MAXIMALLY DIFFERENTIAL IDEALS

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DIFFERENTIAL SIMPLICITY, COHEN-MACAULAYNESS AND FORMAL PRIME DIVISORS

We construct counter-examples, in any characteristic, to the conjecture that differentially simple local domains should be Cohen-Macaulay. Such counter examples can be made henselian.84111

Differential simplicity in polynomial rings and algebraic independence of power series

Let k be a field of characteristic zero, f(X,Y),g(X,Y)is an element ofk[X,Y], g(X,Y)is not an element of(X,Y) and d := g(X,Y) partial derivative/partial derivativeX + f(X,Y) partial derivative/partial derivativeY. A connection is established between the d-simplicity of the local ring k[X,Y]((X,Y)) and the transcendency of the solution in tk[[t]] of the algebraic differential equation g(t,y(t))(.)(partial derivative/partial derivativet)y(t) = f (t,y(t)). This connection is used to obtain some interesting results in the theory of the formal power series and to construct new examples of differentially simple rings.68361563

A note on the Nakai conjecture

The conjectures of Zariski-Lipman and of Nakai are still open in general in the class of rings essentially of finite type over a field of characteristic zero. However, they have long been known to be true in dimension one. Here we give counterexamples to both conjectures in the class of one-dimensional pseudo-geometric local domains that contain a field of characteristic zero. Likewise, in connection with a recent result of Traves on the Nakai conjecture, we also show that their hypothesis of finite generation of the integral closure cannot be removed even in the class of local domains containing a field of characteristic zero.1301152