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A Sweedler coring is an algebraic structure that is roughly the formal dual of the Čech nerve of a cover: it is used to describe descent in algebraic contexts.
See also monadic descent.
Let be the extension of associative unital -algebras (where is a commutative unital ring).
The corresponding canonical coring or Sweedler coring is the -coring
The element is a grouplike element in the Sweedler’s coring.
We give a dual geometric interpretation of the Sweedler coring.
Suppose a context of spaces and function algebras on spaces that satisfies the basic axioms of geometric function theory, in that the algebra of functions on a fiber product
is the tensor product of the functions on the factors:
Then let be a morphism of spaces and set
The morphism induces its augmented Čech nerve
Taking function algebras of this yields, by the above,
Writing again for the Sweedler coring, this is
Relation to ring extensions
Various properties of canonical coring correspond to adequate properties of the ring extension. For example, coseparable Sweedler corings correspond to split extension?s (the -algebra extension is split if there is an -bimodule map with ).
Descent in terms of coring comodules
Given a morphism with corresponding Sweedler coring as above, the category of descent data for the categories of right modules along -algebra extension is precisely the category of right -comodules.
In other words, the objects of are the pairs where is a right -module, and is a right -module morphism and if we write then
In terms of (co)monadic descent
This coring-formulation of descent may be understood as special case of comonadic descent (see also the discussion at Bénabou–Roubaud theorem). See e.g. (Hess 10, section 2) for a review. We spell this out in a bit more detail:
The bifibration in question is
that sends an object in the category Mod of modules to the ring that it is a module over.
A descent datum for a morphism with respect to this bifibration is a (co)algebra object over the comonad induced by this morphism. We have that
the morphism sends an -module to the -module ;
the morphism sends an -module to the -module , where is regarded as a left - and a right -module. So is just the -module with only the right -action remembered.
Accordingly, the comonad with underlying functor sends an -module to the -module .
A (co)algebra object for this comonad is hence a co-action morphism
compatible with the monad action. This is precisely a comodule over the Sweedler coring, as defined above.
Descent for Sweedler corings is a special case of comonadic descent. We describe this in detail and relate it by duality to the geometrically more intuitive monadic descent for codomain fibrations.
Assuming again a suitable geometric context as above, we may identify a module over with (the collection of sections of) a vector bundle (or rather a suitable generalization of that: a coherent sheaf) over . Similarly for . So we write
for the corresponding categories of modules. The assignment of such categories to spaces
extends to a contravariant pseudofunctor
by assigning to a morphism of spaces the corresponding functor
This way becomes a prestack of categories on our category of spaces.
If this prestack satisfies descent along suitable covers, it is a stack.
Geometrically this is the case if for each morphism that is regarded as a cover, the category whose objects are tuples consisting of
Morphism are defined similarly (see stack and descent for details).
To get the geometric pucture that underlies, by duality, the above comodule definition of descent, we need to reformulate this just a little bit more:
every ordinary vector bundle (of finite rank) is the associated bundle of an O(n)-principal bundle , and as such its sections may be identified with -equivariant functions on the total space of .
Using this we may think of the -module of sections of as a submodule of the -module of all functions on
We now reformulate the geometric descent for vector bundles in terms of geometric descent for their underlying principal bundles, and then take functions on everything in sight to obtain the comodule definition of descent that we want to describe:
A descent datum (transition function) for a principal bundle may be thought of as the the morphism in the double pullback diagram
Because here is the space whose points consist of a point in a double overlap of the cover and a point in the fiber of over that with respect to one patch, and the morphism identifies this with a point in the fiber of regarded as sitting over the other patch. Analogously, there is a cocycle condition on on triple overlaps.
Now, blindly applying our functor that takes functions of spaces to the above diagram yields the double pushout diagram
We may restrict to as just discussed and switch to the notation from above to get
obtained this way is the co-action morphism from the above algebraic definition.
The further cocycle condition on similarly translates into the condition that really satisfies the comodule property.
Relation to generalized cohomology and Adams spectral sequence
Applied to E-infinity rings the Sweedler coring construction yields the Hopf algebroids of dual Steenrod algebras and appears in the Adams spectral sequence.
Sweedler corings are named after Moss Sweedler.
A textbook account is in
Section 29 there discusses the relation to the Amitsur complex and the descent theorem.
Discussion in the context of (higher) monadic descent is around example 2.24 of