Chapter 9

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

Last update: 11 June 2013

Cohomology of Affine schemes

Let A be any commutative ring with identity element, and let X=Spec(A). We recall that, for any fA, the 'basic open set' D(f) is the set of all xX such that f(x)0, i.e. such that fjx. Let M be any A-module; then we can form the module of fractions Mf, whose elements are all fractions of the form m/fn (mM n an integer 0). Mf is a module over the ring Af. We may then consider the presheaf D(f)Mf, defined on the basis =D(f)fA of X. We denote by M the presheaf on X which it determines. Since each Mf is an Af-module, M is an A-Module.

Proposition (9.1). M is a sheaf, and hence Γ(D(f),M)=Mf for all fA. In particular Γ(X,M)=M.


Copy of the proof of (5.1) (ii).

Since formation of modules of fractions preserves exactness, it follows that the functor MM (from A-modules to A-Modules) is exact. Moreover,

Corollary (9.2). If M,N are A-modules, then HomA(M,N) HomA(M,N).


φ:MN gives rise to φf:MfNf for each fA, hence to φ:MN. Hence we have a homomorphism HomA(M,N) HomA(M,N). Conversely, given u:MN we have u(X):M(X)N(X), i.e. by (9.1) a homomorphism MN. Hence a map HomA(M,N) HomA(M,N). Verify that the two maps so defined are inverses of each other.

Theorem (9.3). Let be an A-Module. Then the following are equivalent:

(a) M for some A-module M;
(b) there exists a finite open covering of X by basic open sets D(fi) such that |D(fi)Mi for some Afi-module Mi and each index i;
(c) is quasi-coherent;
(d) satisfies the following two conditions:
(d1) for each gA and each s(D(g)) there exists an integer n0 such that gns can be extended to a global section of (i.e. an element of (X));
(d2) for each gA and each t(X) such that t|D(g)=0, there exists n0 such that gnt=0.

Proof according to the scheme
(a) (b) (c). (d)

(a) (b). Take the covering of X consisting of the single set D(1)=X.

(b) (c). Since quasi-coherence is a local property, it is enough to prove (a) (c). We have an exact sequence A(I) A(J)M0, where A(I),A(J) are direct sums of copies of A; hence, since MM is an exact functor, an exact functor, an exact sequence A(I) A(J) M0. Hence M is quasi-coherent.

(c) (b). Each xX has a neighbourhood D(f) over which |D(f) is the cokernel of a homomorphism A(I)|D(f) A(J)|D(f), i.e. a homomorphism Af(I)Af(J). Hence by (9.2) |D(f)=N where N=coker(Af(I)Af(J)). Since X is quasi-compact, (b) is proved.

(a) (d). If gA and s(D(g))=Mg, then s=m/gn for some mM and some integer n0, hence sgn=m/1= image of m in Mg, i.e. sgn is the image of an element of M=(X). If gA, tM and t/1=0 in Mg, then tgn=0 for some integer n0, from the basic properties of modules of fractions.

(b) (d). We have to show that if each |D(fi) satisfies (d), then so does . Take (d2) first. We have then gA, t(X) and t|D(g)=0. Then t|D(gfi)=0 (since D(gfi)=D(f)D(fi)); hence by (d2) applied to |D(fi) there exists an integer ni0 such that (fig)nit|D(fi)=0, i.e. (fig)nit=0 in Mi; now fi is a unit in Afi, hence gnit=0 in Mi. Let n be the largest of the ni, then we have gnt=0 in each |D(fi), hence gnt is the zero section of .

To prove (d1): take gA and s(D(g)). By applying (d1) to |D(fi), there exists an integer ni0 and an element si(D(fi)) which extends (fig)nis|D(fig). Since fi is a unit in Afi, there exists si=finisi, and si extends gnis|D(fig); and we may take all the ni to be equal, say ni=n. By construction, si-sj restricted to D(fifjg) is zero; now since |D(fi)=Mi, it follows that each |D(fi)D(fj) satisfies (a) and therefore (d), hence by (d2) applied to |D(fi)D(fj) there exist integers mij0 such that (fifjg)mij(si-sj) restricted to D(fi)D(fj)=D(fifj) is zero; but fifj is a unit in (D(fifj)), hence gm(si-sj) restricted to D(fi)D(fj) is zero, where m=max(mij). Hence the gmsiΓ(D(fi),) are all of them restrictions of a global section s of . This section s is an extension of gm+ns, hence (d1) is proved.

(d) (a). Let M=(X)=Γ(X,). We shall define a homomorphism u:M and show that it is an isomorphism. For this we must define uf:Mf(D(f)) for each fA, satisfying the usual compatibility conditions. Start with the restriction homomorphism (X)(D(f)), i.e. M(D(f)). Since f is a unit in Af, this homomorphism factorizes through Mf: MMfuf (D(f)). This defines uf. We shall show that (d1) implies uf surjective, and (d2) implies uf injective.

