In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université[-]