3
$\begingroup$

A few commutative algebra questions for which I have no reference

For $P$ = "catenarian", "coherent", " Jacobson":

1- is an arbitrary product of rings satisfying $P$, a ring satisfying $P$?

2- if a ring $A$ satisfies: for every prime ideal $\mathfrak{p}$ of $A$, the localization $A_{\mathfrak{p}}$ satisfies property $P$, is it true that $A$ itself satisfies property $P$?

Improve this question
$\endgroup$
6
  • 3
    $\begingroup$ @nfdc23: your example for number 1 seems to be zero-dimensional, hence catenary and Jacobson. Or am I missing something? (As I read it, the OP asks separate questions for each of these properties.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 1, 2017 at 5:31
  • 1
    $\begingroup$ Yes, the questions are separate. $\endgroup$
    – user95222
    Commented Nov 1, 2017 at 6:52
  • 3
    $\begingroup$ @R.vanDobbendeBruyn: Oops, I thought that catenary and Jacobson rings are noetherian by definition. If not then the definition of catenarity only involves localizations of $A$ at primes, so #2 is tautologically true for "catenary" and #2 for Jacobson is true (and uninteresting) for silly reasons: the only local rings that are Jacobson are those with a unique prime, so if every $A_{\mathfrak{p}}$ is Jacobson then $A$ has no strict containment of primes, so each prime is maximal, so $A$ is trivially Jacobson. For #1 an infinite product of fields fails coherence; I'll mull over the rest of #1. $\endgroup$
    – nfdc23
    Commented Nov 1, 2017 at 14:26
  • 2
    $\begingroup$ #1 fails for $P$ = Jacobson. $\widehat{\mathbf{Z}}$, the profinite completion of the integers, is reduced, but has nontrivial Jacobson radical $J$. On the other hand, $\widehat{\mathbf{Z}} \simeq \prod_p\mathbf{Z}_p$, where each of the factors is Jacobson. To see $J\neq 0$, you may produce lots of nonzero elements in $J$ by defining a "logarithm" $\log : \widehat{\mathbf{Z}}^{\times}\to\mathbf{Z}$ and checking its image is not $(0)\subset\mathbf{Z}$. Say $\log(x) := \lim_{n\to\infty}(x^{n!}-1)/n!$. Its image is in $J$ and $\neq(0)$, hence gives nonzero elements in $J$. In fact, it is $2J$. $\endgroup$
    – user87684
    Commented Nov 2, 2017 at 21:07
  • 2
    $\begingroup$ The problem is that arbitrary products of commutative rings may introduce lots of units, building up elements inside the Jacobson radical. $\endgroup$
    – user87684
    Commented Nov 2, 2017 at 21:11

1 Answer 1

Reset to default
1
$\begingroup$

Concerning the second question about coherence, Theorem 3 in M.E.Harris, Some results on coherent rings II, Glasgow Math. J. 8 (1967) 123-126 says:

There exists a non-coherent ring $R$ such that $R_{\mathfrak{p}}$ is coherent for every prime ideal $\mathfrak{p}$ of $R$.

Concerning the second question about catenarity, the answer is obviously yes.

Improve this answer
$\endgroup$

You must log in to answer this question.