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.
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.
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.
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.