Chapter 7

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 4 June 2013

Operations on sheaves, quasi-coherent and coherent sheaves

Let (X,𝒪) be a ringed space. An 𝒪-Module (note the capital M) is a sheaf of abelian groups such that, for each open set U in X, the group (U) carries a structure of an 𝒪(U)-module, these structures being compatible with the restriction homomorphisms: explicitly, if UV are open sets in X, then the restriction φ:(U)(V) is compatible with the restriction ρ:𝒪(U)𝒪(V), that is to say, if f(U) and a𝒪(U) then φ(af)=ρ(a)·φ(f). Then each stalk x has a natural 𝒪X-module structure, defined as follows: if ax𝒪X, fxx, say ax is the image of a𝒪(U), fx the image of f(U) for some sufficiently small open neighbourhood U of x; then ax·fx is the image of af in x.

In particular, 𝒪 itself is an 𝒪-Module.

Most of the concepts of module theory have their counterparts for Modules: -

(i) An 𝒪-Module homomorphism φ:𝒢 is a sheaf homomorphism (i.e. a family of homomorphisms φ(U):(U)𝒢(U), commuting with the restrictions) such that each φ(U) is an 𝒪(𝔘)-module homomorphism. Then each φx:x𝒢x is an 𝒪X-module homomorphism.

(ii) Sub-Modules. A subsheaf of an 𝒪-Module is a sub-Module of if, for each open set U in X, (U) is a sub-𝒪(U)-module of (U). Then each x is a sub-𝒪X-module of x, and the embedding is an 𝒪-Module homomorphism. In particular, a sub-Module of 𝒪 is called an Ideal (with a capital I).

(iii) Quotient Modules. Let be an 𝒪-Module, a sub-Module of . For each open set U in X, form (U)/(U). U(U)/(U), with the induced restriction homomorphisms, is a presheaf, but not necessarily a sheaf. So we form the sheaf associated with this presheaf: this is the quotient Module =/. Since lim is exact, we have x=x/x.

(iv) Kernel. Let φ:𝒢 be an 𝒪-Module homomorphism. For each open set U in X let (U) be the kernel of φ(U):(U)𝒢(U). Then U(U) is a sheaf , called the kernel of φ. Clearly is an 𝒪-Module. We have x=Ker(φx) for all xX.

(v) Image. For each open set U in X we can form Im(φ(U)), which is a submodule of 𝒢(U). Uim(φ(U)) is a presheaf (not necessarily a sheaf). Let be the sheaf associated with this presheaf. Then is a subsheaf of 𝒢, called the image of 𝒪. Again by the exactness of lim we have x=Im(φx). Also is isomorphic to the quotient /, where is the kernel of φ.

(vi) Cokernel. The cokernel of φ is 𝒢/. We have the formulas

(Ker(φ))x Ker(φx); (Im(φ))x Im(φx); (Coker(φ))x Coker(φx).

The class of 𝒪-Modules is an abelian category. Exact sequences are defined in the usual way.

Lemma (7.1). A sequence φ𝒢ψ is exact if and only if xφx𝒢xψxx is exact for all xX.


𝒢 is exact Im(φ)= Ker(ψ) (Im(φ))x= (Ker(ψ))x for all xXIm(φx)=Ker(ψx) for all xXx𝒢xx is exact for all xX.

Lemma (7.2). The "section functor" Γ() (=Γ(X,)=(X)) is left exact: if 0𝒢 is exact, then 0Γ()Γ(𝒢)Γ() is exact. This follows from (iii) above.

(vii) Direct sum. Let (i)iI be any family of 𝒪-Modules. Their direct sum =iIi is the sheaf UiIi(U). If each i is equal to 𝒪, we write 𝒪(I) for the direct sum. In particular, if I is finite and has n elements, we write 𝒪n for the direct sum of n copies of 𝒪.

(viii) Tensor product. If ,𝒢 are 𝒪-Modules, their tensor product 𝒪𝒢 is defined to be the sheaf associated with the presheaf U(U)𝒪(U)𝒢(U). Since commutes with lim, we have (𝒪𝒢)xx𝒪x𝒢x. This tensor product has all the usual properties: it is commutative, associative, distributive over , and is right exact in each variable (look at the stalks and use (7.1)). Also 𝒪𝒪.

