nLab Ehresmann coring

Idea

Given a $C$-coalgebra-Galois extension $U\hookrightarrow E$ of a $k$-algebra $U$, which is the appropriate generalization of a Hopf-Galois extension, where $E$ is faithfully flat over the base $U$ as a left $E$-module, one constructs a coring, Ehresmann coring, out of these data. Its role is somewhat analogous to the gauge groupoid (cf. Atiyah Lie groupoid), and in the Hopf-Galois case it is an intermediate stage in constructing another analogue, the (Ehresmann-)Schauenburg bialgebroid, see there.

Definition

A right $C$-coalgebra-Galois extension $U\hookrightarrow E$ is a $k$-algebra $E$ together with a right $C$-coaction $\rho:E\to E\otimes C$ where $U = \{ u\in E| \rho(u e) = (u\otimes 1)\rho(e),\forall e\in E\}$ is the subalgebra of coinvariants if the map

$$can : E\otimes_U E\to E\otimes C,\,\,\,\,\,\,e\otimes e'\mapsto (e\otimes 1)\rho(e')$$

is bijective. The map $\tau:C\to E\otimes E$, $c\mapsto can^{-1}(1\otimes c)$ is the translation map:

The underlying bimodule of the Ehresmann coring $D$ is the subbimodule $B\subset E\otimes_k E$

$$B = \{\sum_i e_i\otimes e'_i \in E\otimes_k E | \sum_i e_i \otimes e'_i\otimes_U 1 = \sum_i e_{i(0)}\otimes \tau(e_{i(1)}) e'_i \in E\otimes_k (E\otimes_U E) \}$$

If $E$ is faithfully flat as a left $U$-module then $B$ is a $U$-coring via

$$\Delta_B : \sum_i e_i\otimes e'_i \mapsto \sum_i e_{i(0)}\otimes\tau(e_{i(1)})\otimes e'$$

and comultiplication given by multiplication: $\epsilon_B(\sum_i e_i\otimes e'_i) = \sum_i e_i\cdot_E e'_i$.

Literature

• T. Brzeziński, R. Wisbauer: Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge (2003)