I am wondering if the following rings are catenary:


1. If $k$ is a field, is the ring of formal power series $k[[X\_1,\dots,X\_n]]$ catenary?
2. Is the ring of complex power series with a non-zero radius of convergence $\Bbb C\{X\_1,\dots,X\_n\}$ (*id est* the ring of germs of holomorphic functions at zero) a catenary ring?



Are the ring of power series and the ring of germs of holomorphic functions catenary?
Yes, they are regular (the maximal ideal is generated by a number of elements equal to its dimension) and therefore Cohen-Macaulay (Matsumura, Theorem 17.8). And a Cohen Macaulay ring is catenary (Matsumura, Theorem 17.4).


Matsumura, Commutative Ring Theory, CUP, 1986


