Diagonalisable groups

Stream: toric

[Sophie Morel]

What exactly do you need about diagonalizable groups? I will probably need tori (over a possibly non algebraically closed field) for another project, so maybe it would make sense for me to "claim" the "define diaginalizable groups" task?

[Yaël Dillies]

Oh that is great if you need them too!

[Yaël Dillies]

IANAAG (I Am Not An Algebraic Geometer) and Michal knows the story better than me, but he is sound asleep right now so I get a shot at explaining.

[Yaël Dillies]

The whole of Section 1.1 of CLS is about reducing statements from the world of tori/toric varieties to the world of Hopf algebras.

For this to work, we need to know that, for a torus $T$ and an algebraic group $G$, maps $T \to G$ and $G \to T$ correspond to maps $\Gamma(T) \to \Gamma(G)$ and $\Gamma(G) \to \Gamma(T)$ respectively. This is more or less Cartier duality.

To prove Cartier duality, there are two approaches:

• For Cartier duality over a Spec of a field $k$, we can reduce ourselves to proving simple results about group-like elements (which are now mostly done) if we define "$G$ is diagonalisable" to mean "the group-like elements of $\Gamma(G)$ span it as a $k$-vector space".
• For Cartier duality over a general ring, we can follow SGA III, Exposé VIII, which cheekily defines diagonalisable groups as the Cartier duals of constant group schemes, then proceeds on to showing that constant group schemes are reflexive, meaning that diagonalisable groups too are reflexive, and tada!

In short, the definition of diagonalisable groups depends on which approach we go down. Michal and I started the easy group-like elements one, but if you are interested then the SGA approach is probably the one that mathlib will want to have eventually (I heard you could read French? :wink:)

[Michał Mrugała]

This is not Cartier duality, though Cartier duality can come up here. For the purposes of this project we need tori and 5 facts about them:

• The character lattice of 𝔾ₘ ^ n is ℤ ^ n: 1-1-char-torus.
• The cocharacter lattice is also ℤ ^ n: 1-1-cochar-torus.
• There is a perfect pairing between the character and cocharacter lattices: 1-1-char-cochar-pairing.
• Irreducible subgroups of tori are tori: 1-1-1-subgroup-torus.
• The image of a homomorphism T₁ → T₂ is a closed subtorus of T₂: 1-1-1-group-hom-subtorus.

[Michał Mrugała]

The way I was intending to prove this is to first prove that there is an equivalence of categories between f.g. abelian groups and diagonalizable schemes over an affine base: 0-fg-comm-grp-equiv-diag-grp-sch. We only need this fact over fields, and will probably only try to prove it over fields for now, though I got some suggestions from Jarod Alper for how to go about proving the result over any ring.

[Michał Mrugała]

Cartier duality comes up in Grothendieck's proof that we get a fully faithful functor from f.g. abelian groups to group schemes by showing that constant group schemes are reflexive and so diagonalizable group schemes (the Cartier duals) are also reflexive. I think this would be a good route to prove the result over a general base.

[Michał Mrugała]

At the moment the main obstruction to defining diagonalizable group schemes is lack of the correspondence between affine group schemes and Hopf algebras. Me and Yaël are working on it, but it's proving surprisingly difficult to even find the correct statement!

[Yaël Dillies]

Michał Mrugała said:

it's proving surprisingly difficult to even find the correct statement!

I've now got a statement locally :wink:

[Kevin Buzzard]

I think you might want to talk to Amelia Livingston about this, she might have some useful stuff

[Kevin Buzzard]

She's been thinking about the global Langlands conjectures for tori and it's not a coincidence that she wrote a lot of the hopf algebra stuff in mathlib

[Sophie Morel]

Should I ping her to join this conversation, or is that bad etiquette?

[Patrick Massot]

You can ping her.

[Sophie Morel]

Wait, I apparently need to subscribe her first. At this point I might as well send her a direct message.

[Michał Mrugała]

One result about diagonalizable groups that would be very useful to show is that the functor Grp ⥤ Grp_ Scheme is exact. I'm not exactly sure how this notion should be formalized yet (I guess the group schemes here are abelian, so we should have an abelian category to work with).

[Andrew Yang]

Exact probably means preserves finite limits and colimits.