Let s be any element of (D(f)). Then by (d1) fns lifts to a global section of , for some integer n0, i.e. fns is in the image of M, hence is in uf(Mf). Hence, as f is a unit in Af, we have suf(Mf) and thus uf is surjective.

If z/fnMf is such that uf(z/fn)=0, then uf(z/1)=0 and therefore the restriction of z((X)) to D(f) is zero; hence by (d2) there exists an integer m0 such that zfm=0; hence z/fn=0 in Mf, hence uf is injective.

Corollary (9.4). Γ is exact on quasi-coherent Modules over an affine scheme.


Let 𝒢 be an exact sequence of quasi-coherent A-Modules. By (9.2) and (9.3) this sequence is of the form MuNvP (M=(X), etc.). If Q=Im(u), R=Ker(v) then Q=R (since the functor MM is exact), hence Q=Q(X)=R(X)=R. Hence the sequence MNP is exact, i.e. the sequence Γ(X,) Γ(X,𝒢) Γ(X,) is exact.

Theorem (9.5). Let A be a Noetherian ring, an A-Module. Then the following are equivalent:

(i) is coherent;
(ii) is of finite type and quasi-coherent;
(iii) M for some finitely-generated A-module M.


(i) (ii) is always true (from the definitions).

(ii) (iii): By (9.3) we have M for some A-module M. Since is of finite type and X is quasi-compact, there exists a finite covering of X by basic open sets D(fi), and exact sequences Api0 (over D(fi)), i.e. exact sequences Afipi Mfi0; hence, by (9.4), exact sequences Afipi Mfi0. Thus each Mfi is a finitely-generated Afi-module, generated say by tij/1 (1jpi,tijM). Let N be the submodule of M generated by all the tij. If zM, then z/1Mfi is of the form j(tij/1)· (aij/fimj), hence zfimN for all indices i and some integer m>0. Since the D(fi) cover X, the fim generate the unit ideal, i.e. we have an equation of the form igifim=1, where giA. Hence z=izfimgiN, consequently M=N and therefore M is finitely generated.

(iii) (i). Suppose =M where M is a finitely generated A-module. Then we have an exact sequence of the form ApM0 for some integer p0, hence AM0; thus M is of finite type. It remains to show that if ApM over some open set (which we may take to be D(f) for some fA), then the kernel is of finite type. We have a homomorphism AfpMf, hence a homomorphism AfpMf by (9.2); now Af is Noetherian (since A is), hence the kernel is finitely generated. This completes the proof.

Remark. The Noetherian assumption intervenes only in the proof of (iii) (i).

Corollary (9.6). If A is Noetherian, A is a coherent sheaf of rings.

Proposition (9.7). Let be a quasi-coherent A-Module, and let 𝓊 be a covering of X=Spec(A) by basic open sets D(fi). Then Hp(𝓊,)=0 for all p>0 (and of course H0(𝓊,)= (X)).


By (9.3) we have =M, where M=(X) is an A-module. Consider the Čech resolution of M (Chapter 8):

0MC0 (𝓊,M) C(𝓊,M)

whose sections over X form the Čech complex 0MC0(𝓊,M) C1(𝓊,M) . Recall that Cq(𝓊,M) is the sheaf associated with the pre sheaf

D(g)σM (UσD(g));

now if σ=(i0,,iq) we have UσD(g)=D(fi0)D(fiq)D(g)=D(fi0fiqg)=D(fσg) say; hence Cq(𝓊,M) is the sheaf associated with the presheaf D(g) σMfσg= (σMfσ)g, so that Cq(𝓊,M)=(σMfσ). Hence, by (9.3), the sheaf Cq(𝓊,M) is quasi-coherent; now Γ is exact on quasi-coherent sheaves (9.4), hence the Čech complex is exact, i.e. Hp(𝓊,)=0 for all p>0.

Theorem (9.8). If X is an affine scheme and a quasi-coherent sheaf on X, then Hp(X,)=0 for all p>0.


Since finite basic open coverings are cofinal in the class of all open coverings of X, it follows from (9.7) that Hˇp(X,)=0 for all p>0. Hence for any basic open set U=D(f) we have Hˇq(U,|U)=0 for all q>0 (since U is an affine open set and |U is quasi-coherent). Hence by Cartan's criterion (8.3) we have Hˇp(X,)= Hp(X,) for all p>0. Hence Hp(X,)=0 for all p>0.

Remark. There is another proof, due to Chevalley, of (9.8) avoiding the use of (8.3) (which we didn't prove). Let 001 be an injective resolution of a quasi-coherent sheaf on X. Then we have short exact sequences

(Ep):0 𝒢pp 𝒢p+10

where 𝒢0= and 𝒢p=Im(p-1p) for p>0.

