Description¶

Let me start with a description of categorical Galois theory in a nutshell. One considers a pair of adjoint functors I:C→X and H:X→I (H being the right adjoint) and two classes of arrows F⊂C and Φ⊂X, such that the functors I and H map the arrows in F (Φ) to Φ (F). Secondly, the category C has to admit pullbacks along morphisms from F and the class F has to be pullback stable. One also makes a similar assumption on Φ and X. Given such an adjunction and an object B of C one considers the full subcategory of the comma category C↓B in which the objects are arrows from F. We denote this category by F(B). Then one can construct an adjunction:

(*) F(B)→Φ(I(B)), Φ(I(B))→F(B)

which is induced by I and H. Under some additional assumptions one defines trivial coverings of B in F(B) as those objects (A,α) of F(B), for which the unit map of the adjunction (*) at (A,α) is an isomorphism. An object (A,α) of F(B) is said to be a covering of B if there exists a monadic extension (E,p) of B such that p*(A,α) is a trivial covering (an extension is monadic basically if the functor p*: F(B)→F(E) is monadic, though this is not sufficient condition in this setting). Such a covering (A,α) is also called a split extension over the monadic extension (E,p). The fundamental theorem of categorical Galois theory shows that the category of split extension over a monadic extension (E,p) is equivalent to the category of internal presheaves over a suitably defined internal groupoid (the image under I of the equivalence relation Eq(p)). For an explanation how classical Galois theories fit into this scheme we refer to the literature (G.Janelidze, F.Borceux ‘Galois theories’, Cambridge University Press).

Our first aim is to construct categorical Galois theory for commutative comodule algebras. In this case one could mimic the Magid–Galois theory and consider the Pierce spectrum functor. Since everything is now embedded in the category of comodules of a Hopf algebra the result would be internalised in this category. Instead of going this path we propose to consider a different functor which leads to a new Galois context, and which will give a new perspective into both: categorical Galois theories and the theory of Hopf algebras. Let cAlg(H) be the category of commutative (right) H-comodule algebras, i.e. commutative algebras A which are right H-comodules and such that the comodule structure map: δ:A→A⊗H is a morphism of algebras. Let cAlg be the category of commutative algebras. Then we have a pair of adjoint functors: the right adjoint U:cAlg(H) → cAlg is the coinvariant functor: U(A)={a∈A: δ(a)=a⊗1∈A⊗H}, where δ:A→A⊗H is the comodule structure map of A. The left adjoint is: F:cAlg → cAlg(H), which makes A into H-comodule via ι:A→A⊗H, ι(a)=a⊗1∈A⊗H. We want to show that this adjunction leads to a Galois context by composing it with the adjunction between commutative algebras and profinite spaces via the Pierce spectrum functor. In this composite adjunction a morphism of H-comodule algebras p:(B,δ) → (E,Δ) is called a Galois descent morphism if and only if

1. p is an effective descend (in the category of commutative algebras, which is if and only if p is pure by the unpublished theorem of Joyal–Tirney, see ‘Pure morphisms of commutative rings are effective descent morphisms for modules’ by Bachuki Mesablishvili).
2. the counit of the adjunction between cAlg(H)↓E and Prof↓Sp(U(E)) is an isomorphism
3. for every X the image of X → Sp(U(E)) under this adjunction (which becomes an element of cAlg(H)↓B) is split by p.

From this point we want to start understanding the Galois theory that emerges in this construction. We want to understand not only the fundamental Galois theorem in this context but also find meaningfull statements about the Galois descent morphisms (i.e. Galois extensions in a sense), and the nature of the Galois groupoid.

