# Is there an analog of the Levi–Civita connection for schemes?

Is there an analog of the Levi–Civita connection for schemes?

There exists algebraic de Rham theory, $$n$$-forms on vector bundles (algebraically), and familiar constructions from differential geometry. This made me question whether or not there is an analog of the Levi–Civita connection for schemes.

A motivating example would be the following. Let $$S$$ be a Noetherian $$R$$-algebra of finite type over a commutative ring $$R$$. The $$S$$-module $$\textbf{Der}_R (S)$$ should be finitely generated from the hypothesis. From here, if we fix a set of generators for this $$S$$-module, then we can define a symmetric bilinear $$2$$-form $$\textbf{Der}_R (S) \times \textbf{Der}_R (S) \to S$$ that becomes the analog of a semi-Riemannian metric. Letting $$R$$ be a field $$k$$ of characteristic zero (so that there exists the monomorphism $$\mathbb{Q} \to k$$), then one should be able to concoct similar definitions for an (algebraic) connection and compatibility so that one has an 'algebraic' analog of the Levi–Civita theorem! Please do correct me if I am wrong, but this is just a motivating case for the question!

• A scheme is analogous to a manifold, the latter of which requires a Riemannian metric to define the Levi Civita connection. There is a notion of a connection on a bundle over a scheme due to Grothendieck, the keyword here is crystal. Commented Jan 4, 2020 at 1:06
• If it is not too much to ask, then would you be able to briefly describe how crystals come into play here? Commented Jan 4, 2020 at 15:13
• Unfortunately, I'm a differential geometer with only a buzzword knowledge of crystals, so hopefully someone else will take up such an explanation. My main point in the comment was that expecting a canonical connection on a scheme would be like expecting a canonical connection on a smooth manifold, which is asking for way too much. Whatever crystals are, my impression is that they are at least one answer to the question of what a connection over a scheme is. Commented Jan 4, 2020 at 15:58
• @AndySanders: Crystals are flat connections. Levi-Civita connections are usually not flat. Commented Jan 4, 2020 at 22:57
• Of course you are right Dmitri. Commented Jan 5, 2020 at 5:55