#algebraic-geometry — Public Fediverse posts

An example of a presheaf without an associated sheaf – UPDATED

[This was originally posted on Google+ on 19 June 2013. This has been lightly edited to fit the new format and for clarity. ADDED Dec 2025: it’s also wrong! I explain at the end]

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

(using for a pair of parallel arrows) 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

1. I find this a nicer example, as one then doesn’t have to mess around with descriptions of cardinals and specially constructed bounded functions. Then one should be able to show without too much effort that given a presheaf on , or even , which restricts to the full subcategory as the forgetful functor has no sheafification for the flat pretopology, because it would give one for .
2. It would be nice to say that the presheaf on above was a (nice) Kan extension of the presheaf , then one wouldn’t have to fiddle with existence of presheaves restricting to . This seems not unlikely, given the definition of using bounds on residue fields. Alternatively, we could perhaps extend point 1. to work for any Kan extension of (left/right as appropriate), rather than an extension up to isomorphism.
3. We find that all we are doing is taking the fact that the restriction to of the flat pretopology of is just the maximal pretopology (actually see (*) below), where all maps are covers, and there is no weakly initial set in . Such conditions would probably give a non-sheafifiable presheaf on any for a concrete category whose image in is unbounded.
   (*) 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?
4. On a more foundational/structural note, I would like to be able to define the maps without choosing a well-ordering of every field, but I’m not sure I can do that, as one might not be able to get enough maps . Really one just needs, for any field , a sheaf and a map such that is injective. I don’t know how to supply this if I don’t have AC, but I haven’t thought very hard. Ideas? Perhaps we can prove this by contradiction.

….and why this is wrong.

Sadly, the above reasoning falls over right at the part where I claimed “(this shouldn’t change the calculation above)”. It’s true that one can get a singleton coverage on as described, where every inclusion of fields gives a cover in the opposite category (‘singleton’ in the sense that every covering family consists of a single arrow). The definition of (singleton) coverage doesn’t require pullbacks, merely ‘filler’ squares with the same ‘parallel arrow also is a cover’ condition. For the replacement of the “consider the pullback of along itself”, we can actually fill the square with the identity map of (in the opposite cat) on the remaining two sides. More generally, any maps can be used to fill in the square, using the fact every arrow is a cover. From now on I’ll drop all the ‘op’ business for simplicity.

So when we come to consider the sheaf condition for , the correct definition to use for the general coverage we have here is that for any filling square as suitable, and descent data for the presheaf, it glues uniquely. I have a small quibble in that the Elephant only defines a compatible family (i.e. descent data) for a general coverage as a subordinate clause in the definition of sheaf. But it amounts to this in the current context: descent data for is an element , such that when a span fills the square (i.e. ) then . The sheaf condition is then that given such descent data, we have that there is a unique such that . Since this condition quantifies over fillers of the square, we can consider the subclass where , and thus that . The only way that we can have for all elements of the Galois group is that comes from the (underlying set of the) base field .

Thus while Waterhouse’s example is not a sheaf, and it doesn’t have a sheafification, its restriction to the category of fields is already a sheaf.

#AlgebraicGeometry #CategoryTheory #Mathematics #SheafTheory #Waterhouse

Tenure-track opening @ U. Colorado Boulder Dept. of Math!

Esp. (but not only) looking for:
algebraic geometry
homotopy theory
foundations
functional analysis
number theory
interdisciplinary collab. b/w math & computer science or the math of quantum physics

https://www.mathjobs.org/jobs/list/27231

Please help spread the word!

#AlgebraicGeometry #HomotopyTheory #AlgebraicTopology #FoundationsOfMath #FunctionalAnalysis #NumberTheory #ComputerScience #Quantum #Math

📚🧮 Oh joy, another thrilling installment of 'Algebraic Geometry Notes' with an awe-inspiring #timeline from 2010 to 2024. 🤓✨ Because who wouldn't want to compare outdated math notes like fine wines in a cellar? Cheers to the riveting world of algebraic updates! 🍷📜
https://math.stanford.edu/~vakil/216blog/ #AlgebraicGeometry #MathNotes #2010to2024 #MathUpdates #GeekCulture #HackerNews #ngated

The Rising Sea: Foundations of Algebraic Geometry Notes

https://math.stanford.edu/~vakil/216blog/

#HackerNews #TheRisingSea #AlgebraicGeometry #MathNotes #Stanford #University #Education

A false statement that is easy to believe as true:

Suppose an algebraic group acts on an irreducible variety such that there is a Zariski dense orbit. Then there must be finitely many orbits. No! (But the converse is certainly true).

Example: Let GL(V) act on V times V by simultaneous left multiplication. Then the set of pairs (u,v) with linearly independent u and v forms a dense orbit. But the complement consists of infinitely many GL(V) orbits. Indeed, for different scalars t, the pairs (u, tu) lie in different orbits!

