nLab
connection for a coring

Given an AA-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 AA-coring CC with a grouplike element one associates a semi-free differential graded algebra ΩA=Ω(A,C)\Omega A = \Omega(A,C), sometimes called its (generalized) Amitsur complex. The simplest notion of a connection for the coring CC is a connection for the corresponding Amitsur complex.

Let now AA be a kk-algebra and (C,Δ,ϵ)(C,\Delta,\epsilon) be an AA-coring with grouplike element gg and (ΩA,d)(\Omega A,d) its Amitsur complex.

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

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

determines a flat connection on MM. Conversely, any flat connection determines a right CC-coaction by

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

This amounts to a bijection between CC-coactions and flat connections on MM. 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 AA be an additive category (T,μ,η)(T,\mu,\eta) an additive monad in AA and ν:TMM\nu:TM\to M an action on some object MM in AA. Then a quasi-connection on MM is a map :MTM\nabla:M\to TM such that

νμT()=id TMην:TMTM. \nabla \circ \nu - \mu\circ T(\nabla) = id_{TM} - \eta\circ\nu: TM\to TM.

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

ν=0. \nu\circ\nabla = 0.

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

F :=(id T 2Mη TMμ M)T(). 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.

  • P. Nuss, Noncommutative descent and non-abelian cohomology, KK-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

Revised on July 21, 2010 14:51:15 by Zoran Škoda (161.53.130.104)