2
$\begingroup$

Let $\mathfrak{C}$ be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it?

I am particulary interested in power series rings over a field. Is there a reasonable way to view such a ring as a graded ring?

Let $R = k[x]$ be a graded ring. Let $I=(x)$ be a graded ideal. Is $\varprojlim R/I^n$ an element of $\mathfrak{C}$?

$\endgroup$
  • $\begingroup$ Since this category has product and equalizers, it has limits. $\endgroup$ – YZhou Jul 28 '12 at 2:05
2
$\begingroup$

Are you asking how to put a grading on, say, the ring of p adic integers?

The limit you wrote down is isomorphic in the category of rings to the ring of power series over $k$. A grading is a decomposition into direct sums. But power series have infinitely many terms...This is a kind of vague explanation for why I think it isn't possible.

$\endgroup$
  • $\begingroup$ so you claim that this category does not have limits? $\endgroup$ – the L Oct 28 '11 at 9:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.