(ix) Global Hom. Hom𝒪(,𝒢) is the group of all 𝒪-Module homomorphisms φ:𝒢. It has a natural 𝒪(X)-module structure: if φ:𝒢 and s𝒪(X), define sφ:𝒢 by (sφ)(U)=s|U·φ(U).

(x) Sheaf 𝓂. The presheaf UHom𝒪|U is easily checked to be a sheaf, denoted by 𝓂𝒪(,𝒢). Thus Γ(X,𝓂𝒪(,𝒢)) =Hom𝒪(,𝒢). 𝓂𝒪(,𝒢) has a natural 𝒪-Module structure. Both Hom and 𝓂 are left exact in each variable (contravariant in the 1st variable, covariant in the 2nd). We have 𝓂𝒪(𝒪,𝒢)𝒢.

Let fx(𝓂𝒪(,𝒢))x. Then fx is represented by say f:|U𝒢|U, which gives rise to a homomorphism x𝒢x, i.e. an element of Hom𝒪X(x,𝒢x). Hence we have 𝒪x-module homomorphism

(𝓂𝒪(,𝒢))x Hom𝒪x(x,𝒢x)

which in general is neither injective nor surjective (but see (7.9)).

(xi) Direct image. Let Ψ= (ψ,θ): (X,𝒪X) (Y,𝒪Y) be a morphism of ringed spaces. If is an 𝒪X-Module (thus a sheaf on X), we define its direct image Ψ*, which is an 𝒪Y-Module (thus a sheaf on Y) as follows: Ψ*(V)=(ψ-1(V)) for each open set V in Y; (ψ-1(V)) is an 𝒪X(ψ-1(V))-module, hence an 𝒪Y(V)-module via the homomorphism θ(V)𝒪Y(V)𝒪X(ψ-1(V)).

Ψ* is a left-exact functor from 𝒪X-Modules to 𝒪Y-Modules. For the section functor Γ is left exact by (7.2). Hence if 0 is an exact sequence of 𝒪X-Modules, then 𝒪 Γ(ψ-1(V),) Γ(ψ-1(V),) Γ(ψ-1(V),) is exact for each open VY; hence 0Ψ* Ψ* Ψ* is exact.

In particular, if Y is the ringed space consisting of a single point and the ring 𝒪(X), then Ψ*()=(X)=Γ(X,). Thus Ψ*()=(X)=Γ(X,). Thus Ψ* is a 'relativization' of the section functor Γ.

Quasi-coherent and coherent sheaves

If is an 𝒪-Module, a homomorphism u:𝒪 gives rise to s=u(X)(I)(X), i.e. to a global section of . Conversely, given s(X) we may reconstruct u: if U is open in X and t𝒪(U), then u(t)=t·(s|U). Hence we have a one-one correspondence between 𝒪-Module homomorphisms 𝒪 and global sections of , hence between 𝒪-Module homomorphisms u:𝒪(I) and families (si)iI of global sections of , where I is any index set. u is an epimorphism if and only if each x is generated (as an 𝒪x-module) by the (si)x (for u is an epimorphism if and only if each ux:(𝒪x)(I)x is an epimorphism, by (7.1)).

is said to be quasi-coherent if each xX has an open neighbourhood U such that |U is the cokernel of a homomorphism 𝒪(I)|U𝒪(J)|U, where the index sets I,J are of arbitrary cardinal (and depend on U). Clearly 𝒪 itself is quasi-coherent as an 𝒪-Module.

Thus is quasi-coherent if and only if is locally generated by its sections and if the 'sheaf of relations' is locally generated by its sections.

An 𝒪-module is of finite type if each xX has an open neighbourhood U such that |U is generated by a finite set of sections of over U, i.e. if there exists an epimorphism 𝒪p|U|U for some integer p>0. If ,G are of finite type, then so are 𝒢 and 𝒪𝒢 (the latter because is right exact). If 𝒢 is of finite type and 𝒢 is a homomorphic image of , then 𝒢 is of finite type.

is said to be coherent if

(i) is of finite type;
(ii) for each open set U in X and each homomorphism φ:𝒪n|U|U (n a positive integer), Ker(φ) is of finite type.

