Noncommutative geometry and dual coalgebras

Abstract

In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that the category of noncommutative (thin) schemes is equivalent to the category of coalgebras. We propose that under this correspondence the affine scheme of a k-algebra A is the dual coalgebra A^o and draw some consequences. In particular, we describe the dual coalgebra A^o of A in terms of the A-infinity structure on the Yoneda-space of all the simple finite dimensional A-representations.