Is the direct limit of a direct system of regular rings a regular ring?
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1$\begingroup$ Does "direct system" mean a functor $P \to Ring$ where $P$ is a directed poset? Or just a poset, as here: en.wikipedia.org/wiki/… ? $\endgroup$– Todd Trimble ♦Jul 11, 2011 at 14:54
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3$\begingroup$ And what do you mean by 'regular'? It is one of the most overloaded words in mathematics, even when applied to rings! $\endgroup$– Mariano Suárez-ÁlvarezJul 11, 2011 at 14:57
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4$\begingroup$ Really? The tag "algebraic geometry" probably means that we talk about commutative rings. A commutative ring is regular if it is noetherian and all its localizations are regular, and a regular local ring is defined be a noetherian local ring $(A,\mathfrak{m},k)$ such that $\dim(A)=\dim_k(\mathfrak{m}/\mathfrak{m}^2)$. $\endgroup$– Martin BrandenburgJul 11, 2011 at 15:17
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7$\begingroup$ Martin: if that was the question, the very first example that comes to mind is a counterexample: under the usual definitions the limit of the rings $k[x_1,\dots,x_n]$ under the obvious inclusions is not regular. $\endgroup$– Mariano Suárez-ÁlvarezJul 11, 2011 at 15:21
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1$\begingroup$ Perhaps a better title? $\endgroup$– Steve DJul 11, 2011 at 15:42
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1 Answer
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If you mean regular von Nuemann with regular, then yes, a references should be Berrick & Keating, Categories and Modules.
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3$\begingroup$ In fact, for directed colimits it is obvious that von Neumann regularity passes to the limit: a ring is vN regular iff for every $a$ in the ring there is a $b$ such that $a=aba$, and this kind of condition plays very nicely with colimits. $\endgroup$ Jul 11, 2011 at 17:02