Clearly a coherent sheaf is quasi-coherent. All these properties (quasi-coherence, finite type, coherence) are local with respect to the base-space X.

We shall use the following notation. If U is an open set in X, the phrase 'f:𝒢 (over U)' shall mean f:|U𝒢|U. Similarly for diagrams of sheaves and homomorphisms.

Lemma (7.3). If is a subsheaf of 𝒢, and 𝒢 is of finite type and 𝒢 is coherent, then is coherent.


Let i:𝒢 be the embedding. If we have f:𝒪n (over U), then if:𝒪n𝒢 (over U); but 𝒢 is coherent, hence kerf=kerif is of finite type.

Lemma (7.4). Let 𝒢,H be 𝒪-Modules. If we have a diagram

𝒢 h 0 f 𝒪p (over a neighbourhood ofxX)

with the row exact, then there exists an 𝒪-Module homomorphism g:𝒪p𝒢 (over a (smaller) neighbourhood of x), such that hg=f.


The map f defines p sections si (1ip) belonging to (U) (U some open neighbourhood of x). Explicitly, f(U) maps 𝒪(U)p into (U), and si is the image of the ith generator ei of 𝒪(U)p. Since h is an epimorphism, there exist gi,x𝒢x such that hx(gi,x)=(si)x (1ip). Each gi,x is represented by say gi(Ui); h(gi) agrees with si at x, hence in some open neighbourhood of x, say Vi (UiU). Let V=V1Vp, then the gi=gi|V define g:𝒪p𝒢 (over V), and we have (over V) hg(ei)=h(gi)=si|V=f(ei), hence hg=f.

Theorem (7.5). If 0f𝒢g0 is an exact sequence of 𝒪-Modules on X, and if any two of ,𝒢, are coherent, then so is the third.


(1) 𝒢, coherent. By (7.3) it is enough to show that is of finite type. Let xX. Since 𝒢 is of finite type we have an epimorphism u:𝒪p𝒢 (over some neighbourhood of x). Since is coherent, the kernel of gu is of finite type, hence we have an exact sequence

𝒪qv𝒪p gu0 (over some neighbourhoodUofx).

Hence a commutative diagram with exact rows:

𝒪q v 𝒪p gu 0 w AAAA ww u id 0 f 𝒢 g 0 (overU).

We wish to define w:𝒪q such that fw=uv, and show that w is an epimorphism (over U). Since guv=0, Im(uv)Ker(g)=Im(f), so we can define w to be f-1uv. To show that w is an epimorphism, let yU, consider the corresponding diagram of stalks over y, and verify that wy is an epimorphism by diagram-chasing. Hence by (7.1) w is an epimorphism and therefore is of finite type.

(2) ,𝒢 coherent. 𝒢 is of finite type, hence so is . Let xX and let u:𝒪p be a homomorphism (over an open neighbourhood of x). By (7.4) we can lift u to v:𝒪p𝒢 (over a smaller open neighbourhood of x), so that gv=u. is of finite type, hence we have say e:𝒪q (over some open neighbourhood of x). Hence we have the following diagram:

0 f g 0 e t u 0 𝒪q rh 𝒪p+q sk 𝒪p 0 (over a neighourhood of x).

in which the rows are exact and the bottom row is split: rh=1, ks=1, hr+sk=1. Define t=fer+vk:𝒪p+q𝒢, then the diagram is commutative. Since 𝒢 is coherent, the kernel of t is of finite type and we can therefore enlarge the diagram:

0 f g 0 e t u 0 𝒪q h 𝒪p+q k 𝒪p 0 k kw 𝒪n id 𝒪n (over a neighourhood of x).

Verify that the right-hand column is exact, e.g. by considering the corresponding diagram of stalks over a point yU. Hence is coherent.

(3) , coherent. Since and are of finite type we have

0 f g 0 u v AAAA vv w 0 𝒪q 𝒪p+q 𝒪p 0 (over a neighourhood of x).

with u,w epimorphisms; hence as in (2) we can define v:𝒪p+q𝒢. Since u,w are epi, so is v (by the 5 lemma). Hence 𝒢 is of finite type.

