nLab Sweedler coring

Idea

A canonical or 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 $f : R \hookrightarrow S$ be the extension of associative unital $k$-algebras (where $k$ is a commutative unital ring).

The corresponding canonical coring or Sweedler coring is the $S$-coring

$$C = S\otimes_R S$$

with coproduct

$$\Delta : C\to C\otimes_S C \cong S\otimes_R S\otimes_R S$$

given by the bilinear extension of the formula

$$\Delta: s_1\otimes s_2 \mapsto s_1\otimes 1 \otimes s_2$$

and counit

$$\epsilon : C\to S,\,\,\,\,\,\,\, \epsilon: s_1 \otimes s_2 \mapsto s_1 s_2 \,.$$

The element $1\otimes 1$ 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 $C(Y_1 \times_X Y_2)$ on a fiber product

$$\array{ Y_1 \times_X Y_2 &\to& Y_2 \\ \downarrow && \downarrow \\ Y_1 &\to& X }$$

is the tensor product of the functions on the factors:

$$C(Y_1 \times_X Y_2) = C(Y_1) \otimes_{C(X)} C(Y_2) \,.$$

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

$$R := C(X)$$

and

$$S = C(Y)$$

and

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

The morphism $\pi$ induces its augmented Čech nerve

$$\left( \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} Y \times_X Y \times_X Y \stackrel{\to}{\stackrel{\to}{\to}} Y \times_X Y \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X \right) \,.$$

Taking function algebras of this yields, by the above,

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

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

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

Properties

Relation to ring extensions

Various properties of canonical coring correspond to adequate properties of the ring extension. For example, coseparable canonical corings correspond to split extension?s (the $k$-algebra extension $R\to S$ is split if there is an $R$-bimodule map $h: S\to R$ with $h(1_S) = 1_R$).

In the case of the trivial ring extension $R\to R$ the coproduct of the canonical coring is the canonical isomorphism $R\to R\otimes_R R$ and the counit is the identity $R\to R$. Thus, every right $R$-module is a comodule over the canonical coring. Such a canonical coring is called a trivial coring. This example puts the rings (or associative unital algebras) and their categories of modules as a special case of corings and their categories of comodules. In view of the next paragraph, this is a generalization of a coring of a cover for the case of a trivial (identity) cover.

Descent in terms of coring comodules

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

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

• $\sum_i \alpha(m_i)\otimes s_i = \sum_i m_i\otimes 1\otimes s_i$,

• $\sum_i m_i s_i = m$.

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

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

• the morphism $f_*$ sends an $R$-module $N$ to the $S$-module $N \otimes_R S$;

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

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

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

$$P \to P \otimes_S C$$

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

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

and

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

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

$$Vec : Z \mapsto Vec(Z)$$

extends to a contravariant pseudofunctor

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

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

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

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

• an object $a \in Vec(Y)$

• an isomorphism $g : \pi_1^* a \to \pi_2^* a$

• such that

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

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

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

$$\Gamma(E) \subset C(P) \,.$$

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 $Q \to Y$ may be thought of as the the morphism $g$ in the double pullback diagram

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

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

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

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

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

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

The morphism

$$\alpha := g^* : N \to N \otimes_S C$$

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

The further cocycle condition on $g$ similarly translates into the condition that $\alpha$ 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.

Related concepts

• Amitsur complex, Hopf algebroid

• descent

  • cover

  • cohomological descent

  • monadic descent,

    • Sweedler coring

    • monadic descent, higher monadic descent

    • descent in noncommutative algebraic geometry

• sheaf, (2,1)-sheaf, 2-sheaf (∞,1)-sheaf

References

Sweedler corings are named after Moss Sweedler.

A textbook account is in

• Tomasz Brzezinski, Robert Wisbauer, section 29 of Corings and Comodules, Cambridge University Press, London Math. Soc. LN 309 (2003), (errata pdf)

Section 29 there discusses the relation to the Amitsur complex and the descent theorem.

• MO: how-is-a-descent-datum-the-same-as-a-comodule-structure

Discussion in the context of (higher) monadic descent is around example 2.24 of

• Kathryn Hess, section 6 of A general framework for homotopic descent and codescent (arXiv:1001.1556)