nLab
Grothendieck connection

Idea

A Grothendieck connection $\nabla$ is a way to encode the parallel transport of a flat connection along infinitesimal paths.

In the language of ∞-Lie theory it is a morphism from the infinitesimal path ∞-groupoid? ${\Pi }^{\inf }(X)$ of a suitable space $X$ (a scheme in algebraic geometry or, more generally, a space in synthetic differential geometry) to some Lie ∞-groupoid $A$:

$$\nabla :{\Pi }^{\inf }(X)\to A.$$\nabla : \Pi^{inf}(X) \to A \,.

Here ${\Pi }^{\inf }(X)$ is effectively the infinitesimal singular simplicial complex of $X$ and $A$ is modeled typically as a sheaf of ∞-groupoids as described at models for ∞-stack (∞,1)-toposes.

This was originally considered by Grothendieck for schemes in the context of algebraic geometry for the case that $A$ is a 1-groupoid – but the generalization to ∞-groupoids in synthetic differential geometry is immediate.

Details

In his study of crystalline cohomology, Grothendieck has noticed that flat connections correspond to the descent data for deRham descent; he called them the costratification of the $n$-th order if one considers the $n$-the infinitesimal neighborhoods.

Already in EGA, Grothendieck has introduced a notion of regular differential operator and of jet-spaces, which were later by Malgrange and Spencer transferred into the study of differential equations and analytic deformation theory.

De Rham descent for a relative scheme (= morphism of schemes) $f:X\to S$ is formulated in terms of resolutions or infinitesimal thickenings of the diagonal for the simplicial scheme

$$\cdots X{\times }_{S}X{\times }_{S}X\overrightarrow{\overrightarrow{\to }}X{\times }_{S}X\overrightarrow{\to }X$$\cdots X \times_S X \times_S X \stackrel{\to}{\stackrel{\to}{\to}} X \times_S X \stackrel{\to}{\to} X

corresponding to $f$. Usually one assumes that $X$ is a smooth scheme of finite type over $S$. By separatedness the diagonal $\Delta :X\hookrightarrow X{\times }_{S}X$ is a closed immersion of schemes. Let then $ℐ$ be the corresponding defining ideal sheaf (locally it is generated by the elements of the form $t\otimes 1-1\otimes t$, cf. Kähler differential). The definining ideal ${ℐ}^{n+1}$ defines for the $n$-th infinitesimal neighborhood $X^{(n)}$ and the diagonal subscheme $X\cong \Delta (X)\hookrightarrow X{\times }_{S}X$; there is a series of inclusions

$$X\hookrightarrow X^{(1)}\hookrightarrow X^{(2)}\hookrightarrow \dots \hookrightarrow X^{(\mathrm{\infty })}=\hat{X}$$X\hookrightarrow X^{(1)}\hookrightarrow X^{(2)}\hookrightarrow \ldots \hookrightarrow X^{(\infty)}=\hat{X}

where $\hat{X}$ is the completion (the corresponding formal scheme). Let $ℱ\to \mathrm{Sch}$ be a Grothendieck fibration over the category of schemes (or the subcategory of the category of schemes containing all schemes in our consideration) classified under the Grothendieck construction by some pseudofunctor $A:{\mathrm{Sch}}^{\mathrm{op}}\to \mathrm{Cat}$. A typical examples would be the stack of quasicoherent sheaves of $𝒪$-modules.

Consider now the projections $d_0,d_1:X{\times }_{S}X\to X$. Then

Definition (Grothendieck connection)

A Grothendieck connection on an object $\rho \in ℱ$ is a descent datum of the form $(\rho ,\theta )$ where $\theta :d_0^{*}\rho \to d_1^{*}\rho$ is an isomorphism satisfying the cocycle (and the normalization) condition and such that the restriction (pullback along inclusion) on the diagonal is the identity.

The constructions can be almost literally transferred to the synthetic differential geometry using infinitesimal neighborhoods in that setup.

More recently, some partial generalizations were found in the purely algebraic framework. One could take any coring (or even an additive comonad) with a grouplike element and define corresponding connections and descent data and generalize the correspondence found by Grothendieck (see semi-free dga and Menini-Stefan reference below).

Literature and links

Related entries include deRham space, crystal, crystalline cohomology

• A. Grothendieck, Crystals and the de Rham cohomology of schemes, p. 306–358 of Dix exposes sur la cohomologie des schemas, North Holland 1968, Dix exp. pdf

• P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9

• N. Katz, Nilpotent connections and the monodromy theorem : applications of a result of Turrittin, Publications Mathématiques de l’IHÉS, 39 (1970), p. 175-232 numdam

• A. Grothendieck, (EGA IV.16) Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie. Publications Mathématiques de l’IHÉS, 32 (1967), p. 5-361 numdam

• SGA III, vol. 1, Expose VIIa (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES, 409-473

• wikipedia: Grothendieck connection

• B. Osserman, Connections, curvature and $p$-curvature, pdf (expositional preprint)

• A. Beilinson, I. N. Bernstein, A proof of Jantzen conjecture, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR95a:22022

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

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

• L. Breen, Messing, Combinatorial differential forms, arxiv:math/0005087

• Notes in Gaitsgory’s grad student seminar pdf