To explain the second idea I have to introduce some of the results from my PhD thesis. For this sake, let A be an H-comodule algebra, i.e. an algebra over a commutative base ring R, which is also an H-comodule via an algebra homormophism δ:A→A⊗H. Then A is called Hopf-Galois extension (or H-Galois for short) if the canonical map: A⊗A→A⊗H: which sends a⊗b to aδ(b)∈A⊗H is an isomorphism. Let me note that the tensor product in A⊗A is over the coinvariants subalgebra B={a∈A: δ(a)=a⊗1∈A⊗H}, and A is also called H-Galois extension of B. This condition generalises Galois extension property of fields, in fact a field extension is Galois if and only if it is Hopf Galois over the dual of the group Hopf algebra induced by the Galois group, at least for finite field extensions. From the other side this condition generalises the condition met in principal bundles: that the structure group acts freely and transitively on fibers - like in the theory of covering spaces: the fundamental group acts freely and transitively on fibers of the universal cover. The important difference from this two classical Galois theories is that here the symmetry object - the Hopf algebra H is given, while there it is constructed from the extension (think of deck transformations rathern than the fundamental group in the case of covering spaces). This observation is also important for the first part of our project where an additional Galois object will emerge despite H it self. In my thesis I constructed a Galois correspondence between intermediate subalgebras of A/B and the quotients of H (which generalised the classical Galois correspondence for field extensions). When A is over arbitrary commutative base ring my construction is purely poset theoretical, but when A is over a field I also constructed a Hopf algebraic formula for it. This made possible to show more delicate Hopf algebraic statements. Schauenburg considered a very similar situation, though he had stronger assumptions: A was as here a possibly noncommutative algebra over a commutative base ring R, but the coinvariants were supposed to be equal R (thus commutative). Then he constructed another Hopf algebra which made A into a bicomodule algebra and a Hopf–Galois extension (this is another point where I relaxed the assumptions: I did not have to require that the comodule algebra A was initially an H-Hopf Galois extension, and even I did not required H to be a Hopf algebra, a bialgebra was enough). Schauenburg constructed correspondence between intermediate subalgebras A/R and the additionally constructed Hopf algebra L. He proved that under some flatness/coflatness assumptions subalgebras of A/quotients of L are closed in the Schauenburg–Galois correspondence. I proved the same result in a broader setting, with weaker requirements and also I simplified the proof using poset–algebraic techniques. I also managed to prove two, completley new, criteria for closedness of subalgebras of A and quotients of H: if Q a quotient of H is such that A as an Q-comodule algebra is a Hopf–Galois extension then Q must be closed (under some mild ring theoretical assumptions). The other criterion is very similar: a subalgebra S of A/B is closed if Schneider’s canonical map is an isomorphism. Using this theorems I proved some deep results. For example I found a new proof of the Takeuchi correspondence between (right) coideal subalgebras and left ideal coideals of any finite dimensional Hopf algebra (a result proven independently by S.Skryabin few years earlier with much more sophisticated machinery of ring theory). In the case of Takeuchi correspondence (for arbitrary Hopf algebras) I was able to show that a quotient Q is closed if and only if H is Q-Galois, also the other criteria has a stronger form here: a (left) coideal subalgebra of H is closed if and only if the Schneider’s canonical map is a bijection. From my results I was also able to show that for an A/B H-comodule algebra (with coinvariants B) over a commutative base field, every closed quotient Q of H is of the form H/K^+H for some left coideal subalgebra K of H - let me note that the association K→H/K^+H is one of the maps in the Takeuchi correspondence and I have managed to give a condition (*) for which the bijectivity of the canonical map for a quotient Q is equivalent to (*) plus Q being closed. Furthermore I generalised the results of Takeuchi correspondence to cleft extensions. In this proposal I want to carry this line of research to bialgebroids, first through the case of cleft extensions and Takeuchi like correspondence for bialgebroids (and Hopf algebroids) and then possibly with more general extensions. Let me note that the step from extensions based on bialgebras (Hopf algebras) to bialgebras (Hopf algebroids) is not merely a technical detail. Hopf algebroids are much more sophisticated objects, and there have not been as much research on them as on Hopf algebras, mostly because they appeared in the last decade, while Hopf algebras are in mathematical attention for much longer.

Let me breifly explain the last objective. Stasheff in ‘Parallel transport and classification of fibrations’ (LNM, Vol.428, Springer, 1974) reconstructs a fibration E over B with fiber F from F as an Omega(B)-set in the homotopy category, where Omega(B) is the group of closed path in B. Let us note that in any category of fibrant objects Omega(B) becomes an internal group in the category homotopy category and F holds an Omega(B)-action in homotopy category. We believe that Stasheff’s construction leads to a Galois context which we hope to generalise to more abstract model categories. In the first place we would like to investigate the Stasheff construction and then go to simplicial model categories since they have a reach structure and this might already lead to new important developments. We want to start from simplicial categories also because knowing how to build a Galois theory therein it should gives incentives how to build Galois theory in more general model structures.