Now let u:𝒪r𝒢 be a homomorphism (over some open neighbourhood of x); we have to show that Ker(u) is of finite type. Since is coherent we have an exact sequence of the form 𝒪sv𝒪rgu (over some open neighbourhood u of x), hence a diagram

0 f g 0 w AAAA ww u id 0 𝒪s v 𝒪r gu 0 (overU).

here we have guv=0, hence Im(uv)Ker(g)=Im(f), so we can define w:𝒪s (over U) so that uv=fw. Now is coherent, hence the kernel of w is of finite type, hence we can enlarge the diagram:

0 f g 0 w u id 0 𝒪s v 𝒪r gu p vp 𝒪t id 𝒪t (over a neighourhood of x).

Here the first column (as well as the top row) is exact, and we verify (e.g. by diagram-chasing in the stalks) that the second column is exact. Hence Ker(u) is of finite type and therefore Q is coherent.

Corollary (7.6). and 𝒢 are coherent if and only if 𝒢 is coherent.


If ,𝒢 are coherent, the exact sequence 0𝒢𝒢0 shows that 𝒢 is coherent. If 𝒢 is coherent then is of finite type because it is a homomorphic image of 𝒢; it is also a subsheaf of 𝒢, hence coherent by (7.3).

Corollary (7.7). If φ:𝒢 is a homomorphism of coherent 𝒪-Modules, then the kernel, image and cokernel of φ are all coherent.


Im(φ) is a homomorphic image of , hence is of finite type; it is also a sub-Module of 𝒢, hence coherent by (7.3). Now apply (7.5) to the exact sequences

0Ker(φ) Im(φ)0 0Im(φ)𝒢 Coker(φ)0.

Corollary (7.8). If 1φ2 34 ψ5 is an exact sequence in which all but 3 are coherent, then 3 is coherent.


From (7.7) and the exact sequence 0Coker(φ)3Ker(ψ)0.

Proposition (7.9). If ,𝒢 are coherent 𝒪-Modules, then 𝒪𝒢 and ℋℴ𝓂𝒪(,𝒢) are coherent.


Consider 𝒪𝒢. Let xX; since is coherent there is an exact sequence.

() 𝒪q𝒪p0 (over some open neighourboodUof x);

hence, as tensoring with 𝒢 is right exact and 𝒪𝒢𝒢, an exact sequence

𝒢q𝒢p𝒢 0(overU);

since 𝒢 is coherent, so are 𝒢p,𝒢q by (7.6), hence 𝒢 is coherent by (7.7) and the fact that coherence is a local property.

For ℋℴ𝓂𝒪(,𝒢), operate on () ℋℴ𝓂𝒪(,𝒢). The argument is similar.

Proposition (7.10). If ,𝒢 are 𝒪-Modules and is coherent, the mapping

(ℋℴ𝓂𝒪(,𝒢))x Hom𝒪x(x,𝒢x)

is an isomorphism.


From () we have 𝒪xq 𝒪xp x0 exact, hence by the left exactness of ℋℴ𝓂𝒪 and Hom𝒪x we have exact sequences

0 (ℋℴ𝓂𝒪(,𝒢))x (ℋℴ𝓂𝒪(𝒪p,𝒢))x (ℋℴ𝓂𝒪(𝒪q,𝒢))x 0 Hom𝒪x(x,𝒢x) Hom𝒪x(𝒪xp,𝒢x) Hom𝒪x(𝒪xq,𝒢x).

Since ℋℴ𝓂𝒪(𝒪p,𝒢)=𝒢p, the second and third vertical arrows are isomorphisms, hence so is the first.

If 𝒪 itself is coherent as an 𝒪-Module, we shall say that 𝒪 is a coherent sheaf of rings.

Proposition (7.11). Let 𝒪 be a coherent sheaf of rings and let 𝒢 be an 𝒪-Module. Then is coherent if and only if it is locally finitely presented, i.e. for each xX there is an exact sequence 𝒪q𝒪p0 over some neighbourhood of x.


If is coherent it is locally finitely presented (whether 𝒪 is coherent or not). Conversely, if 𝒪 is coherent, so are 𝒪p and 𝒪q by (7.6), hence so is by (7.7) (since coherence is a local property).

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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