nLab connection for a coring

Given an $A$-coring or even a more general additive comonad with a grouplike element there are several related (but in general nonequivalent) notions of connections.

As explained in grouplike element, to an $A$-coring $C$ with a grouplike element one associates a semi-free differential graded algebra $\Omega A = \Omega(A,C)$, sometimes called its (generalized) Amitsur complex. The simplest notion of a connection for the coring $C$ is a connection for the corresponding Amitsur complex.

Let now $A$ be a $k$-algebra and $(C,\Delta,\epsilon)$ be an $A$-coring with grouplike element $g$ and $(\Omega A,d)$ its Amitsur complex.

A connection $\nabla:M\otimes_A\Omega^\bullet\to M\otimes_A\Omega^{\bullet+1}$ on a module $M$ over a semifree dga (in the sense of the entry connection for a differential graded algebra) is determined by its value $\nabla_M|_M$ on $M\cong M\otimes_A A$. If $\rho^M:M\to M\otimes_A C$ is a right $C$-coaction then the formula

$$\nabla|_M:m\mapsto \rho^M(m)-m\otimes g$$

determines a flat connection on $M$. Conversely, any flat connection determines a right $C$-coaction by

$$\rho^M(m)=\nabla(m)+m\otimes g.$$

This amounts to a bijection between $C$-coactions and flat connections on $M$. Regarding that coactions correspond to descent data in the context of comonadic descent, this gives the flat connection interpretation of such descent data. A first instance is probably Grothendieck’s identification of flat connections and the first order costratifications in Grothendieck’s theory of differential calculus on schemes (foundations of crystalline cohomology, see book by Berthelot and Ogus; cf. also Grothendieck connection).

Connections on comodules directly

One the other hand, one can consider more generally additive comonads, and define connections on comodules over them rather directly. Or dually one can work with connection on modules over additive monads.

Menini and Ştefan first define an intermediate notion of a quasi-connection for monads. Let $A$ be an additive category $(T,\mu,\eta)$ an additive monad in $A$ and $\nu:TM\to M$ an action on some object $M$ in $A$. Then a quasi-connection on $M$ is a map $\nabla:M\to TM$ such that

$$\nabla \circ \nu - \mu\circ T(\nabla) = id_{TM} - \eta\circ\nu: TM\to TM.$$

A quasi-connection is a connection if, in addition,

$$\nu\circ\nabla = 0.$$

For every connection in Menini–Ştefan sense, one defines its curvature $F_\nabla : M\to T^2 M$ by the formula

$$F_\nabla := (id_{T^2 M} - \eta_{TM}\circ\mu_M)\circ T(\nabla)\circ\nabla.$$

As usually, we define a flat connection as a connection whose curvature vanishes.

In this setting one again has a bijection between flat connections and descent data.

Literature and links

• P. Nuss, Noncommutative descent and non-abelian cohomology, $K$-Theory 12 (1997), no. 1, 23–74.

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

• C. Menini, Talk at MSRI: Connections, symmetry operators and descent data for triples, 1999, link

• C. Menini, D. Ştefan, Descent theory and Amitsur cohomology of triples, J. Algebra 266 (2003), no. 1, 261–304.

• T. Brzeziński, Flat connections and (co)modules, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 arxiv:math.QA/0608170