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How would you show that for a field $k$, the rings $k[a,b,c,d,e,f]/(ab+cd+ef)$ and $k[x_1,x_2,x_3,x_4,x_5]$ are not isomorphic, using methods that are algebraic?

To be quite honest, I have no idea how to approach this problem.

2 Answers 2

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Note that both rings are integral domains of Krull dimension 5, so you have to look at slightly more subtle invariants to tell them apart. Here is one way:

If you localize the first ring at the maximal ideal $(a,b,c,d,e,f)$, the resulting local ring is not regular. On the other hand, all localizations of $k[x_1,\ldots,x_5]$ at its maximal ideals are regular local rings.


In geometric terms, the first ring corresponds to a cone in $6$-dim'l affine space, and in particular, it is singular at its cone point. The second ring corresponds to $5$-dim'l affine space, which is smooth at each of its points.

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    Someone told me the solution in terms of smoothness, but did not tell me the algebraic analogue (regularity), which is what I was looking for. I am still learning the geometric side of the algebraic constructions, and I am not that proficient at it. Thank you!2011-11-06
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The first variety has both singular and nonsingular $k$-points, while the second is homogeneous and nonsingular at all points. The dimensions of the local cotangent spaces can be defined algebraically from the coordinate rings as $\dim \mathfrak{m}/\mathfrak{m^2}$ and this local invariant will be greater than $5$ at the point $(0,0,0,0,0,0)$ (ie., for $\mathfrak{m}$ the ideal generated by all the variables) for the first ring, but equal to $5$ for all maximal ideals in the second ring.