# Derivations in Connes' noncommutative geometry

In Connes' noncommutative geometry, the starting point is a spectral triple $$(A,D,H)$$ where $$A$$ is a commutative C* algebra, e.g. as in Connes "ON THE SPECTRAL CHARACTERIZATION OF MANIFOLDS" , where 5 conditions are stated which the triple must satisfy.

In order to construct coordinate charts on the $$X = Spec(A)$$, Connes proceeds with analysis of derivations $$\delta_j$$ on A, which satisfy exponentiability conditions (Lemma 3.3 in the above paper).

Question: where do these derivations come from?

They do not seem to be part of the package of the 5 conditions.

Is my understanding correct that Connes uses the spectrum of these derivations to identify local charts on X with $$\mathbb{R}^p$$?

For user's convenience, here are the 5 conditions:

(1) The n-th characteristic value of the resolvent of D is $$O(n^{-1/p}$$.

(2) $$[[D,a],b]=0$$ $$\forall a,b \in A$$

(3) For any $$a\in A$$, both $$a$$ and $$[D,a]$$ belong to the domain of $$\delta^m$$ for any integer m and $$\delta(T) = [|D|,T]$$

(4) There exists a Hochschild cycle $$c\in Z_p(A,A)$$ such that $$\pi(c)=1$$ for p odd, and $$\pi(c)=\gamma$$ is $$\mathbb{Z}/2$$ grading for p even

(5) The A-module $$H_\infty = \cap Dom(D^m)$$ is finite and projective. It also has hermitian structure $$(|)$$ defined as $$<\xi,a\eta> = \int a(\xi|\eta)|D|^{-p}$$

If I'm not mistaken, the derivations in questions are those of the form of equation (22) in Lemma 4.3, i.e., derivations of the form $$a \mapsto \mathrm{i}(\xi \vert [D,a]\xi)$$ for $$\xi \in \mathcal{H}^\infty$$. The paper is a difficult one, but the introduction does provide a decent roadmap to understanding where and how the construction of the coordinate charts plays out in the body of the paper.