theHigherGeometerAn example of a presheaf without an associated sheaf
[This was originally posted on Google+ on 19 June 2013. This has been lightly edited to fit the new format and for clarity]
I may have shared this blog posting before, but this is a really good example of when we have a large site which isn’t constrained by a small amount of data for each object: the category of (affine) schemes with the pretopology of flat surjections. One can define a presheaf on which has no sheafification for this pretopology, but its definition in the linked blog post explicitly uses von Neumann ordinals. I should like to write down a more structural version of this. I have some points I’d like to clear up, if you want to chip in, namely 2.-4. under ‘Some final comments’. The original source for this material is
- William C. Waterhouse, Basically bounded functors and flat sheaves.
Pacific J. Math. Volume 57, Number 2 (1975), 597-610, http://projecteuclid.org/euclid.pjm/1102906018
The example as given by Waterhouse
Given an affine scheme , assign to it the set of locally constant functions from to the von Neumann cardinal of the set
the supremum of the cardinalities of residue fields at points of , such that the value at any point (which is a cardinal less than ) is smaller than the cardinality of the residue field at that point. This gives a functor , using the fact maps of fields are injective.
Simplifying the example
In fact, one can take the site to be merely the full subcategory of affine schemes which are spectra of fields, since one arrives at a contradiction assuming the existence of a sheafification by using a flat covering for and fields (in fact any field extension gives such a cover). Then locally constant functions are merely elements of , or in other words, can be taken as itself. In other words, restricts to the forgetful functor . This is a very natural presheaf to consider (see comment 3 below regarding the pretopology on this subcategory).
Calculation
Let denote the constant sheaf on corresponding to a well-ordered set of the same name. Then there is a map of presheaves which is just the inclusion for and the retract sending to the bottom element of otherwise. Then by the universal property of sheafification, there must be a unique map making the obvious triangle commute, where is the sheafification of , for any . In particular, factors through . Now for any given take so that is injective, which implies that is injective, and hence that is a mono.
Now we use the fact that for any map (necessarily a flat cover) we have that the equaliser of the two maps [here we’ve embedded into , see comments below]
injects into (We can check this by applying the natural transformation to the diagram
and remembering is an equaliser.) But is a point, so this equaliser is itself (can we see this directly without going via the spectrum?)
The upshot of the preceding two paragraphs is this: must have an injective set map into , but can be as large as we like, independent of (take say the function field over on the power set of , which is certainly a field larger that ). Thus cannot have a sheafification.
Some final comments
(*) This is being a little slack, as isn’t a field, but can find a map from it to a field (as -algebras), and so we get a coverage on , rather than a Grothendieck pretopology (this shouldn’t change the calculation above). The inclusion functor is flat, so I think this means it is a morphism of sites as in Remark 2.3.7 of Sketches of an Elephant (certainly covers in are sent to covers in ). Any thoughts on this?
#algebraicGeometry #categoryTheory2 #mathematics #sheafTheory #waterhouse
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