Mathematics Asked on December 6, 2020

While reading a book, I found an example that said that the ring $K[w,x,y,z]/(wy,wz,xy,xz)$ is not Cohen-Macaulay. In order to check this, it is stated to take the quotient by the ideal generated by the non-zero divisor $w-y$, that gives the ring $K[x,y,z]/(y^2,yz,xy,xz)$. The book assures that all the elements of *its maximal ideal are zero divisors*.

I couldn’t follow through this last part: I don’t see why this ring would be local (do I need to localize in a maximal ideal?)

You can find this section of the pdf file by looking up "25. Regular local rings Corollary 25.1".

Observe that $(wy, wz, xy, xz) = (w, x) cap (y, z),$ hence the height of $(wy, wz, xy, xz)$ is $2.$ We have therefore that $R = k[w, x, y, z] / (wy, wz, xy, xz)$ has dimension $2.$ On the other hand, the element $w - y$ is neither a zero divisor nor a unit, hence the sequence $(w - y)$ is $R$-regular. Going modulo $(w - y),$ we have that $R / (w - y) cong k[x, y, z] / (y^2, yz, xy, xz),$ and each of the generators of the unique homogeneous maximal ideal $mathfrak m = (x, y, z)$ is a zero divisor. Consequently, there are no regular sequences in $R / (w - y),$ hence it follows that $operatorname{depth} R = 1 < 2 = dim R$ so that $R$ is not Cohen-Macaulay.

Answered by Carlo on December 6, 2020

If it were local, all its quotients would be local, but factoring out by the ideal generated by $y$ and $z$, you get $K[x]$, which isn't local.

Secondly, the minimal primes of the second quotient are pretty clearly the ideals $(x,y)$ and $(y,z)$, quotienting by them we get $K[z]$ and $K[x]$ respectively, so it becomes clear that among the maximal ideals there are $(x,y,p(z))$ and $(p(x), y, z)$, where the $p$ represents an irreducible polynomial.

It isn't clear to me how $x-1$ would be a zero divisor in $(x-1,y,z)$ in the second quotient.

That doesn't completely rule out that the example isn't Cohen-Macaulay, though, only that the hint appears to have problems. Unfortunately, I'm unfamiliar with Cohen-Macaulay rings and can't tell you straightaway. Perhaps if you provide more info about the work this appeared in.

Answered by rschwieb on December 6, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Peter Machado on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP