# Connections and curvature in commutative algebra

Since on any commutative algebra $R$ over ring $S$ we have module of Kahler differentials $(\Omega_{R/S},d)$ which extends to the algebraic de-Rham complex $(\Omega^\bullet,d),$ it is natural to define connection on $R$-module $P$ as any $S$-linear map $\nabla:P\to\Omega_{R/S}\otimes P$ such that $$\nabla(rp)=dr\otimes p + r\nabla(p).$$ As in geometry we can extend $\nabla$ to $d^\nabla:\Omega^\bullet\otimes P\to\Omega^{\bullet+1}\otimes P$ and define its curvature $F^\nabla.$

I look for some source which analyze such approach. The only thing I found are Sardanashvilys Lectures on Differential Geometry of Modules and Rings which define connection via jet bundle, but I guess it is equivalent definition. However this source is from 2009 and in 1995 Dubois-Violette and Michor wrote Connections on central bimodules in which they investigate connections in noncommutative case. They also refer to Koszules Lectures on fibre bundles and differentials, but Koszul does not write anything about kahler differentials.

Question. Is there any source which analyze connections from purely commutative algebraic point of view?

Or maybe there is no point for such approach because there is no interesting connections beyond algebra of smooth functions on smooth manifold?

Some footnote. In commutative algebra connections can be defined even for any dga as in ncatlab, but there is no reference so again I do not know where such case is investigated.

• This is probably not what you are looking for, but there is Bertram's monograph "Differential Geometry over General Base Fields and Rings" (2005) and Betram, Glöckner and Neeb's "Differential Calculus, Manifolds, and Lie Groups over Arbitrary Infinite Fields" (2003). I don't know if their approach can be of any interest to you. – M.G. Dec 12 '16 at 17:27
• If your major concern is the rings of mappings, Peter Michor's works are mostly concerned with such manifolds corresponding to Lie-algebra/rings. But typically the focus is on Lie algebra, that is why the motivation comes from noncommutative cases. But nice question! – Henry.L Dec 12 '16 at 19:28
• @July Thanks for the first one. I found something interesting in this book, but not related to the question. I looked through the book and I doubt I will find such construction there. Besides it is relatively new book. And I suspect the construction I wrote above should have been done in early 70s. – Fallen Apart Dec 12 '16 at 19:28
• @Henry.L I am also interested in manifolds. I love Jet Nestruevs book and he (they) proved that $(\Omega^1(M),d)$ is a module of kahler differentials in the category of so called geometric modules. So any result from this abstract perspective would apply to manifold case. – Fallen Apart Dec 12 '16 at 19:31
• Perhaps you should check out the paper Local Cohomological Dimension of Algebraic Varieties by Arthur Ogus, (Annals of Mathematics Second Series, Vol. 98, No. 2 (Sep., 1973), pp. 327-365). Not sure if that's what you are looking for but it may be relevant. – Mahdi Majidi-Zolbanin Dec 13 '16 at 19:34