Last update: 11 June 2013
Let be any commutative ring with identity element, and let We recall that, for any the 'basic open set' is the set of all such that i.e. such that Let be any then we can form the module of fractions whose elements are all fractions of the form an integer is a module over the ring We may then consider the presheaf defined on the basis of We denote by the presheaf on which it determines. Since each is an is an
Proposition (9.1). is a sheaf, and hence for all In particular
Copy of the proof of (5.1) (ii).
Since formation of modules of fractions preserves exactness, it follows that the functor (from to is exact. Moreover,
Corollary (9.2). If are then
gives rise to for each hence to Hence we have a homomorphism Conversely, given we have i.e. by (9.1) a homomorphism Hence a map Verify that the two maps so defined are inverses of each other.
Theorem (9.3). Let be an Then the following are equivalent:
|(b)||there exists a finite open covering of by basic open sets such that for some and each index|
satisfies the following two conditions:
|Proof according to the scheme|
(a) (b). Take the covering of consisting of the single set
(b) (c). Since quasi-coherence is a local property, it is enough to prove (a) (c). We have an exact sequence where are direct sums of copies of hence, since is an exact functor, an exact functor, an exact sequence Hence is quasi-coherent.
(c) (b). Each has a neighbourhood over which is the cokernel of a homomorphism i.e. a homomorphism Hence by (9.2) where Since is quasi-compact, (b) is proved.
(a) (d). If and then for some and some integer hence image of in i.e. is the image of an element of If and in then for some integer from the basic properties of modules of fractions.
(b) (d). We have to show that if each satisfies (d), then so does Take first. We have then and Then (since hence by applied to there exists an integer such that i.e. in now is a unit in hence in Let be the largest of the then we have in each hence is the zero section of
To prove take and By applying to there exists an integer and an element which extends Since is a unit in there exists and extends and we may take all the to be equal, say By construction, restricted to is zero; now since it follows that each satisfies (a) and therefore (d), hence by applied to there exist integers such that restricted to is zero; but is a unit in hence restricted to is zero, where Hence the are all of them restrictions of a global section of This section is an extension of hence is proved.
(d) (a). Let We shall define a homomorphism and show that it is an isomorphism. For this we must define for each satisfying the usual compatibility conditions. Start with the restriction homomorphism i.e. Since is a unit in this homomorphism factorizes through This defines We shall show that implies surjective, and implies injective.
Let be any element of Then by lifts to a global section of for some integer i.e. is in the image of hence is in Hence, as is a unit in we have and thus is surjective.
If is such that then and therefore the restriction of to is zero; hence by there exists an integer such that hence in hence is injective.
Corollary (9.4). is exact on quasi-coherent Modules over an affine scheme.
Let be an exact sequence of quasi-coherent By (9.2) and (9.3) this sequence is of the form etc.). If then (since the functor is exact), hence Hence the sequence is exact, i.e. the sequence is exact.
Theorem (9.5). Let be a Noetherian ring, an Then the following are equivalent:
|(ii)||is of finite type and quasi-coherent;|
|(iii)||for some finitely-generated|
(i) (ii) is always true (from the definitions).
(ii) (iii): By (9.3) we have for some Since is of finite type and is quasi-compact, there exists a finite covering of by basic open sets and exact sequences (over i.e. exact sequences hence, by (9.4), exact sequences Thus each is a finitely-generated generated say by Let be the submodule of generated by all the If then is of the form hence for all indices and some integer Since the cover the generate the unit ideal, i.e. we have an equation of the form where Hence consequently and therefore is finitely generated.
(iii) (i). Suppose where is a finitely generated Then we have an exact sequence of the form for some integer hence thus is of finite type. It remains to show that if over some open set (which we may take to be for some then the kernel is of finite type. We have a homomorphism hence a homomorphism by (9.2); now is Noetherian (since 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 is Noetherian, is a coherent sheaf of rings.
Proposition (9.7). Let be a quasi-coherent and let be a covering of by basic open sets Then for all (and of course
By (9.3) we have where is an Consider the resolution of (Chapter 8):
whose sections over form the complex Recall that is the sheaf associated with the pre sheaf
now if we have say; hence is the sheaf associated with the presheaf so that Hence, by (9.3), the sheaf is quasi-coherent; now is exact on quasi-coherent sheaves (9.4), hence the complex is exact, i.e. for all
Theorem (9.8). If is an affine scheme and a quasi-coherent sheaf on then for all
Since finite basic open coverings are cofinal in the class of all open coverings of it follows from (9.7) that for all Hence for any basic open set we have for all (since is an affine open set and is quasi-coherent). Hence by Cartan's criterion (8.3) we have for all Hence for all
Remark. There is another proof, due to Chevalley, of (9.8) avoiding the use of (8.3) (which we didn't prove). Let be an injective resolution of a quasi-coherent sheaf on Then we have short exact sequences
where and for
Lemma (9.9). Let and let be any finite covering of by basic open sets. Then for all and all
Proof by induction on True for by (9.7). Let and assume (9.9) true for this value of (and all Then for any finite covering of by basic open sets. Since such open coverings of are cofinal in the class of all open coverings of it follows that and therefore that (since always). Hence, from the exact cohomology sequence of we have an exact sequence
Since this sequence is exact for. every it follows that the sequence of complexes
is exact. Now is injective, hence its restriction to the open set is injective and therefore the complex is acyclic; consequently, from the cohomology exact sequence of we get
and the term on the right is zero by the inductive hypothesis.
Taking in (9.9), we have for all hence hence But from the exact sequences we get (since each is injective)
Theorem (9.10). If is a scheme and is a quasi-coherent then for any covering of by affine open sets.
Let be an affine open covering of Since, is a scheme, each is affine and hence by (9.8) for all and all Hence by the comparison theorem (8.2) we have for all
Corollary (9.11). under the hypotheses of (9.10).
There is a converse of (9.8):
Theorem (9.12). (Serre's criterion.) Let be either a quasi-compact scheme or a prescheme whose underlying space is Noetherian. If for every quasi-coherent (or even only for every quasi-coherent Ideal of then 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 for and quasi-coherent characterizes affine schemes.
Let be a projective algebraic variety over an algebraically closed field and let be a coherent where is the sheaf of local rings on Serre proved that
|(ii)||is a finite-dimensional space for|
The proof of (i) is easy: by (9.10) (or rather its counterpart for algebraic varieties) it is enough to find a covering of by affine open sets, where and this can be achieved by intersecting by suitably chosen hyperplanes in the projective space in which is embedded. (ii) is proved by reducing to the case where and calculating the quite explicitly.
Grothendieck subsequently generalized this theorem, firstly to the case where is complete (but not necessarily projective) and then to a statement about proper morphisms. If is a morphism of algebraic varieties, then (Chapter 7) is a left-exact functor from to hence has right derived functors Explicitly, if is an is the sheaf on associated to the presheaf open in Then:
If are algebraic varieties over a proper morphism, a coherent then the 'higher direct images' are coherent (The statement for a complete variety is obtained by taking to consist of a single point.)
Finally, this theorem generalizes to the case of a proper morphism of preschemes:
Let be preschemes, locally Noetherian (this means that can be covered by affine open sets each of which is the scheme of a Noetherian ring). If is a proper morphism and a coherent then the are coherent (E.G.A., III, 3.2.1).
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".