Lemma (9.9). Let fA and let 𝓊 be any finite covering of D(f) by basic open sets. Then Hq(𝓊,𝒢p|D(f))=0 for all p0 and all q>0.


Proof by induction on p. True for p=0 by (9.7). Let p0 and assume (9.9) true for this value of p (and all q>0). Then H1(𝓊,𝒢p|D(f))=0 for any finite covering of D(f) by basic open sets. Since such open coverings of D(f) are cofinal in the class of all open coverings of D(f) it follows that Hˇ1(D(f),𝒢p|D(f))=0 and therefore that H1(D(f),𝒢p|D(f))=0 (since H1=Hˇ1 always). Hence, from the exact cohomology sequence of (Ep), we have an exact sequence

0𝒢p(D(f)) p(D(f)) 𝒢p+1(D(f)) 0.

Since this sequence is exact for. every fA, it follows that the sequence of Čech complexes

() 0C (𝓊,p|D(f)) C (𝓊,p|D(f)) C(𝓊,𝒢p+1|D(f)) 0

is exact. Now p is injective, hence its restriction to the open set D(f) is injective and therefore the complex C(𝓊,p|D(f)) is acyclic; consequently, from the cohomology exact sequence of (), we get

Hq(𝓊,𝒢p+1|D(f)) Hq+1 (𝓊,𝒢p|D(f)) (q>0)

and the term on the right is zero by the inductive hypothesis.

Taking f=1, q=1 in (9.9), we have H1(𝓊,𝒢p)=0 for all p0, hence Hˇ1(X,𝒢p)=0, hence H1(X,𝒢p)=0. But from the exact sequences (Ep) we get (since each p is injective)

Hp(X,)= Hp(X,𝒢0) Hp-1 (X,𝒢1) H1(X,𝒢p-1) =0(p>0).

Theorem (9.10). If (X,𝒪X) is a scheme and is a quasi-coherent 𝒪X-Module, then Hq(X,)Hq(𝓊,) for any covering 𝓊 of X by affine open sets.


Let 𝓊=(Ui)iI be an affine open covering of X. Since, X is a scheme, each Uσ=Ui0Uiq is affine and hence by (9.8) Hp(Uσ,|Uσ)=0 for all σ and all p>0. Hence by the comparison theorem (8.2) we have Hp(X,) Hp(𝓊,) for all p0.

Corollary (9.11). Hp(X,) Hˇp(X,) under the hypotheses of (9.10).

There is a converse of (9.8):

Theorem (9.12). (Serre's criterion.) Let X be either a quasi-compact scheme or a prescheme whose underlying space is Noetherian. If H1(X,)=0 for every quasi-coherent 𝒪X-Module (or even only for every quasi-coherent Ideal of 𝒪X), then X is an affine scheme.

For the proof we refer to (E.G.A., II, 5.2.1). (9.8) and (9.12) show that the vanishing of the Hp(X,) for p>0 and quasi-coherent characterizes affine schemes.

Let X be a projective algebraic variety over an algebraically closed field k, and let be a coherent 𝒪X-Module, where 𝒪X is the sheaf of local rings on X. Serre proved that

(i) Hq(X,)=0 for q>dimX;
(ii) Hq(X,) is a finite-dimensional k-vector space for 0qdimX.

The proof of (i) is easy: by (9.10) (or rather its counterpart for algebraic varieties) it is enough to find a covering of X by d+1 affine open sets, where d=dimX, and this can be achieved by intersecting X by suitably chosen hyperplanes in the projective space P in which X is embedded. (ii) is proved by reducing to the case where X=P and calculating the Hq(P,) quite explicitly.

Grothendieck subsequently generalized this theorem, firstly to the case where X is complete (but not necessarily projective) and then to a statement about proper morphisms. If f:XY is a morphism of algebraic varieties, then f (Chapter 7) is a left-exact functor from 𝒪X-Modules to 𝒪Y-Modules, hence has right derived functors Rpf(p0). Explicitly, if is an 𝒪X-Module, Rpf() is the sheaf on Y associated to the presheaf UHp(f-1(U),) (U open in Y). Then:

If X,Y are algebraic varieties over k,f:XY a proper morphism, a coherent 𝒪X-Module, then the 'higher direct images' Rpf() are coherent 𝒪Y-Modules. (The statement for a complete variety X is obtained by taking Y to consist of a single point.)

Finally, this theorem generalizes to the case of a proper morphism of preschemes:

Let X,Y be preschemes, Y locally Noetherian (this means that Y can be covered by affine open sets each of which is the scheme of a Noetherian ring). If f:XY is a proper morphism and a coherent 𝒪X-Module, then the Rpf() are coherent 𝒪Y-Modules (E.G.A., III, 3.2.1).

Notes and References

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

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