[Michał Mrugała]

The functor is exact in terms of exact sequences of fpqc sheaves. See Theorem 3.1 in SGA 3 expose 8.

[Michał Mrugała]

I think for the purposes of toric we would like to have this
image.png

[Andrew Yang]

I don't think it mentions fpqc sheaves? It just says that $D_S$ takes injections to fpqc morphisms.

[Michał Mrugała]

I didn't post the fpqc sheaf part, since I don't think we'll need it. Here is the full theorem:
image.png

[Michał Mrugała]

I updated the blueprint with the statements we need and included proof sketches.

[Sophie Morel]

I introduced a definition of diagonalizable Hopf algebras and I'm planning to prove stuff for them first, to avoid getting lost in category theory stuff that is being worked out independently. Btw, the characterization of diagonalizable group schemes in the project is wrong as stated, because we need to assume that the group schemes are affine. (Otherwise, an abelian variety would be diagonalizable.)

[Michał Mrugała]

Nice catch! This seems to be a mistake in Milne's notes as well. Right now the blueprint for diagonalizable group schemes is a mess and needs a rehaul. Feel free to make changes to the blueprint in general!

[Sophie Morel]

Another question: why do we assume that the group A is finitely generated in the definition of a diagonalizable group scheme (as something isomorphism to a Spec R[A])? SGA 3 does not make any such assumption.

[Michał Mrugała]

Another consequence of me reading Milne uncritically :sweat_smile:
Thought I still imagine that we'd want tori to come from finitely generated groups.

[Sophie Morel]

Yes, we want tori to come from finitely generated free groups. Are you planning the theory of locally diagonalizable groups in various topologies? Tori are not diagonalizable over a non-algebraically closed field (and a fortiori a general base), only fpqc locally diagonalizable.

[Sophie Morel]

Or maybe all tori will be split to simplify?

[Michał Mrugała]

At the moment we weren't planning on developing those, but I'm open to develop that as part of the project. So far we're only working with split tori, I think we will eventually assume that we're working over an algebraically closed field of characteristic 0 so this shouldn't be a problem.

[Michał Mrugała]

But of course mathlib eventually needs to have non-split tori

[Sophie Morel]

I might need non-split tori, but diagonalizable groups are a good start.

[Michał Mrugała]

Working with other topologies might not be feasible right now, because afaik we have almost nothing regarding flat topologies and descent in mathlib.

[Christian Merten]

Michał Mrugała said:

Working with other topologies might not be feasible right now, because afaik we have almost nothing regarding flat topologies and descent in mathlib.

We have #23547 and more on fpqc descent coming in a follow-up.

[Michał Mrugała]

And here I was, hoping to maybe finally learn how descent works by formalizing it :pensive:

[Sophie Morel]

Okay, I have the characterization of diagonalizable Hopf algebras over fields, modulo some "easy" lemmas (like "comultiplication sends inverses to inverses"). The idea is very simple but the proof I wrote is disgusting.

[Sophie Morel]

Had to define the (commutative) group structure on grouplike elements, and prove the universal property of the monoid algebra as a bialgebra.

[Yaël Dillies]

That's great! Eagerly waiting for the PR :grinning_face_with_smiling_eyes:

[Sophie Morel]

I have to do PRs? You don't have elves to take care of that? :unamused:

Yeah okay, but it will probably be a few PRs, and I should polish the proofs a little bit.

[Michał Mrugała]

I want to hijack this thread to discuss how we should proceed with definitions and instances on diagonalizable groups @Andrew Yang @Sophie Morel

[Michał Mrugała]

Andrew is advocating not getting instances on things such as Spec ℤ[ℕ] to come from Hopf algebras because it will result in poor defeqs when we want to get ring scheme structures on them

[Sophie Morel]

Andrew has more mathlib experience than me, so I'll defer to him. Does that mean that we want the instances to come from the presheaves that these objects represent?

[Andrew Yang]

The conclusion I reached with @Yaël Dillies today is that the instance of Spec ℤ[G] could come from hopf algebras and the yoneda thing should take in the multiplication maps and proofs that they are compatible with the group structure on $\mathrm{Hom}(-, G)$. I am not 100% sure this is possible but I have yet to see an obstacle.

[Michał Mrugała]

I'm not sure I understand, would you mind elaborating?