Topics in Non commutative algebraic geometry and congruent zeta functions (Part IV). Theory of connections.

Yoshifumi Tsuchimoto

Jujitsu? I'm going to learn Jujitsu?

Neo, ca 2199

In this part we present basic tools of ``differential geometry on schemes.'' Since we mainly deal with local theories, we describe them in terms of rings and modules. Even so, students who are interested may note that the technique employed here (except for last few sections where the story is specific for characteristic $p\neq 0$ ) is also applicable for ordinary differential geometry. In fact, for any differentiable manifold $M$ , we have

• The ring $B=C^\infty (M)$ of $C^\infty$ -functions on $M$ plays the role of ``affine coordinate ring''. that is, if $M$ is compact, the set $\operatorname{Spm}B$ of maximal ideals of $B$ with the Zariski topology is homeomorphic to $M$ . If $M$ is not compact, $\operatorname{Spm}(B)$ is a bit larger, containing ``ideal boundaries'' of $M$ , but even then $B$ carries virtually all the information to study $M$ .
• The Lie algebra of derivations $\operatorname{Der}_\mathbb{C}(B)$ is identified with the Lie algebra of $C^\infty$ vector field on $M$ .
• The module of 1-forms $\Omega^1_{B/\mathbb{C}}$ is identified with the module of $C^\infty$ -1-forms over $M$ .

And so on. So the theory here gives results on differential manifold without serious modifications.