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Sweedler coring

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Idea

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.

Definition

In components

Let f:RSf : R \hookrightarrow S be the extension of associative unital kk-algebras (where kk is a commutative unital ring).

The corresponding canonical coring or Sweedler coring is the SS-coring

C=S RS C = S\otimes_R S

with coproduct

Δ:CC SCS RS RS \Delta : C\to C\otimes_S C \cong S\otimes_R S\otimes_R S

given by

Δ:s 1s 2s 11s 2 \Delta: s_1\otimes s_2 \mapsto s_1\otimes 1 \otimes s_2

and counit

ϵ:CS \epsilon : C\to S

given by

ϵ:s 1s 2s 1s 2. \epsilon: s_1 \otimes s_2 \mapsto s_1 s_2 \,.

The element 111\otimes 1 is a grouplike element in the Sweedler’s coring.

Geometric interpretation

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 C(Y 1× XY 2)C(Y_1 \times_X Y_2) on a fiber product

Y 1× XY 2 Y 2 Y 1 X \array{ Y_1 \times_X Y_2 &\to& Y_2 \\ \downarrow && \downarrow \\ Y_1 &\to& X }

is the tensor product of the functions on the factors:

C(Y 1× XY 2)=C(Y 1) C(X)C(Y 2). C(Y_1 \times_X Y_2) = C(Y_1) \otimes_{C(X)} C(Y_2) \,.

Then let π:YX\pi : Y \to X be a morphism of spaces and set

R:=C(X) R := C(X)

and

S=C(Y) S = C(Y)

and

(RS):=π *:C(X)C(Y). (R \hookrightarrow S) := \pi^* : C(X) \to C(Y) \,.

The morphism π\pi induces its augmented Čech nerve

(Y× XY× XYY× XYYπX). \left( \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} Y \times_X Y \times_X Y \stackrel{\to}{\stackrel{\to}{\to}} Y \times_X Y \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X \right) \,.

Taking function algebras of this yields, by the above,

(S RS RSS RSSπ *R). \left( \cdots \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}} S \otimes_R S \otimes_R S \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}} S \otimes_R S \stackrel{\leftarrow}{\leftarrow} S \stackrel{\pi^*}{\leftarrow} R \right) \,.

Writing again C=S RSC = S \otimes_R S for the Sweedler coring, this is

(C SCCSπ *R). \left( \cdots \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}} C \otimes_S C \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}} C \stackrel{\leftarrow}{\leftarrow} S \stackrel{\pi^*}{\leftarrow} R \right) \,.

Properties

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 kk-algebra extension RSR\to S is split if there is an RR-bimodule map h:SRh: S\to R with h(1 S)=1 Rh(1_S) = 1_R).

Descent in terms of coring comodules

Given a morphism f:RSf : R \to S with corresponding Sweedler coring (C=S RS,Δ,ϵ)(C = S \otimes_R S,\Delta,\epsilon) as above, the category of descent data Desc(S/R)\mathrm{Desc}(S/R) for the categories of right modules along kk-algebra extension RSR\to S is precisely the category of right CC-comodules.

In other words, the objects of Desc(S/R)\mathrm{Desc}(S/R) are the pairs (N,α)(N,\alpha) where NN is a right SS-module, and α:NN RS\alpha: N\to N\otimes_R S is a right SS-module morphism and if we write α(m)= im is i\alpha(m) = \sum_i m_i \otimes s_i then

  • iα(m i)s i= im i1s i\sum_i \alpha(m_i)\otimes s_i = \sum_i m_i\otimes 1\otimes s_i,

  • im is i=m\sum_i m_i s_i = m.

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

p:p : Mod \to CRing

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 f:RSf : R \to S with respect to this bifibration is a (co)algebra object over the comonad f *f *f_* f^* induced by this morphism. We have that

  • the morphism f *f_* sends an RR-module NN to the SS-module N RSN \otimes_R S;

  • the morphism f *f^* sends an SS-module PP to the RR-module P SS RP \otimes_S S_R, where S RS_R is SS regarded as a left SS- and a right RR-module. So P SS RP \otimes_S S_R is just the SS-module PP with only the right RR-action remembered.

Accordingly, the comonad with underlying functor f *f *f_* f^* sends an SS-module PP to the SS-module P SS RS=P SCP \otimes_S S \otimes_R S = P \otimes_S C.

A (co)algebra object for this comonad is hence a co-action morphism

PP SC P \to P \otimes_S C

compatible with the monad action. This is precisely a comodule over the Sweedler coring, as defined above.

