Algebraic groups and representations

Stream: toric

[Kevin Buzzard]

How are you defining algebraic groups, let alone their representations?

I also need algebraic groups (over a general field, mostly affine ones) for FLT (and also finite group schemes over a general base).

[Michał Mrugała]

Channel for discussing implementation of algebraic groups and reps

[Notification Bot]

A message was moved here from #toric > Current tasks by Michał Mrugała.

[Michał Mrugała]

I was planning to define algebraic groups as group objects in the category of schemes (so {G : Scheme} [Grp_Class G]).

[Michał Mrugała]

When it comes to representations the plan is to define it as a morphism of functors G → GL V

[Michał Mrugała]

I'm very open to suggestion on how to approach defining both of these though, especially since they should be useful for other people.

[Kevin Buzzard]

Michał Mrugała said:

I was planning to define algebraic groups as group objects in the category of schemes (so {G : Scheme} [Grp_Class G]).

That will only give you group schemes over Spec(Z). You don't want group varieties over e.g. a field?

[Michał Mrugała]

We’ve also defined them in the Over category as group objects

[Sophie Morel]

Michał Mrugała said:

I'm very open to suggestion on how to approach defining both of these though, especially since they should be useful for other people.

I've definitely used infinite-dimensional representations before (the regular representation for example), but I'm trying to think how these could fit in your definition.

[Sophie Morel]

For an arbitrary representation of G on V, you want every element of V to be contained in a G-stable finite type submodule.

[Sophie Morel]

(So that it corresponds to a comodule structure on V over the Hopf algebra of G, when G is affine.)

[Sophie Morel]

So you want at least the elements of GL V to be locally finite automorphisms of V, i.e. automorphisms g of V such that every element of V is contained in a g-invariant submodule of finite type.

[Sophie Morel]

But I'm not sure that's enough to be able to define a group scheme structure on GL V. I want to say something like "GL V should be the colimit over all decompositions V = \oplus V i of V as a direct sum of finite type submodules of the product of the GL (V i) (maybe it's safer to make V projective too, I need to think about it more). We do want a property like that, but I find that definition ugly.

[Michał Mrugała]

The definition I found in the literature : GL V is defined to be the group valued functor on the category of R-algebras mapping A ↦ Aut_A(V ⊗ A). As far as I can tell this satisfies the conditions you outlined, but as far as I can tell GL V is not representable in general when V is infinite-dimensional.

[Sophie Morel]

I don't think that satisfies the condition you need for representations. For example, if V is the free k-vector space on $\mathbb{Z}$ and g acts by translation by 1 on basis elements, then it is an automorphism of V hence defines a k-point of your functor GL V, but it does not stabilize any finite-dimensional subspace of V, so we don't want a group scheme to act on V through that element.

[Sophie Morel]

Your functor GL V is almost certainly not representable by a scheme if V is infinite-dimensional, but neither is the functor I suggested, I think. The difference is that the functor I suggested is an inductive limit of schemes, i.e. an ind-schemes, and these are still reasonably well-behaved.

[Sophie Morel]

In order to define morphisms from G to GL V, we don't need GL V to be representable anyway, because we can just take morphisms of sheaves in groups, and by Yoneda this will give the correct result for V finite-dimensional, where there is no doubt what the definition of GL V should be.

[Sophie Morel]

The thing is, I never saw a definition via "morphism into GL V" in the literature for general representations, because, as you pointed out, with the classical definition of GL V it won't give the right notion in the infinite-dimensional case (which is definitely important). The only definition in that generality I saw is for affine group schemes and goes through the Hopf algebra.
However, this does not mean that such a definition is not possible/desirable, just that we have to be careful about GL V should be.

[Sophie Morel]

Ah, no, it's fine! If G is an affine group scheme, then using the definition of GL V that you suggest will lead to the correct notion of representation, though that is not totally obvious. Whew! (I keep forgetting that the discrete group scheme $\mathbb{Z}$ is not affine...)

[Michał Mrugała]

Great! I was starting to get worried. In any case I should track down how they're defined in SGA3 to compare.

[Sophie Morel]

My guess is that they define a representation of a group scheme $G$ over $k$ (where $k$ is a commutative ring) on a $k$-module $M$ as an action of the functor in groups $A\mapsto G(A)$ on the functor $A\mapsto M\otimes_k A$ (where $A$ is a commutative $k$-algebra) that preserves the linear structure on each $M\otimes_k A$.

[Sophie Morel]

Which is equivalent to a morphisms of functors in groups from $A\mapsto G(A)$ to $A\mapsto\mathrm{Aut}_A(M\otimes_k A)$.