#math #maths #example #algebraicgeometry

Two postdoc positions (2-5 years) at our institute (at the Potsdam site) in an ERC Synergy Grant project on mathematical structures of scattering amplitudes.

https://astrodon.social/@mpi_grav/113504052382582384

#Mathematics #AlgebraicGeometry #NumberTheory #Mathstodon #Postdoc

Algebraische Geometrie und ihre Schichten von Definitionen ...

#mathmemes #algebraicgeometry

Are the following vector fields (section of a vector bundle) somehow special?

Consider the space V of k x n matrices. I want to consider those sections of a trivial vector bundle on V that are of the form: there exists a vector u such that the vector over M in V is Mu.

#geometry #AlgebraicGeometry #algebra

the cross product shouldn't exist. wedge products all the way!

#math #maths #vector #AlgebraicGeometry

Is there a name for surfaces in 4D that contain 2 different families of elliptic curves? #math #algebraicgeometry

Nine years ago on a defunct platform I wrote this (and I made a big statement, now-deleted, more below)

==== (start quote)
What do you think of this explanation of the Hodge conjecture by Melbourne mathematician Arun Ram?

https://theconversation.com/millennium-prize-the-hodge-conjecture-4243

I think it's great: the right level of metaphor and concrete examples that arise from simple cases of the problem.

[...]

#mathematics #millenniumprize #hodge #algebraicgeometry
==== (end quote)

Last week I was asked to explain off the cuff the Hodge conjecture by a colleague who is an engineer of some kind. I had recalled from a more recent peep into a certain pop-maths book (mentioned in the [...] above) that the author kinda wimped out, leaving the Hodge conjecture for last and then admitting it was rather complicated. I think I was a bit harsh in what I said in the redacted bit. But that may have been because I don't have the expertise in writing for a popular audience, and this certain person very much did, and held themselves to a higher standard of exposition in both detail and clarity than I would have, nine years ago.

I'm also slightly cooler on the metaphor in the linked post — it's still a good metaphor, but it doesn't work so well for me now as when I first wrote the above.

If R is a filtered ring, but filtered by an ordered commutative monoid (not necessarily N), is the associated graded a flat deformation?

#algebra #math #AlgebraicGeometry

I posted a brief note about Hensel lifting for systems of polynomial equations when number of eqns ≠ number of vars.

Purely expository b/c I couldn't find it anywhere, even though it's elementary. Feedback appreciated before I decide about posting on arXiv!

https://home.cs.colorado.edu/~jgrochow/hensel-notes.pdf

#AlgebraicGeometry #algebra

#Macaulay2 is a #ComputerAlgebra System devoted to supporting research in #AlgebraicGeometry and commutative algebra, whose creation has been funded by the National Science Foundation since 1992.

Macaulay2 includes core algorithms for computing Gröbner bases and graded or multi-graded free resolutions of modules over quotient rings of graded or multi-graded polynomial rings with a monomial ordering. The core algorithms are accessible through a versatile high level interpreted user language with a powerful debugger supporting the creation of new classes of mathematical objects and the installation of methods for computing specifically with them. Macaulay2 can compute Betti numbers, Ext, cohomology of coherent sheaves on projective varieties, primary decomposition of ideals, integral closure of rings, and more.

https://macaulay2.com/

An affine variety can be embedded into proj space in many ways. Do they all give the same Betti tables? Or are the Betti tables somehow related? So much I've found starts w projective varieties / graded rings. Are there good keywords to search for to understand 👆?

#AlgebraicGeometry #algebra

Anyone know a good source for multivariate Hensel lifting w/ more equations than variables?

I wrote up some brief notes for myself on it a while back, and I'd post them, but I'd feel silly doing so if there's a good source out there already.

#Algebra #AlgebraicGeometry #NumberTheory

Is there a term for when a class of equations has sol'ns over an extension field iff it has sol'ns over the ground field?

Examples:
1. Linear eqns
2. Eqns saying that two fin. dim. rep'ns of a fin. generated algebra are equivalent

Non-ex: eqns saying two 3-tensors are isomorphic

#Algebra #AlgebraicGeometry

Question for #AlgebraicGeometry #AlgebraicTopology folk:

Suppose I have a Zariski-constructible set S defined over the reals R (or even over the integers Z). Is there a nice relationship between the reduced singular (co)homology of its real points and its complex points?

While preparing for #juliacon :julia: 2023, #throwback to last year and one of the #projects I presented.

GeometricTheoremProver.jl: simple package for automated theorem proving in Euclidean geometry, powered by #julialang amazing metaprogramming and Ritt-Wu method from #algebraicgeometry.

Still a lot of room for improvement and development, hopefully over time...

https://github.com/lucaferranti/GeometricTheoremProver.jl

Dang, this #AlgebraicGeometry text has a very poetic title, "The Rising Sea", which refers to an oneiric metaphor of Grothendieck about how a nut of knowledge is slowly cracked by the slowness of an incoming ocean tide.

It also cites Morpheus from The Matrix in its preface, haha.

Now I really want to read this. I like poetical mathematics. Kinda like Ada Lovelace.

http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf