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$\begingroup$ Martin, this is a useful comment, but not an answer to his question: is it possible to axiomatize the theory of local rings as a coherent theory or at least a geometric theory in the technical sense of the word. $\endgroup$

Zoran Skoda
– Zoran Skoda

2012-02-28 10:23:11 +00:00
Commented Feb 28, 2012 at 10:23
3

$\begingroup$ You are right. I just wanted to mention that with this axiomatization, 1+2 are fulfilled, but 3 is not. Zhen already asked specifically about the equation $R = R^* \cup \mathfrak{m}_R$ (in logical language), therefore I've added this not just as a comment. I hope it's ok ... $\endgroup$

Martin Brandenburg
– Martin Brandenburg

2012-02-28 10:36:52 +00:00
Commented Feb 28, 2012 at 10:36
2

$\begingroup$ I'm not sure your definition of $\mathfrak{m}$ is geometric/coherent. Certainly, $\exists z . \; z - x y z = 1$ is a coherent formula, but the problem is that what we want is $(\forall y. \; \exists z . \; z - x y z = 1) \vdash x \in \mathfrak{m}$, and this is not a coherent sequent. $\endgroup$

Zhen Lin
– Zhen Lin

2012-02-28 23:25:52 +00:00
Commented Feb 28, 2012 at 23:25

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