# When does Tannakian theory work over affine schemes besides fields?

By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case.

Specifically, if $A$ is an affine ring, and if $\operatorname{Proj}(A)$ is the category of finitely generated projective A-modules, when can we say that a fiber functor $w:\mathcal{T}\to\operatorname{Proj}(A)$ corresponds to an algebraic group over $A$, where $\mathcal{T}$ is an $A$-linear tensor category.

-

If I understand your question correctly you are asking whether or not there is a characterization of those A-linear functors C-->Proj(A) which are equivalent to the forgetful functor Rep(G)-->Proj(A), where G is an affine group scheme over A and Rep(G) is the category of representations of G whose underlying A-module is finitely generated projective.

In this case the categories Rep(G) need not be abelian because Proj(A) is in general not abelian, so the classical Tannakian formalism probably won't help you. However, as in the classical case we can rephrase the problem in terms of comodules and Hopf algebras: An affine group scheme G over A is of the form Spec(H) for some Hopf algebra H, and a representation of G is the same as an H-comodule.

The classical characterization of Deligne (found in "Categories Tannakiennes", Grothendieck Festschrift Vol. II) is split up into the following parts:

1) Every faithful exact functor w:C-->Vect(k) is equivalent to a forgetful functor Comod(L(w))-->Vect(k) for some k-coalgebra L(w).

2) If the category C has a symmetric monoidal structure and if w is a strong monoidal functor, then L(w) is a bialgebra.

3) If C is rigid, then L(w) is a Hopf algebra.

For some time now I've been working on a generalization of step 1) to the case of arbitrary rings A, where we replace Vect(k) by Proj(A). I have recently uploaded a paper (here) which contains the following result (see Corollary 9.8):

Let C be an A-linear category (with finite direct sums) and w:C-->A-Mod an A-linear functor whose image is contained in Proj(A). We say that a diagram F:D-->C is w-rigid if the colimit of wF:D-->A-Mod is finitely generated and projective. If

a) w reflects isomorphisms,

b) the category el(w) of elements of w is cofiltered (equivalently, w is flat),

c) C has colimits of w-rigid diagrams and w preserves them,

then there is a flat coalgebra L(w) such that w:C-->Proj(A) is equivalent to the forgetful functor Comod(L(w))-->A-Mod, where Comod(L(w)) denotes the category of L(w)-comodules whose underlying A-module is finitely generated and projective. Condition a) and c) are necessary conditions. I don't know if b) is a necessary condition.

I have convinced myself that result 2) from above should work at this level of generality, and I think that 3) should not cause any trouble either. In other words, if your category C is a symmetric monoidal category where every object has a dual and if w is a strong monoidal functor satisfying a)-c), then the coalgebra L(w) should be a Hopf algebra, and C is in fact a category of representations of an affine group scheme.

-

There is a nice recent paper by Michael Broshi on the arxiv which is related to this theme when the base is a Dedekind scheme (such as Dedekind domain, or regular proper curve over a field).

-

My understanding from Dennis Gaitsgory's seminar this semester http://www.math.harvard.edu/~gaitsgde/grad_2009/ is that one has the following statement:

Principal G-torsors over X ~~ Symmetric monoidal functors from Rep(G) to Coh(X).

As special case of this is if your scheme is a field-valued point, then Coh(X) is just vector spaces over the ground field. I believe, once it is stated this way, it can be applied to non-affine schemes as well.

Unfortunately, the notes which I attached are not incredibly well-organized and I couldn't, in a ten minute look-through, find the exact statement you're asking about.

-