Timeline for $K[[X_1,...]]$ is a UFD (Nishimura's Theorem)
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 20, 2018 at 16:44 | comment | added | Harry Richman | @IgorKhavkine yes, I was thinking of polynomials not power series, my mistake | |
| Feb 20, 2018 at 13:22 | comment | added | Pierre | SashaP, great thanks! from Rinmyaku | |
| Feb 20, 2018 at 12:09 | comment | added | Peter Heinig | [...] of this cone is a mono $\phi$. By the universal property of the limit, there is a unique arrow $\Psi\colon K[[x_1]]\to\varprojlim_{m\to\infty}K[[X_1,\dotsc,X_m]]$ making the diagram commute, hence $\phi$ factorizes as $\pi_1 \circ \Psi = \phi$, where $\pi_1$ is the canonical projection from the limit to $K[[x_1]]$. In the category of rings, a non-mono cannot become mono by post-composition, hence it is impossible that $\Psi$ is non-mono. In classical logic, $\Psi$ is therefore mono, giving a claimed copy of $K[[X_1,\dotsc,X_m]]$ inside the limit, and hence the claimed 'element' exists. | |
| Feb 20, 2018 at 12:08 | comment | added | Peter Heinig | @HarryRichman: I second Igor Khavkine's comment: statement "the projective limit is the set of power series such that, if you set sufficiently high-index variables all to zero, then you're left with a polynomial in the remaining variables." is false and it'd improve the thread if it'd be deleted. Claim: $\sum_{i=0}^\infty X^i \in \varprojlim_{m\to\infty} K[[X_1,\dotsc,X_m]]$. Proof. Consider the cone over the diagram $\dotsm\twoheadrightarrow K[[X_1,X_2]]\overset{p_1}{\twoheadrightarrow} K[[X_1]]$ given by the canonical injections of $K[[X_1]]$ to those rings. The 'first' leg [...] | |
| Feb 20, 2018 at 9:38 | comment | added | Igor Khavkine | @HarryRichman, hmm, I think what you wrote can't be strictly correct, since it corresponds to $\varprojlim_{m\to\infty} K[X_1,\ldots,X_m]$, which would exclude elements like $(1+X_1+X_1^2+\cdots) + X_2$. Deciphering Bourbaki, as per abx's reference, I think it's more accurate to say that any element of the projective limit is a sum of infinitely many homogeneous components, where each homogeneous component is a polynomial after setting all but finitely many variables to zero. | |
| Feb 20, 2018 at 5:12 | comment | added | abx | @IgorKhavkine: This is the standard definition of the ring of formal series in infinitely many variables, see for instance Bourbaki's Algebra IV, §4. | |
| Feb 20, 2018 at 4:34 | comment | added | Harry Richman | @IgorKhavkine the projective limit is the set of power series such that, if you set sufficiently high-index variables all to zero, then you're left with a polynomial in the remaining variables. (This is assuming the projection maps are defined by $x_i = 0$ at each step, which I believe is standard.) | |
| Feb 20, 2018 at 3:38 | comment | added | Pierre | I suppose that we are engaged in Projective limit. So such an element as X_1 + X_2 + ... ∈ K[[X_1,...]]. K[[X_1,...]] is much larger than the inductive limit of K[[X_1,...,X_m]]. | |
| Feb 19, 2018 at 22:54 | comment | added | Igor Khavkine | @Gro-Tsen Aha, it's quite possible that I mixed them up. Even though I should know better! The inductive limit is easy to understand: it consists of the power series that depend on finitely many variables (without a limit on the number of variables). The projective limit seems rather weirder. What is its pedestrian description? | |
| Feb 19, 2018 at 12:49 | comment | added | Gro-Tsen | Wait, are we talking about the inductive or the projective limit of the finite-variable power series rings? The question says projective (I didn't look up Nishimura's result to check). | |
| Feb 19, 2018 at 12:48 | history | answered | Igor Khavkine | CC BY-SA 3.0 |