In this post, we give a categorical proof of Lemma 33.2 (Tag 00AK) by showing that a glued sheaf is defined as an equaliser. This equaliser provides a tool for calculating global sections of glued schemes. We later present some examples for calculating global sections and glueing constructions using (co)limits descriptions. (Sometimes people say they may lose insights about the details and the real maths behind abstraction. But it really depends on one’s approach and ways of thinking. ) The goal of this post is to give a structured summary of glueing constructions of schemes after meditating on explicit constructions, using categorical language. It is not meant to replace explicit arguments for schemes, but to give some ideas on how general a construction is and whether it can be transferred to a different setting. For example, we will know what to do if we are working on a site with a different Grothendieck topology instead of the Zariski topology. Note that some consequences of the categorical facts we use are indeed straightforward for schemes, for example, the two pullback squares in Example 4. This post is open-ended and more examples of glueing will be added.
Remark. Note that for a sheaf on a topological space and an open cover
where is the covering sieve generated by the covering , namely, is the collection of all those with for some This limit is equivalent to the following equaliser diagram in the usual sheaf definition:
where for , and for a family
For a scheme with an open cover in the Zariski topology, is the colimit indexed over the covering sieve generated by the covering . This colimit can also be simplified to be a coequaliser diagram.
First, we include the explicit constructions for glueing sheaves and some sources for details check of these constructions. The categorical proof we are going to show is a categorical rephrasing by meditating on these constructions, which gives a more structured presentation.
Proposition A.1 [Tag 00AK] Let be a topological space. Let be an open covering. Let be sheaves of sets on . Given a collection
of maps of sheaves such that for all the maps restrict to the same map , then there exists a unique map of sheaves
whose restriction to each agrees with .
Proof. Take any , where is open, and let . Then we have an element and by the glueing condition. Thus by the sheaf condition for , the sections patch together to give a section in , define this section to be . (We omitted the checking details. )
Explicit construction of glueing sheaves is given in Lemma 6.33.2, Tag 00AK, but the details of checking have been omitted. For some details of a reality check, see this post.
Proof of Lemma 6.33.2: (Pdf version: Proof of sheaf glueing)
To glue schemes, one needs to define the glued topological spaces which will be a quotient space of the disjoint union of the glued spaces and then verify the structure sheaves of the glued schemes satisfy the condition of Lemma 6.33.2 [see Tag 01JA].
Example 1. (The affine line with doubled origin is not affine). Let be a field. Let , . Let . Consider the ismorphism given by . Let be the glued scheme, from the equaliser diagram in the proof of Lemma 6.33.2, we see that the structure sheaf is given by
Thus the global section . From this we see that is not affine, since which is not the case: the underlying topological space has one more point- the doubled origin.
Added on 21/01/2020
Example 2. (Quasiseparated scheme is glued from affine schemes). Note that every scheme is a colimit of affine schemes. This is true in general by the fact that every sheaf is a colimit of representables and that the Zariski topology is subcanonical (see here for more details).
For the case that a scheme is separated (for which the intersection
of any two affine open sets is affine), take an affine open cover such that each intersection is affine, then is just the coequaliser of the diagram . This is the glueing construction as we described above.
If is not separated, one can still write as a colimit of affines but not with the same diagram as we used for glueing, see here for a description of the diagram.
Remark. One implication of viewing as a colimit is: let Sch and Rings be the categories of schemes and rings respectively, given that , one can deduce that for any scheme , by the fact Hom-functor preserves (co)limits.
Added on 01/02/2020
Example 3 (Proj construction). [Section 4.5.7, Vakil]
Let be a -graded ring and be the irrevalant ideal. Suppose is homogeneous, there is a bijection between the prime ideals of and the homogeneous prime ideal of . The projective distinguished open set is identified with . If are homogeneous and nonzero, is isomorphic to the distinguished open subset of , similarly for . is glued from various along the pairwise intersections .
Example 4. (Fibre product of schemes)[For detailes of a categorical proof of this construction, see the post: Construction of Fibre Product of Schemes or the pdf here: Construction of fibre product.]
Given arbitrary schemes , let and be the given morphisms. Let be an open affine cover of . Let , , choose an affine open cover for and an affine open cover for . The fibre product is constructed by glueing various together.