=0\) near a point, then at points with \(z_1\neq0\) we get \(g_1=-g_2z_2/z_1\). At \(0\), \(z_1\) divides \(g_2z_2\), and since \(z_1,z_2\) are coprime in the factorial ring \(\mathcal O_2\), \(z_1\) divides \(g_2\): \(g_2=-hz_1\), and then \(g_1=hz_2\). So every relation is \(h\,(z_2,-z_1)\), and \(h\) is unique because \(\mathcal O\) has no zero divisors. At other points the same holds since one of \(z_1,z_2\) is a unit there.

Exercise 4.2. On \(\mathbf C\), let \(\mathcal S=\mathcal O/(z)\), the skyscraper sheaf with stalk \(\mathbf C\) at \(0\). Show that \(\mathcal S\) is coherent but not locally free, and identify its support and its annihilator.

Solution. The presentation \(\mathcal O\xrightarrow{z}\mathcal O\to\mathcal S\to0\) shows coherence. The stalk at \(0\) is \(\mathbf C\), which is not free over \(\mathcal O_1\), while the stalks elsewhere are \(0\); so \(\mathcal S\) is not locally free. Its support is \(\{0\}\), the zero set of the \(1\times1\) minor \(z\), and its annihilator is the ideal sheaf \((z)\).

Exercise 4.3. Let \(H=\{z\in\mathbf C:\ \operatorname{Re}z\geq0\}\) and let \(\mathcal J\subset\mathcal O_{\mathbf C}\) be the subsheaf of germs vanishing on \(H\). Show that \(\mathcal J\) is not of finite type, although every stalk of \(\mathcal J\) is a finitely generated ideal.

Solution. If \(\operatorname{Re}x<0\), a small disc about \(x\) misses \(H\), so \(\mathcal J_x=\mathcal O_x\). If \(\operatorname{Re}x\geq0\), every neighbourhood of \(x\) meets \(H\) in a set with interior points, so a germ at \(x\) vanishing on \(H\) is zero by the identity theorem, and \(\mathcal J_x=0\). Both kinds of stalks are finitely generated. The support of \(\mathcal J\) is the open half-plane \(\{\operatorname{Re}z<0\}\), which is not closed; by Lemma 1.3 a sheaf of finite type has closed support. So \(\mathcal J\) is not of finite type. The set \(H\) is not an analytic subset, and Lesson 5 shows that for analytic subsets the corresponding ideal sheaf is coherent.

Exercise 4.4. Let \(\mathcal S\) be coherent on \(M\) and \(s\in\mathcal S(M)\). Show that \(\{x:\ s_x=0\}\) is open and that \(\{x:\ s_x\neq0\}\) is an analytic subset.

Solution. If \(s_x=0\), then \(s\) vanishes on a neighbourhood of \(x\) by the definition of germs; so the first set is open. The second set is the support of the image sheaf \(\mathcal O\cdot s\subset\mathcal S\), which is a finitely generated, hence coherent, subsheaf; by Theorem 3.1(3) its support is analytic.

References