Timeline for Adic filtration and integral closure
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2017 at 15:48 | comment | added | Neil Epstein | The reason this counterexample exists is non-standard grading. I think that if $A$ is a finitely generated standard-graded algebra over a field (i.e. $A$ contains a field $k$ consisting of its homogeneous degree zero elements, and $A$ is generated by degree one elements over $k$), and, $\frak m$ its homogeneous maximal ideal, and $R=A_{\frak m}$, then the answer to your question should be 'yes'. In the example above, $A = \mathbb R[x,y] / (x^2 + y^4)$, there is no way to give a grading to $A$ in such a way that it is generated over $\mathbb R$ by degree one elements; deg(x) must be 2deg(y) | |
| Jun 24, 2017 at 18:44 | comment | added | Avi Steiner | For posterity: Here's why this answer satisfies the requirements on $S$: The $\Bbb R$-algebra homomorphism $R[u]\to \Bbb C[[y]]$ taking $y$ to itself and $u$ to $i$ is an isomorphism. Since $\Bbb C[[y]]$ is a DVR, it's integrally closed, and therefore $R[u]=S$. We win. | |
| Jun 24, 2017 at 18:39 | vote | accept | Avi Steiner | ||
| Jun 22, 2017 at 0:00 | history | answered | Mohan | CC BY-SA 3.0 |