Geometric interpretation

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 R=C(X)R = C(X) with (the collection of sections of) a vector bundle (or rather a suitable generalization of that: a coherent sheaf) over XX. Similarly for YY. So we write

Vec(X):=RMod Vec(X) := R Mod

and

Vec(Y):=SMod Vec(Y) := S Mod

for the corresponding categories of modules. The assignment of such categories to spaces

Vec:ZVec(Z) Vec : Z \mapsto Vec(Z)

extends to a contravariant pseudofunctor

Vec:Spaces opCat Vec : Spaces^{op} \to Cat

by assigning to a morphism f:YXf : Y \to X of spaces the corresponding functor

Vec(X)C(X)Mod fC(Y)C(Y)ModVec(Y). Vec(X) \simeq C(X) Mod \stackrel{- \otimes_{f} C(Y)}{\to} C(Y) Mod \simeq Vec(Y) \,.

This way VecVec 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 π:YX\pi : Y\to X that is regarded as a cover, the category Desc(Y,Vec)Desc(Y,Vec) whose objects are tuples consisting of

  • an object aVec(Y)a \in Vec(Y)

  • an isomorphism g:π 1 *aπ 2 *ag : \pi_1^* a \to \pi_2^* a

  • such that

    π 2 *a π 12 *g π 23 *g π 1 *a π 13 * π 3 * \array{ && \pi_2^* a \\ & {}^{\mathllap{\pi_{12}^* g}}\nearrow && \searrow^{\mathrlap{\pi_{23}^* g}} \\ \pi_1^* a &&\stackrel{\pi_{13}^*}{\to}&& \pi_3^* }

    commutes.

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 EXE \to X (of finite rank) is the associated bundle EP× O(n)VE \simeq P \times_{O(n)} V of an O(n)-principal bundle PXP \to X, and as such its sections may be identified with O(n)O(n)-equivariant functions PV nP \to V \simeq \mathbb{R}^n on the total space of PP.

Using this we may think of the C(X)C(X)-module of sections of EE as a submodule of the C(X)C(X)-module of all functions on PP

Γ(E)C(P). \Gamma(E) \subset C(P) \,.

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 QYQ \to Y may be thought of as the the morphism gg in the double pullback diagram

Y× XY× YQ Q g Y× XY Y π Q Y π X. \array{ &&Y\times_X Y \times_Y Q &\to& Q \\ &&\downarrow && \downarrow \\ &{}^{\mathllap{g}}\swarrow&Y \times_X Y &\to& Y \\ &&\downarrow && \downarrow^{\mathrlap{\pi}} \\ Q &\to& Y &\stackrel{\pi}{\to}& X } \,.

Because here Y× XY× YQY \times_X Y \times_Y Q is the space whose points consist of a point in a double overlap of the cover and a point in the fiber of QQ over that with respect to one patch, and the morphism identifies this with a point in the fiber of QQ regarded as sitting over the other patch. Analogously, there is a cocycle condition on gg on triple overlaps.

Now, blindly applying our functor that takes functions of spaces to the above diagram yields the double pushout diagram

C(Y× XY× YQ) C(Q) g * C(Y× XY) C(Y) π * C(Q) C(Y) π * C(X). \array{ &&C(Y\times_X Y \times_Y Q) &\leftarrow& C(Q) \\ &&\uparrow && \uparrow \\ &{}^{\mathllap{g}^*}\nearrow&C(Y \times_X Y) &\leftarrow& C(Y) \\ &&\uparrow && \uparrow^{\mathrlap{\pi^*}} \\ C(Q) &\leftarrow& C(Y) &\stackrel{\pi^*}{\leftarrow}& C(X) } \,.

We may restrict to N:=Γ(E)C(Q)N := \Gamma(E) \subset C(Q) as just discussed and switch to the notation from above to get

N SC N g * C S π * N S π * R. \array{ &&N \otimes_{S} C &\leftarrow& N \\ &&\uparrow && \uparrow \\ &{}^{\mathllap{g}^*}\nearrow& C &\leftarrow& S \\ &&\uparrow && \uparrow^{\mathrlap{\pi^*}} \\ N &\leftarrow& S &\stackrel{\pi^*}{\leftarrow}& R } \,.

The morphism

α:=g *:NN SC \alpha := g^* : N \to N \otimes_S C

obtained this way is the co-action morphism from the above algebraic definition.

The further cocycle condition on gg similarly translates into the condition that α\alpha 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.

References

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

Revised on February 22, 2014 05:16:45 by Zoran Škoda (95.168.125.34)